Research Article Second Order Duality in Multiobjective Fractional Programming...

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Research Article Second Order Duality in Multiobjective Fractional Programming with Square Root Term under Generalized Univex Function Arun Kumar Tripathy Department of Mathematics, Trident Academy of Technology, F2/A, Chandaka Industrial Estate, Bhubaneswar, Odisha 751024, India Correspondence should be addressed to Arun Kumar Tripathy; arun tripathy06@rediffmail.com Received 12 March 2014; Accepted 10 April 2014; Published 7 July 2014 Academic Editor: Majid Soleimani-damaneh Copyright © 2014 Arun Kumar Tripathy. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ree approaches of second order mixed type duality are introduced for a nondifferentiable multiobjective fractional programming problem in which the numerator and denominator of objective function contain square root of positive semidefinite quadratic form. Also, the necessary and sufficient conditions of efficient solution for fractional programming are established and a parameterization technique is used to establish duality results under generalized second order -univexity assumption. 1. Introduction A fractional programming problem arises in many types of optimization problems such as portfolio selection, pro- duction, information theory, and numerous decision mak- ing problems in management science. More specifically, it can be used in engineering and economics to minimize a ratio of physical or economical function or both, such as cost/time, cost/volume, and cost/benefit, in order to measure the efficiency or productivity of the system. Many economic, noneconomic, and indirect applications of fractional pro- gramming problem have also been given by Bector [1], Bector and Chandra [2], Craven [3], Mond and Weir [4], Stancu- Minasian [5], Schaible and Ibaraki [6], Ahmad et al. [7], Ahmad and Sharma [8], and Gulati et al. [9]. e central concept in optimization is known as the duality theory which asserts that, given a (primal) mini- mization problem, the infimum value of the primal problem cannot be smaller than the supermom value of the associated (dual) maximization problem and the optimal values of primal and dual problems are equal. Duality in fractional programming is an important class of duality theory and several contributions have been made in the past [1, 5, 8, 10– 14]. Second order duality provides a tighter bound for the value of the objective function when approximations are used. For more details, one can consult [15, page 93]. Another advantage of second order duality when applicable is that if a feasible point in the primal is given and first order duality does not apply, then we can use second order duality to provide a lower bound of the value of the primal problem (see [4]). Multiobjective fractional programming duality has been of much interest in the recent past. Schaible [16] and Bector et al. [11] derived Fritz John and Karush-Kuhn Tucker necessary and sufficient optimality condition for a class of nondifferen- tiable convex multiobjective fractional programming prob- lems and established some duality theorems. Liang et al. [17, 18] discussed the optimality condition and duality for non- linear fractional programming. Santos et al. [19] discussed the optimality and duality for nonsmooth multiobjective fractional programming with generalized convexity. Bector et al. [20] and Xu [21] gave a mixed type duality for fractional programming, established some sufficient conditions, and obtained various duality results between the mixed dual and primal problem. Zhou and Wang [22] introduced a class of mixed type dual for nonsmooth multiobjective fractional programming and established the duality results under (, ) invexity assumption. Duality for various forms of mathematical problems involving square roots of positive semidefinite quadratic forms has been discussed by many authors [10, 23–25]. Mond [25] considered a nonlinear fractional programming problem Hindawi Publishing Corporation International Scholarly Research Notices Volume 2014, Article ID 541524, 8 pages http://dx.doi.org/10.1155/2014/541524

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Research ArticleSecond Order Duality in Multiobjective Fractional Programmingwith Square Root Term under Generalized Univex Function

Arun Kumar Tripathy

Department of Mathematics Trident Academy of Technology F2A Chandaka Industrial EstateBhubaneswar Odisha 751024 India

Correspondence should be addressed to Arun Kumar Tripathy arun tripathy06rediffmailcom

Received 12 March 2014 Accepted 10 April 2014 Published 7 July 2014

Academic Editor Majid Soleimani-damaneh

Copyright copy 2014 Arun Kumar Tripathy This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Three approaches of second order mixed type duality are introduced for a nondifferentiable multiobjective fractional programmingproblem inwhich the numerator and denominator of objective function contain square root of positive semidefinite quadratic formAlso the necessary and sufficient conditions of efficient solution for fractional programming are established and a parameterizationtechnique is used to establish duality results under generalized second order 120588-univexity assumption

1 Introduction

A fractional programming problem arises in many typesof optimization problems such as portfolio selection pro-duction information theory and numerous decision mak-ing problems in management science More specifically itcan be used in engineering and economics to minimize aratio of physical or economical function or both such ascosttime costvolume and costbenefit in order to measurethe efficiency or productivity of the system Many economicnoneconomic and indirect applications of fractional pro-gramming problem have also been given by Bector [1] Bectorand Chandra [2] Craven [3] Mond and Weir [4] Stancu-Minasian [5] Schaible and Ibaraki [6] Ahmad et al [7]Ahmad and Sharma [8] and Gulati et al [9]

The central concept in optimization is known as theduality theory which asserts that given a (primal) mini-mization problem the infimum value of the primal problemcannot be smaller than the supermom value of the associated(dual) maximization problem and the optimal values ofprimal and dual problems are equal Duality in fractionalprogramming is an important class of duality theory andseveral contributions have been made in the past [1 5 8 10ndash14] Second order duality provides a tighter bound for thevalue of the objective function when approximations areused For more details one can consult [15 page 93] Another

advantage of second order duality when applicable is that ifa feasible point in the primal is given and first order dualitydoes not apply then we can use second order duality toprovide a lower bound of the value of the primal problem (see[4])

Multiobjective fractional programming duality has beenofmuch interest in the recent past Schaible [16] and Bector etal [11] derived Fritz John and Karush-Kuhn Tucker necessaryand sufficient optimality condition for a class of nondifferen-tiable convex multiobjective fractional programming prob-lems and established some duality theorems Liang et al [1718] discussed the optimality condition and duality for non-linear fractional programming Santos et al [19] discussedthe optimality and duality for nonsmooth multiobjectivefractional programming with generalized convexity Bectoret al [20] and Xu [21] gave a mixed type duality for fractionalprogramming established some sufficient conditions andobtained various duality results between the mixed dual andprimal problem Zhou and Wang [22] introduced a classof mixed type dual for nonsmooth multiobjective fractionalprogramming and established the duality results under (119881 120588)invexity assumption

Duality for various forms of mathematical problemsinvolving square roots of positive semidefinite quadraticforms has been discussed bymany authors [10 23ndash25] Mond[25] considered a nonlinear fractional programming problem

Hindawi Publishing CorporationInternational Scholarly Research NoticesVolume 2014 Article ID 541524 8 pageshttpdxdoiorg1011552014541524

2 International Scholarly Research Notices

involving square roots of positive semidefinite quadratic formin the numerator and denominator and proved the neces-sary and sufficient condition for optimality Kim et al [2627] formulated a nondifferentiable multiobjective fractionalproblem in which numerators contain support function Oneof the most known approaches used for solving nonlin-ear fractional programming problem is called parametricapproach Dinklebaeh [28] and Jagannathan [12] introducedthis approach that was used later byOsuna-Gomez et al [13]to characterize solution of a multiobjective fractional prob-lem under generalized convexity Tripathy [14] introducedthree approaches given by Dinklebaeh [28] and Jagannathan[12] for both primal and mixed type dual of a nondifferen-tiablemultiobjective fractional programming and establishedthe duality results under generalized 120588-univexity

To relax convexity assumption imposed on the functionin theorems on optimality and duality various generalizedconvexity concepts have been proposed Hanson [29] intro-duced the class of invex functions Bector et al [30] intro-duced univex function Mishra [31] derived the optimalitycondition for multiobjective programming with generalizedunivexity Jayswal [32] presented minimax fractional pro-gramming under generalized 120588-univexity assumption

Motivated by the earlier authors in this paper we haveintroduced three approaches given by Dinklebaeh [28] andJagannathan [12] for both primal and second order mixedtype dual of a nondifferentiable multiobjective fractionalprogramming problem in which the numerator and denom-inator of objective function contain square root of positivesemidefinite quadratic form Also we have established thenecessary and sufficient optimality condition and used aparameterization technique to establish duality results undergeneralized 120588-univexity assumption

2 Notations and Preliminaries

Let R119899 be the 119899-dimensional Euclidean space and R119899+its

nonnegative orthant The following conventions for inequal-ity will be used throughout this paper For any 119909 =

(1199091 1199092 119909

119899) 119910 = (119910

1 1199102 119910

119899) we denote the following

(i) 119909 gt 119910 hArr 119909119894gt 119910119894 for all 119894 = 1 2 119899

(ii) 119909 ge 119910 hArr 119909119894ge 119910119894 for all 119894 = 1 2 119899

Throughout the paper let 119883 be a nonempty open subset ofR119899

Consider the following nondifferentiable multiobjectivefractional programming problem

21 Multiobjective Fractional Primal Problem

(i) MFP0 Minimize

119891 (119909) + (119909119879

119861119909)12

119892 (119909) minus (119909119879119862119909)12

= (1198701(119909) 119870

2(119909) 119870

119896(119909)) (1)

where

119870119894(119909) =

119891119894(119909) + (119909

119879

119861119894119909)12

119892119894(119909) minus (119909119879119862

119894119909)12

119894 = 1 2 119896 (2)

(ii) MFP1 Minimize

119865 (119909) = (1198651(119909) 119865

2(119909) 119865

119896(119909)) (3)

where

119865119894(119909) = 119891

119894(119909) + (119909

119879

119861119894119909)12

minus ]119894119892119894(119909) minus (119909

119879

119862119894119909)12

119894 = 1 2 119896

(4)

]119894are fixed parameters

(iii) 119872119865119875120582 Minimize 120582119865(119909) 120582 is 119896-dimensional strictlypositive vector

all subject to same constraint

ℎ (119909) le 0 119909 isin 119883 sube R119899

(5)

where 119891119894 R119899 rarr R 119892

119894 R119899 rarr R 119894 = 1 2 119896 and ℎ =

(ℎ1 ℎ

119898) ℎ119895 R119899 rarr R 119895 = 1 2 119898 are differentiable

functions 119861119894and 119862

119894 119894 = 1 2 119896 are positive semidefinite

matrices of order 119899 In the sequel we assume that 119891119894(119909) ge 0

and 119892119894(119909) gt 0 on R119899 for 119894 = 1 2 119896

Let 1198830= 119909 isin 119883 sube R119899 ℎ

119895(119909) le 0 119895 = 1 2 119898 for all

feasible solutions of MFP0 MFP1 and MFP120582 and denote 119868 =1 2 3 119896119872 = 1 2 3 119898 119869

1= 119895 isin 119872 ℎ

119895(119909) = 0

and 1198692= 119895 isin 119872 ℎ

119895(119909) lt 0 It is obvious that 119869

1cup 1198692= 119872

Throughout the paper consider 119891119894 119883 rarr R 120578 119883 times

119883 rarr R119899 119901 isin R119899 120588 isin RAssume that 120595 R rarr R satisfying 119886 le 0 rArr 120595(119886) le 0 or

120595(119886) le 0 rArr 119886 le 0 and 120595(minus119886) = minus120595(119886) 119870 119883 times 119883 rarr R+

For 119909 119909 isin 119883 we can write119870(119909 119909) = lim120582rarr0119887(119909 119909 120582) ge 0

Definition 1 The real differentiable function 119891119894is said to be

second order 120588-univex at 119909 isin 119883 with respect to 120578 120595 and 119870if

119870 (119909 119909) 120595 [119891119894(119909) minus 119891

119894(119909) +

1

2119901119879

(nabla2

119891119894(119909) 119901)]

ge 120578(119909 119909)119879

[nabla119891119894(119909) + nabla

2

119891119894(119909) 119901] + 120588119909 minus 119909

2

forall119909 isin 119883

(6)

Definition 2 The real differentiable function 119891119894is said to be

second order 120588-pseudounivex at 119909 isin 119883 with respect to 120578 120595and119870 if

120578(119909 119909)119879

[nabla119891119894(119909) + nabla

2

119891119894(119909) 119901] + 120588119909 minus 119909

2

ge 0

997904rArr 119870 (119909 119909) 120595 [119891119894(119909) minus 119891

119894(119909) +

1

2119901119879

(nabla2

119891119894(119909) 119901)] ge 0

forall119909 isin 119883

(7)

International Scholarly Research Notices 3

Definition 3 The real differentiable function 119891119894is said to be

second order 120588-quasiunivex at 119909 isin 119883 with respect to 120578 120595and119870 if

119870 (119909 119909) 120595 [119891119894(119909) minus 119891

119894(119909) +

1

2119901119879

(nabla2

119891119894(119909) 119901)] le 0

997904rArr 120578(119909 119909)119879

[nabla119891119894(119909) + nabla

2

119891119894(119909) 119901] + 120588119909 minus 119909

2

le 0

forall119909 isin 119883

(8)

Remark 4 If 119901 = 0 the above definitions reduce to thedefinitions of 120588-univex 120588-pseudounivex and 120588-quasiunivexas introduced in [14]

Definition 5 A feasible point119909 is said to be efficient forMFP1if there exists no other feasible point 119909 in MFP1 such that119865119894(119909) le 119865

119894(119909) 119894 = 1 2 119896 and 119865

119903(119909) lt 119865

119903(119909) for some

119903 isin 1 2 119896

Definition 6 (see [33]) A feasible point119909 is said to be properlyefficient for MFP1 if it is efficient and there exist 119872 gt 0

such that for each 119894 isin 1 2 119896 and for all feasible point119909 in MFP1 satisfying 119865

119894(119909) lt 119865

119894(119909) we have 119865

119894(119909) minus 119865

119894(119909) le

119872(119865119903(119909) minus 119865

119903(119909)) for some 119903 isin 1 2 119896 such that 119865

119903(119909) gt

119865119903(119909)

We assume that 119891119894(119909) + (119909

119879

119861119894119909)12

ge 0 119892119894(119909) minus

(119909119879

119862119894119909)12

gt 0 119894 = 1 2 119896 for all 119909 isin 119883

Definition 7 (generalized Schwarz Inequality) Let 119861 be apositive semidefinite matrix of order 119899 Then for all 119909 119908 isinR119899 119909119879119861119908 le (119909119879119861119909)12(119908119879119861119908)12

The equality holds if 119861119909 = 120582119861119908 for some 120582 ge 0Let 1198691(119909) = 119895 isin 119872 = 1 2 119898 ℎ

119895(119909) = 0 and

V119894= (119891119894(119909) + (119909

119879

119861119894119909)12

)(119892119894(119909) minus (119909

119879

119862119894119909)12

)Then define the set119882(119909) = 119908 isin R119899 119908119879nablaℎ

119895(119909) le 0 119895 isin

119869(119909) satisfying any one of the following conditions

(a) 119909119879119861119894119909 gt 0 119909119879119862

119894119909 = 0 rArr 119908

119879

(nabla119891119894(119909) + 119861

119894119909

(119909119879

119861119894119909)12

minus V119894nabla119892119894(119909)) + (119908

119879

(V2119894119862119894)119908)12

ge 0 119908 isin119882(119909) 119894 = 1 2 119896

(b) 119909119879119861119894119909 = 0 119909119879119862

119894119909 gt 0 rArr 119908

119879

(nabla119891119894(119909) minus V

119894nabla119892119894(119909) minus

119862119894119909(119909119879

119862119894119909)12

) + (119908119879

119861119894119908)12

ge 0 119908 isin 119882(119909) 119894 =1 2 119896

(c) 119909119879119861119894119909 = 0 119909119879119862

119894119909 = 0 rArr 119908

119879

(nabla119891119894(119909) minus V

119894nabla119892119894(119909)) +

(119908119879

119861119894119908)12

+ (119908119879

(V2119894119862119894)119908)12

ge 0 119908 isin 119882(119909) 119894 =1 2 119896

(d) 119909119879119861119894119909 gt 0 119909119879119862

119894119909 gt 0 rArr 119908

119879

(nabla119891119894(119909) +

119861119894119909(119909119879

119861119894119909)12

minus V119894nabla119892119894(119909) minus 119862

119894119909(119909119879

119862119894119909)12

) ge 0119908 isin 119882(119909) 119894 = 1 2 119896

Lemma 8 (see [33]) If 1199090 isin 1198830is an optimal solution of

119872119865119875120582 then 1199090 is properly efficient for MFP1

Lemma 9 (see [12]) 1199090 isin 1198830is an efficient solution for MFP0

if and only if it is an efficient solution of MFP1 with 119865(1199090) = 0

Lemma 10 (see [10] necessary optimality condition) If 119909 isin1198830is an optimal solution of (119872119865119875120582) such that119882(119909) = 120601 then

there exist V119894isin R+ 119908 119911 isin R119899 and 119910 isin R119898 such that

nabla120582119865 (119909) + 119910119879

nablaℎ (119909)

=

119896

sum

119894=1

120582119894[nabla119891119894(119909) + 119861

119894119908 minus V119894nabla119892119894(119909) minus 119862

119894119911]

+

119898

sum

119895=1

119910119895nablaℎ119895(119909) = 0

(9)

119865119894(119909) = 119891

119894(119909) + (119909

119879

119861119894119909)12

minus V119894(119892119894(119909) minus (119909

119879

119862119894119909)12

) = 0

119894 = 1 2 119896

(10)

119910119879

ℎ (119909) = 0 (11)

119908119879

119861119894119908 le 1 119911

119879

119862119894119911 le 1 119894 = 1 2 119896 (12)

(119909119879

119861119894119909)12

= 119909119879

119861119894119908 (119909

119879

119862119894119909)12

= 119909119879

119862119894119911

119894 = 1 2 119896

(13)

119910 ge 0 (14)

V119894ge 0 119894 = 1 2 119896 (15)

Theorem 11 (sufficient optimality condition) Let 119909 isin 1198830be

a feasible solution of MFP1 and there exist 120582119894isin R+119908 119911 isin R119899

V119894isin R+ and 119910 isin R119898 satisfying the condition in Lemma 10 at

119909 Furthermore suppose that the following conditions hold

(i) 119875(119909) = sum119896

119894=1120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus

(119909119879

119862119894119909)12

] + sum119898

119895=1119910119895ℎ119895(119909) is second order 120588-

pseudounivex with respect to 120578 120595 and 119870 at 119909 isin 1198830

with (nabla2119875(119909))119901 = 0 where119891119894 119883 rarr R119892

119894 119883 rarr R

ℎ119895 119883 rarr R 119894 = 1 2 119896 119895 = 1 2 119898 119901 isin R119899

120578 119883times119883 rarr R119899119870 119883times119883 rarr R+ and 120595 R rarr R

satisfying 120595(119886) le 0 rArr 119886 le 0

(ii) 120588 ge 0

Then 119909 is an efficient solution of MFP1

Proof Suppose that the hypothesis holdsSince the conditions of Lemma 10 are satisfied from (9)

and (13) we have

nabla119875 (119909)

= nabla(

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minusV119894119892119894(119909) minus (119909

119879

119862119894119909)12

] +

119898

sum

119895=1

119910119895ℎ119895(119909))

4 International Scholarly Research Notices

= nabla(

119896

sum

119894=1

120582119894[119891119894(119909) + 119909

119879

119861119894119908 minus V119894119892119894(119909) minus 119909

119879

119862119894119911]

+

119898

sum

119895=1

119910119895ℎ119895(119909)) = 0

(16)

Also from hypothesis (i) we have nabla2119875(119909)119901 = 0So we can write nabla119875(119909) + nabla2119875(119909)119901 = 0Now for 120578(119909 119909) isin R119899 we can write 120578(119909 119909)119879(nabla119875(119909) +

nabla2

119875(119909)119901) = 0For 120588 ge 0 we have 120578(119909 119909)119879(nabla119875(119909) + nabla2119875(119909)119901) +

120588119909 minus 1199092

ge 0Since 119875(119909) is second order 120588-pseudounivex with respect

to 120578 120595 and 119870 at 119909 isin 1198830 we have 119870(119909 119909)120595119875(119909) minus 119875(119909) +

(12)119901119879

(nabla2

119875(119909))119901 ge 0 and using the properties of 119870 120595 itgives

119875 (119909) minus 119875 (119909) +1

2119901119879

nabla2

119875 (119909) 119901

ge 0 997904rArr 119875 (119909) ge 119875 (119909) minus1

2119901119879

nabla2

119875 (119909) 119901

(17)

Since nabla2119875(119909)119901 = 0 the above inequality implies 119875(119909) ge 119875(119909)

997904rArr

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minusV119894119892119894(119909) minus (119909

119879

119862119894119909)12

] +

119898

sum

119895=1

119910119895ℎ119895(119909)

ge

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

] +

119898

sum

119895=1

119910119895ℎ119895(119909)

(18)

Suppose that 119909 is not efficient solution of MFP1 then thereexist 119909 isin 119883

0such that

119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

le 119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

119894 = 1 2 119896

(19)

and 119891119905(119909) + (119909

119879

119861119905119909)12

minus V119905119892119905(119909) minus (119909

119879

119862119905119909)12

le 119891119905(119909) +

(119909119879

119861119905119909)12

minus V119905119892119905(119909)minus (119909

119879

119862119905119909)12

for some 119905 isin 1 2 119896

The above relation together with the relation 120582119894gt 0

implies that

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

]

lt

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

]

(20)

From the relations (5) (11) and (14) we get

119898

sum

119895=1

119910119895ℎ119895(119909) le

119898

sum

119895=1

119910119895ℎ119895(119909) (21)

Consequently (20) and (21) yield

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

]

+

119898

sum

119895=1

119910119895ℎ119895(119909)

lt

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

]

+

119898

sum

119895=1

119910119895ℎ119895(119909)

(22)

This contradicts (18) Hence 119909 is an efficient solution forMFP1

Theorem 12 (sufficient optimality condition) Let 119909 isin 1198830be

a feasible solution of MFP1 and there exist 120582119894isin R+119908 119911 isin R119899

V119894isin R+ and 119910 isin R119898 satisfying the condition in Lemma 10 at

119909 Furthermore suppose that the following conditions hold

(i) 119876(119909) = sum119896

119894=1120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus

(119909119879

119862119894119909)12

] is second order 120588-pseudounivex withrespect to 120578 120595

0 and 119870 at 119909 isin 119883

0and 119867(119909) =

sum119898

119895=1119910119895ℎ119895(119909) is second order 120590-quasiunivex with

respect to 120578 1205951 and 119870 at 119909 isin 119883

0with (nabla2119876(119909))119901 = 0

and (nabla2119867(119909))119901 = 0 where 119891119894 119883 rarr R 119892

119894 119883 rarr R

ℎ119895 119883 rarr R 119894 = 1 2 119896 119895 = 1 2 119898 119901 isin R119899

120578 119883 times 119883 rarr R119899 119870 119883 times 119883 rarr R+ and

1205950 1205951 R rarr R satisfying 120595

0(119886) ge 0 rArr 119886 ge 0 and

1205951(119886) le 0 rArr 119886 le 0

(ii) 120588 + 120590 ge 0

Then 119909 is an efficient solution of1198721198651198751

Proof Suppose hypothesis holds

International Scholarly Research Notices 5

From the relations (5) (11) and (14) we get119898

sum

119895=1

119910119895ℎ119895(119909)

le

119898

sum

119895=1

119910119895ℎ119895(119909) 997904rArr 119867(119909) le 119867 (119909) 997904rArr 119867(119909) minus 119867 (119909) le 0

(23)

Also from hypothesis (i) we get (nabla2119867(119909))119901 = 0So we have the following

119867(119909) minus 119867 (119909) +1

2119901119879

(nabla2

119867(119909)) 119901 le 0

997904rArr 119870 (119909 119909) 1205951119867 (119909) minus 119867 (119909) +

1

2119901119879

(nabla2

119867(119909)) 119901 le 0

(24)

Hence the 120590-quasiunivexity of 119867(119909) with respect to 120578 1205951

and119870 implies the following

120578(119909 119909)119879

nabla119867 (119909) + (nabla2

119867(119909)) 119901 + 120590119909 minus 1199092

le 0

997904rArr 120578(119909 119909)119879

nabla119867 (119909) + 120590119909 minus 1199092

le 0

(25)

From (9) we get

119896

sum

119894=1

120582119894[nabla119891119894(119909) + 119861

119894119908 minus V119894nabla119892119894(119909) minus 119862

119894119911]

+

119898

sum

119894=1

119910119894nablaℎ119894(119909) = 0

997904rArr nabla119876 (119909) + nabla119867 (119909) = 0

997904rArr 120578(119909 119909)119879

[nabla119876 (119909) + nabla119867 (119909)] = 0

997904rArr 120578(119909 119909)119879

nabla119876 (119909) + 120578(119909 119909)119879

nabla119867 (119909)

+ 120590119909 minus 1199092

minus 120590119909 minus 1199092

= 0

(26)

Using (25) in (26) we get

120578(119909 119909)119879

nabla119876 (119909) minus 120590119909 minus 1199092

ge 0 (27)

Since 120588 + 120590 ge 0 we get

120588119909 minus 1199092

ge minus120590119909 minus 1199092

(28)

So we have

120578(119909 119909)119879

nabla119876 (119909) + 120588119909 minus 1199092

ge 0

997904rArr 120578(119909 119909)119879

nabla119876 (119909) + nabla2

119876 (119909) 119901 + 120588119909 minus 1199092

ge 0

(29)

Since 119876(119909) is 120588-pseudounivex with respect to 120578 1205950 and 119870

we obtained

119870 (119909 119909) 1205950[119876 (119909) minus 119876 (119909) +

1

2119901119879

nabla2

119876 (119909) 119901] ge 0 (30)

Using the property of119870 and 1205950 we get

119876 (119909) minus 119876 (119909) +1

2119901119879

nabla2

119876 (119909) 119901 ge 0 997904rArr 119876 (119909) ge 119876 (119909)

997904rArr

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minusV119894119892119894(119909) minus (119909

119879

119862119894119909)12

]

ge

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

]

(31)

If 119909 were not an efficient solution to MFP1 then from (20)we have

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

]

lt

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minusV119894119892119894(119909) minus (119909

119879

119862119894119909)12

]

(32)

This contradicts (31)Therefore 119909 is an efficient solution for MFP1

3 Second Order Mixed Type MultiobjectiveFractional Duality

(i) MMFD0 Maximize

119871 (119906)

= (1198711(119906) minus

1

2119901119879

nabla2

1198711(119906) 119901 119871

2(119906)

minus1

2119901119879

nabla2

1198712(119906) 119901 119871

119896(119906) minus

1

2119901119879

nabla2

119871119896(119906) 119901)

(33)

where

119871119894(119906) =

119891119894(119906) + 119910

119879

1198691

ℎ1198691(119906) + 119906

119879

119861119894119908

119892119894(119906) minus 119906119879119862

119894119911

119894 = 1 2 119896 (34)

(ii) MMFD1 Maximize

119866 (119906)

= (1198661(119906) minus

1

2119901119879

nabla2

1198661(119906) 119901 119866

2(119906)

minus1

2119901119879

nabla2

1198662(119906) 119901 119866

119896(119906) minus

1

2119901119879

nabla2

119866119896(119906) 119901)

(35)

where 119866119894(119906) = 119891

119894(119906) + 119910

119879

1198691

ℎ1198691

(119906) + 119906119879

119861119894119908 minus ]119894119892119894(119906) minus

119906119879

119862119894119911 119894 = 1 2 119896 ]

119894are fixed parameters

6 International Scholarly Research Notices

(iii) 119872119872119865119863120582 Maximize 120582119866(119906) 120582 is 119896-dimensionalstrictly positive vector

all subject to same constraints

119896

sum

119894=1

120582119894[nabla119866119894(119906) + nabla

2

119866119894(119906) 119901] + 119910

119879

1198692

[nablaℎ1198692(119906) + nabla

2

nablaℎ1198692(119906) 119901]

= 0

(36)

119891119894(119906) + 119910

119879

1198691

ℎ1198691(119906) + 119906

119879

119861119894119908 minus V119894119892119894(119906) minus 119906

119879

119862119894119911 ge 0

for 119894 = 1 2 119896(37)

119910119879

1198692

ℎ1198692(119906) minus

1

2119901119879

nabla2

(119910119879

1198692

ℎ1198692(119906)) 119901 ge 0

1199101198692

isin R119898minus|1198691|

(38)

119908119879

119861119894119908 le 1 119911

119879

119862119894119911 le 1 119894 = 1 2 119896 (39)

119910 ge 0 V119894ge 0 119894 = 1 2 119896 (40)

where119891119894 119883 rarr R119892

119894 119883 rarr R ℎ

119895 119883 rarr R 119894 = 1 2 119896

119895 = 1 2 119898 are differentiable functions119908 119911 isin R119899119901 isin R119899119861119894 and 119862

119894 119894 = 1 2 119896 are positive semidefinite matrices of

order 119899For the following theorems we assume that 120578 119883 times119883 rarr

R119899 119870 119883 times 119883 rarr R+ and 120595

0 1205951 R rarr R satisfying

1205950(119886) ge 0 rArr 119886 ge 0 and 119887 le 0 rArr 120595

1(119887) le 0 and 120588 120590 isin R

Theorem 13 (weak duality) Let 119909 be a feasible solution for theprimalMFP120582 and let (119906 119910 V 119908) be feasible for dual SMMFD120582If

(i) sum119896119894=1120582119894119866119894(sdot) is second order 120588-pseudounivex with

respect to 120578 1205950 119870 and for 119910

1198692

isin R119898minus|1198691| 1199101198791198692

ℎ1198692

(sdot) issecond order 120590-quasiunivex with respect to 120578 120595

1 and

119870 along with

(ii) 120588 + 120590 ge 0 then Inf(120582119865(119909)) ge Sup(120582119866(119906))

Proof Now from the primal and dual constraints we have

ℎ (119909) le 0

119910119879

1198692

ℎ1198692(119906) minus

1

2119901119879

nabla2

(119910119879

1198692

ℎ1198692(119906)) 119901 ge 0

(41)

So

119910119879

1198692

ℎ1198692(119909) minus 119910

119879

1198692

ℎ1198692(119906) +

1

2119901119879

nabla2

(119910119879

1198692

ℎ1198692(119906)) 119901 le 0

997904rArr 119870 (119909 119906) 1205951[119910119879

1198692

ℎ1198692(119909) minus 119910

119879

1198692

ℎ1198692(119906)

+1

2119901119879

nabla2

(119910119879

1198692

ℎ1198692(119906)) 119901] le 0

(42)

Since 1199101198791198692

ℎ1198692

is second order 120590-quasiunivex with respect to120578 1205951 and119870 and in view of (42) for 119909 119906 isin R119899 we have

120578(119909 119906)119879

nabla [119910119879

1198692

ℎ1198692(119906)] + nabla

2

[119910119879

1198692

ℎ1198692(119906)] 119901 + 120590119909 minus 119906

2

le 0

(43)

Again from the dual constraint (36) we have119896

sum

119894=1

120582119894[nabla119866119894(119906) + nabla

2

119866119894(119906) 119901]

+ 119910119879

1198692

119879

[nablaℎ1198692(119906) + nabla

2

ℎ1198692(119906) 119901] = 0

(44)

Since 120578(119909 119906) isin R119899 we have

120578(119909 119906)119879

119896

sum

119894=1

120582119894[nabla119866119894(119906) + nabla

2

119866119894(119906) 119901]

+ 120578(119909 119906)119879

119910119879

1198692

[nablaℎ1198692(119906) + nabla

2

ℎ1198692(119906) 119901] = 0

997904rArr 120578(119909 119906)119879

119896

sum

119894=1

120582119894[nabla119866119894(119906) + nabla

2

119866119894(119906) 119901]

+ 120578(119909 119906)119879

119910119879

1198692

[nablaℎ1198692(119906) + nabla

2

ℎ1198692(119906) 119901]

+ 120590119909 minus 1199062

minus 120590119909 minus 1199062

= 0

(45)

Using (43) in above equation we get 120578(119909 119906)119879sum119896119894=1120582119894

[nabla119866119894(119906) + nabla

2

119866119894(119906)119901] minus 120590119909 minus 119906

2

ge 0Since 120588 + 120590 ge 0 we get 120588119909 minus 1199062 ge minus120590119909 minus 1199062So we have

120578(119909 119906)119879

119896

sum

119894=1

120582119894[nabla119866119894(119906) + nabla

2

119866119894(119906) 119901] + 120588119909 minus 119906

2

ge 0

(46)

Since sum119896

119894=1120582119894119866119894(119906) is second order 120588-pseudounivex

with respect to 120578 1205950 and 119870 by Definition 2 and

(46) we get 119870(119909 119906)1205950sum119896

119894=1120582119894119866119894(119909) minus sum

119896

119894=1120582119894119866119894(119906) +

(12)119901119879

sum119896

119894=1120582119894119866119894(119906)119901 ge 0

Using the property of 1205950and119870 we get

119896

sum

119894=1

120582119894119866119894(119909) minus

119896

sum

119894=1

120582119894119866119894(119906) +

1

2119901119879

119896

sum

119894=1

120582119894119866119894(119906) 119901 ge 0

997904rArr

119896

sum

119894=1

120582119894119866119894(119909) ge

119896

sum

119894=1

120582119894119866119894(119906)

997904rArr

119896

sum

119894=1

120582119894[119891119894(119909) + 119910

119879

1198691

ℎ1198691(119909)

+119909119879

119861119894119908 minus V119894119892119894(119909) minus 119909

119879

119862119894119911]

ge

119896

sum

119894=1

120582119894[119891119894(119906) + 119910

119879

1198691

ℎ1198691(119906)

+119906119879

119861119894119908 minus V119894119892119894(119906) minus 119909

119879

119862119894119911]

(47)

Equation (5) gives ℎ(119909) le 0 rArr 1199101198791198691

ℎ1198691

(119909) le 0 for 1199101198691

ge 0

International Scholarly Research Notices 7

So (47) implies that

119896

sum

119894=1

120582119894[119891119894(119909) + 119909

119879

119861119894119908 minus V119894119892119894(119909) minus 119909

119879

119862119894119911]

ge

119896

sum

119894=1

120582119894[119891119894(119906) + 119910

119879

1198691

ℎ1198691(119906) + 119906

119879

119861119894119908

minusV119894119892119894(119906) minus 119906

119879

119862119894119911]

(48)

Now by Schwarz Inequality and (39) we have

119909119879

119861119894119908 le (119909

119879

119861119894119909)12

(119908119879

119861119894119908)12

le (119909119879

119861119894119909)12

119909119879

119862119894119911 le (119909

119879

119862119894119909)12

(119911119879

119862119894119911)12

le (119909119879

119862119894119909)12

119894 = 1 2 119896

(49)

So both (48) and (49) imply that

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

]

ge

119896

sum

119894=1

120582119894[119891119894(119906) + 119910

119879

1198691

ℎ1198691(119906) + 119906

119879

119861119894119908

minus V119894119892119894(119906) minus 119906

119879

119862119894119911]

997904rArr Inf (120582119865 (119909)) ge Sup (120582119865 (119906))

(50)

Theorem 14 (strong duality) Let 119909 be optimal solution forMFP120582 and let 119882(119909) = 120601 Then there exist 120582

119894isin R+ 119908 119911 isin

R119899 V119894isin R+ and 119910 isin R119898 such that (119906 120582 119910 V 119908 119911 119901 =

0) is a feasible solution for dual and the objective values ofboth primal and dual are equal to zero Furthermore if (i)(119906 120582 119910 V 119908 119911) is feasible for dual (ii) sum119896

119894=1120582119894119866119894(sdot) is second

order 120588-pseudounivex with respect to 120578 1205950 and 119870 and for

1199101198692

isin R119898minus|1198691| 1199101198791198691

ℎ1198692

(sdot) is second order 120590-quasiunivex withrespect to 120578 120595

1 and 119870 along with (iii) 120588 + 120590 ge 0 then

(119906 120582 119910 V 119908 119911 119901 = 0) is properly efficient for SMMFD0

Proof Since 119909 is optimal solution for (MFP120582) by Lemma 10there exist 120582

119894isin R+119908 119911 isin R119899 V

119894isin R+ and 119910 isin R119898 such that

119896

sum

119894=1

120582119894[nabla119891119894(119909) + 119861

119894119908 minus V119894nabla119892119894(119909) minus 119862

119894119911] + 119910

119879

nablaℎ (119909) = 0

119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894(119892119894(119909) minus (119909

119879

119862119894119909)12

) = 0

119894 = 1 2 119896

119910119879

ℎ (119909) = 0

119908119879

119861119894119908 le 1 119911

119879

119862119894119911 le 1 119894 = 1 2 119896

(119909119879

119861119894119909)12

= 119909119879

119861119894119908 (119909

119879

119862119894119909)12

= 119909119879

119862119894119911

119894 = 1 2 119896

119910 ge 0

V119894ge 0 119894 = 1 2 119896

(51)

which can be written as119896

sum

119894=1

120582119894[nabla119891119894(119909) + 119910

119879

1198691

nablaℎ1198691(119909) + 119861

119894119908 minus V119894nabla119892119894(119909) minus 119862

119894119911]

+ 119910119879

1198691

nablaℎ1198692(119909) = 0

119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

+ 119910119879

1198691

ℎ1198691(119909) = 0

119910119879

1198691

ℎ1198692(119909) = 0

119908119879

119861119894119908 le 1 119911

119879

119862119894119911 le 1 119894 = 1 2 119896

(119909119879

119861119894119909)12

= 119909119879

119861119894119908 (119909

119879

119862119894119909)12

= 119909119879

119862119894119911

119894 = 1 2 119896

119910 ge 0

V119894ge 0 119894 = 1 2 119896

(52)

These are nothing but the dual constraintsSo (119906 120582 119910 V 119908 119911 119901 = 0) is feasible solution for dual

problemAnd the objective values of MFP120582 and SMMFD120582 are

equal to zero It follows fromTheorem 13 and for any feasiblesolution (119906 120582 119910 V 119908 119911 119901 = 0) of dual that 120582119866(119906) le 120582119866(119909)

So (119906 120582 119910 V 119908 119911 119901 = 0) is optimal solution ofSMMFD120582 Then applying Lemmas 8 and 9 we concludethat (119906 120582 119910 V 119908 119911 119901 = 0) is properly efficient for(SMMFD0)

4 Special Case

If 119901 = 0 119862119894= 0 119894 = 1 2 119896 then our dual programming

reduces to the dual programming proposed by Tripathy [14]

5 Conclusion

In this paper three approaches given by Dinklebaeh [28] andJagannathan [12] for both primal and second order mixedtype dual of a nondifferentiable multiobjective fractionalprogramming problem are introduced and the necessaryand sufficient optimality conditions are established and aparameterization technique is used to establish duality results

8 International Scholarly Research Notices

under generalized second order 120588-univexity assumptionTheresults developed in this paper can be further extendedto higher order mixed type fractional problem containingsquare root term Also the present work can be furtherextended to a class of nondifferentiable minimax mixedfractional programming problems

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] C R Bector ldquoDuality in nonlinear fractional programmingrdquoZeitschrift fur Operations Research vol 17 no 5 pp 183ndash1931973

[2] C R Bector and S Chandra ldquoFirst and Second order duality fora class of non differentiable fractional programmingrdquo Journal ofInformation amp Optimization Sciences vol 7 pp 335ndash348 1986

[3] B D Craven ldquoFractional programmingrdquo in Sig Ma Series inApplied Mathematics vol 4 pp 145ndash152 Heldermann BerlinGermany 1998

[4] B Mond and T Weir ldquoDuality for fractional programmingwith generalized convexity conditionrdquo Journal of Information ampOptimization Sciences vol 3 pp 105ndash124 1982

[5] I M Stancu-Minasian Fractional Programming TheoryMethod and Application Kluwer Academic PublishersDordrecht The Netherland 1997

[6] S Schaible and T Ibaraki ldquoFractional programmingrdquo EuropeanJournal of Operational Research vol 12 no 4 pp 325ndash338 1983

[7] I Ahmad S K Gupta N Kailey and R P Agarwal ldquoDualityin nondifferentiable minimax fractional programming with B-(pr)-invexityrdquo Journal of Inequalities and Applications vol 2011article 75 15 pages 2011

[8] I Ahmad and S Sharma ldquoSecond-order duality for nondif-ferentiable multiobjective programming problemsrdquo NumericalFunctional Analysis and Optimization vol 28 no 9-10 pp 975ndash988 2007

[9] T R Gulati I Ahmad and D Agarwal ldquoSufficiency andduality in multiobjective programming under generalized typei functionsrdquo Journal of Optimization Theory and Applicationsvol 135 no 3 pp 411ndash427 2007

[10] I Ahmad Z Husain and S Al-Homidan ldquoSecond-order dual-ity in nondifferentiable fractional programmingrdquo NonlinearAnalysis Real World Applications vol 12 no 2 pp 1103ndash11102011

[11] C R Bector S Chandra and I Husain ldquoOptimality conditionsand duality in subdifferentiable multiobjective fractional pro-grammingrdquo Journal of Optimization Theory and Applicationsvol 79 no 1 pp 105ndash125 1993

[12] R Jagannathan ldquoDuality for nonlinear fractional programsrdquoZeitschrift fur Operations Research vol 17 no 1 pp 1ndash3 1973

[13] R Osuna-Gomez A Ruız-Canales and P Rufian-Lizana ldquoMul-tiobjective fractional programmingwith generalized convexityrdquoTrabajos de Investigacion Operativa vol 8 no 1 pp 97ndash1102000

[14] A K Tripathy ldquoMixed type duality in multiobjective fractionalprogramming under generalized 120588-univex functionrdquo Journal ofMathematical Modelling and Algorithms in Operations Research2013

[15] S K Mishra S Wang and K K Lai ldquoRole of 120572-pseudo-univex functions in vector variational-like inequality problemsrdquoJournal of Systems Science and Complexity vol 20 no 3 pp344ndash349 2007

[16] S Schaible ldquoFractional programmingrdquo in Handbook of GlobalOptimization R Horst and P M Pardalos Eds pp 495ndash608 KluwerAcademic Publishers DordrechtTheNetherlands1995

[17] Z A Liang H X Huang and P M Pardalos ldquoOptimalityconditions and duality for a class of nonlinear fractionalprogramming problemsrdquo Journal of Optimization Theory andApplications vol 110 no 3 pp 611ndash619 2001

[18] Z-A Liang H-X Huang and P M Pardalos ldquoEfficiencyconditions and duality for a class of multiobjective fractionalprogramming problemsrdquo Journal of Global Optimization vol27 no 4 pp 447ndash471 2003

[19] L B Santos R Osuna-Gomez and A Rojas-Medar ldquoNons-moothmultiobjective fractional programmingwith generalizedconvexityrdquo Revista Integracion vol 26 no 1 pp 1ndash12 2008

[20] C R Bector S Chandra and A Abha ldquoOn mixed duality inmathematical programmingrdquo Journal of Mathematical Analysisand Applications vol 259 no 1 pp 346ndash356 2001

[21] Z Xu ldquoMixed type duality in multiobjective programmingproblemsrdquo Journal of Mathematical Analysis and Applicationsvol 198 no 3 pp 621ndash635 1996

[22] H Zhou and Y Wang ldquoOptimality and mixed duality for non-smooth multi-objective fractional programmingrdquo Pure Mathe-matics and Applications vol 14 no 3 pp 263ndash274 2003

[23] I Ahmad andZHusain ldquoDuality in nondifferentiableminimaxfractional programming with generalized convexityrdquo AppliedMathematics and Computation vol 176 no 2 pp 545ndash5512006

[24] I Ahmad and Z Husain ldquoOptimality conditions and dualityin nondifferentiable minimax fractional programming withgeneralized convexityrdquo Journal of Optimization Theory andApplications vol 129 no 2 pp 255ndash275 2006

[25] B Mond ldquoA class of nondifferentiable fractional program-ming problemrdquo Zeitschrift fur Angewandte Mathematik undMechanik vol 58 no 8 pp 337ndash341 1978

[26] D S Kim S J Kim and M H Kim ldquoOptimality andduality for a class of nondifferentiable multiobjective fractionalprogramming problemsrdquo Journal of Optimization Theory andApplications vol 129 no 1 pp 131ndash146 2006

[27] D S Kim Y J Lee and K D Bae ldquoDuality in nondifferentiablemultiobjective fractional programs involving conesrdquo TaiwaneseJournal of Mathematics vol 13 no 6 A pp 1811ndash1821 2009

[28] W Dinklebaeh ldquoOn nonlinear fractional programmingrdquoMan-agement Science vol 13 no 7 pp 492ndash498 1967

[29] M A Hanson ldquoOn sufficiency of the Kuhn-Tucker conditionsrdquoJournal of Mathematical Analysis and Applications vol 80 no2 pp 545ndash550 1981

[30] C R Bector S K Suneja and S Gupta ldquoUnivex functionsand univex nonlinear programmingrdquo in Proceeding of theAdministrative Science Association of Canada pp 115ndash124 1992

[31] S K Mishra ldquoOn multiple-objective optimization with gener-alized univexityrdquo Journal of Mathematical Analysis and Applica-tions vol 224 no 1 pp 131ndash148 1998

[32] A Jayswal ldquoNon-differentiable minimax fractional program-ming with generalized 120572-univexityrdquo Journal of Computationaland Applied Mathematics vol 214 no 1 pp 121ndash135 2008

[33] A M Geoffrion ldquoProper efficiency and the theory of vectormaximizationrdquo Journal of Mathematical Analysis and Applica-tions vol 22 no 3 pp 618ndash630 1968

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Stochastic AnalysisInternational Journal of

Page 2: Research Article Second Order Duality in Multiobjective Fractional Programming …downloads.hindawi.com/archive/2014/541524.pdf · 2019-07-31 · Research Article Second Order Duality

2 International Scholarly Research Notices

involving square roots of positive semidefinite quadratic formin the numerator and denominator and proved the neces-sary and sufficient condition for optimality Kim et al [2627] formulated a nondifferentiable multiobjective fractionalproblem in which numerators contain support function Oneof the most known approaches used for solving nonlin-ear fractional programming problem is called parametricapproach Dinklebaeh [28] and Jagannathan [12] introducedthis approach that was used later byOsuna-Gomez et al [13]to characterize solution of a multiobjective fractional prob-lem under generalized convexity Tripathy [14] introducedthree approaches given by Dinklebaeh [28] and Jagannathan[12] for both primal and mixed type dual of a nondifferen-tiablemultiobjective fractional programming and establishedthe duality results under generalized 120588-univexity

To relax convexity assumption imposed on the functionin theorems on optimality and duality various generalizedconvexity concepts have been proposed Hanson [29] intro-duced the class of invex functions Bector et al [30] intro-duced univex function Mishra [31] derived the optimalitycondition for multiobjective programming with generalizedunivexity Jayswal [32] presented minimax fractional pro-gramming under generalized 120588-univexity assumption

Motivated by the earlier authors in this paper we haveintroduced three approaches given by Dinklebaeh [28] andJagannathan [12] for both primal and second order mixedtype dual of a nondifferentiable multiobjective fractionalprogramming problem in which the numerator and denom-inator of objective function contain square root of positivesemidefinite quadratic form Also we have established thenecessary and sufficient optimality condition and used aparameterization technique to establish duality results undergeneralized 120588-univexity assumption

2 Notations and Preliminaries

Let R119899 be the 119899-dimensional Euclidean space and R119899+its

nonnegative orthant The following conventions for inequal-ity will be used throughout this paper For any 119909 =

(1199091 1199092 119909

119899) 119910 = (119910

1 1199102 119910

119899) we denote the following

(i) 119909 gt 119910 hArr 119909119894gt 119910119894 for all 119894 = 1 2 119899

(ii) 119909 ge 119910 hArr 119909119894ge 119910119894 for all 119894 = 1 2 119899

Throughout the paper let 119883 be a nonempty open subset ofR119899

Consider the following nondifferentiable multiobjectivefractional programming problem

21 Multiobjective Fractional Primal Problem

(i) MFP0 Minimize

119891 (119909) + (119909119879

119861119909)12

119892 (119909) minus (119909119879119862119909)12

= (1198701(119909) 119870

2(119909) 119870

119896(119909)) (1)

where

119870119894(119909) =

119891119894(119909) + (119909

119879

119861119894119909)12

119892119894(119909) minus (119909119879119862

119894119909)12

119894 = 1 2 119896 (2)

(ii) MFP1 Minimize

119865 (119909) = (1198651(119909) 119865

2(119909) 119865

119896(119909)) (3)

where

119865119894(119909) = 119891

119894(119909) + (119909

119879

119861119894119909)12

minus ]119894119892119894(119909) minus (119909

119879

119862119894119909)12

119894 = 1 2 119896

(4)

]119894are fixed parameters

(iii) 119872119865119875120582 Minimize 120582119865(119909) 120582 is 119896-dimensional strictlypositive vector

all subject to same constraint

ℎ (119909) le 0 119909 isin 119883 sube R119899

(5)

where 119891119894 R119899 rarr R 119892

119894 R119899 rarr R 119894 = 1 2 119896 and ℎ =

(ℎ1 ℎ

119898) ℎ119895 R119899 rarr R 119895 = 1 2 119898 are differentiable

functions 119861119894and 119862

119894 119894 = 1 2 119896 are positive semidefinite

matrices of order 119899 In the sequel we assume that 119891119894(119909) ge 0

and 119892119894(119909) gt 0 on R119899 for 119894 = 1 2 119896

Let 1198830= 119909 isin 119883 sube R119899 ℎ

119895(119909) le 0 119895 = 1 2 119898 for all

feasible solutions of MFP0 MFP1 and MFP120582 and denote 119868 =1 2 3 119896119872 = 1 2 3 119898 119869

1= 119895 isin 119872 ℎ

119895(119909) = 0

and 1198692= 119895 isin 119872 ℎ

119895(119909) lt 0 It is obvious that 119869

1cup 1198692= 119872

Throughout the paper consider 119891119894 119883 rarr R 120578 119883 times

119883 rarr R119899 119901 isin R119899 120588 isin RAssume that 120595 R rarr R satisfying 119886 le 0 rArr 120595(119886) le 0 or

120595(119886) le 0 rArr 119886 le 0 and 120595(minus119886) = minus120595(119886) 119870 119883 times 119883 rarr R+

For 119909 119909 isin 119883 we can write119870(119909 119909) = lim120582rarr0119887(119909 119909 120582) ge 0

Definition 1 The real differentiable function 119891119894is said to be

second order 120588-univex at 119909 isin 119883 with respect to 120578 120595 and 119870if

119870 (119909 119909) 120595 [119891119894(119909) minus 119891

119894(119909) +

1

2119901119879

(nabla2

119891119894(119909) 119901)]

ge 120578(119909 119909)119879

[nabla119891119894(119909) + nabla

2

119891119894(119909) 119901] + 120588119909 minus 119909

2

forall119909 isin 119883

(6)

Definition 2 The real differentiable function 119891119894is said to be

second order 120588-pseudounivex at 119909 isin 119883 with respect to 120578 120595and119870 if

120578(119909 119909)119879

[nabla119891119894(119909) + nabla

2

119891119894(119909) 119901] + 120588119909 minus 119909

2

ge 0

997904rArr 119870 (119909 119909) 120595 [119891119894(119909) minus 119891

119894(119909) +

1

2119901119879

(nabla2

119891119894(119909) 119901)] ge 0

forall119909 isin 119883

(7)

International Scholarly Research Notices 3

Definition 3 The real differentiable function 119891119894is said to be

second order 120588-quasiunivex at 119909 isin 119883 with respect to 120578 120595and119870 if

119870 (119909 119909) 120595 [119891119894(119909) minus 119891

119894(119909) +

1

2119901119879

(nabla2

119891119894(119909) 119901)] le 0

997904rArr 120578(119909 119909)119879

[nabla119891119894(119909) + nabla

2

119891119894(119909) 119901] + 120588119909 minus 119909

2

le 0

forall119909 isin 119883

(8)

Remark 4 If 119901 = 0 the above definitions reduce to thedefinitions of 120588-univex 120588-pseudounivex and 120588-quasiunivexas introduced in [14]

Definition 5 A feasible point119909 is said to be efficient forMFP1if there exists no other feasible point 119909 in MFP1 such that119865119894(119909) le 119865

119894(119909) 119894 = 1 2 119896 and 119865

119903(119909) lt 119865

119903(119909) for some

119903 isin 1 2 119896

Definition 6 (see [33]) A feasible point119909 is said to be properlyefficient for MFP1 if it is efficient and there exist 119872 gt 0

such that for each 119894 isin 1 2 119896 and for all feasible point119909 in MFP1 satisfying 119865

119894(119909) lt 119865

119894(119909) we have 119865

119894(119909) minus 119865

119894(119909) le

119872(119865119903(119909) minus 119865

119903(119909)) for some 119903 isin 1 2 119896 such that 119865

119903(119909) gt

119865119903(119909)

We assume that 119891119894(119909) + (119909

119879

119861119894119909)12

ge 0 119892119894(119909) minus

(119909119879

119862119894119909)12

gt 0 119894 = 1 2 119896 for all 119909 isin 119883

Definition 7 (generalized Schwarz Inequality) Let 119861 be apositive semidefinite matrix of order 119899 Then for all 119909 119908 isinR119899 119909119879119861119908 le (119909119879119861119909)12(119908119879119861119908)12

The equality holds if 119861119909 = 120582119861119908 for some 120582 ge 0Let 1198691(119909) = 119895 isin 119872 = 1 2 119898 ℎ

119895(119909) = 0 and

V119894= (119891119894(119909) + (119909

119879

119861119894119909)12

)(119892119894(119909) minus (119909

119879

119862119894119909)12

)Then define the set119882(119909) = 119908 isin R119899 119908119879nablaℎ

119895(119909) le 0 119895 isin

119869(119909) satisfying any one of the following conditions

(a) 119909119879119861119894119909 gt 0 119909119879119862

119894119909 = 0 rArr 119908

119879

(nabla119891119894(119909) + 119861

119894119909

(119909119879

119861119894119909)12

minus V119894nabla119892119894(119909)) + (119908

119879

(V2119894119862119894)119908)12

ge 0 119908 isin119882(119909) 119894 = 1 2 119896

(b) 119909119879119861119894119909 = 0 119909119879119862

119894119909 gt 0 rArr 119908

119879

(nabla119891119894(119909) minus V

119894nabla119892119894(119909) minus

119862119894119909(119909119879

119862119894119909)12

) + (119908119879

119861119894119908)12

ge 0 119908 isin 119882(119909) 119894 =1 2 119896

(c) 119909119879119861119894119909 = 0 119909119879119862

119894119909 = 0 rArr 119908

119879

(nabla119891119894(119909) minus V

119894nabla119892119894(119909)) +

(119908119879

119861119894119908)12

+ (119908119879

(V2119894119862119894)119908)12

ge 0 119908 isin 119882(119909) 119894 =1 2 119896

(d) 119909119879119861119894119909 gt 0 119909119879119862

119894119909 gt 0 rArr 119908

119879

(nabla119891119894(119909) +

119861119894119909(119909119879

119861119894119909)12

minus V119894nabla119892119894(119909) minus 119862

119894119909(119909119879

119862119894119909)12

) ge 0119908 isin 119882(119909) 119894 = 1 2 119896

Lemma 8 (see [33]) If 1199090 isin 1198830is an optimal solution of

119872119865119875120582 then 1199090 is properly efficient for MFP1

Lemma 9 (see [12]) 1199090 isin 1198830is an efficient solution for MFP0

if and only if it is an efficient solution of MFP1 with 119865(1199090) = 0

Lemma 10 (see [10] necessary optimality condition) If 119909 isin1198830is an optimal solution of (119872119865119875120582) such that119882(119909) = 120601 then

there exist V119894isin R+ 119908 119911 isin R119899 and 119910 isin R119898 such that

nabla120582119865 (119909) + 119910119879

nablaℎ (119909)

=

119896

sum

119894=1

120582119894[nabla119891119894(119909) + 119861

119894119908 minus V119894nabla119892119894(119909) minus 119862

119894119911]

+

119898

sum

119895=1

119910119895nablaℎ119895(119909) = 0

(9)

119865119894(119909) = 119891

119894(119909) + (119909

119879

119861119894119909)12

minus V119894(119892119894(119909) minus (119909

119879

119862119894119909)12

) = 0

119894 = 1 2 119896

(10)

119910119879

ℎ (119909) = 0 (11)

119908119879

119861119894119908 le 1 119911

119879

119862119894119911 le 1 119894 = 1 2 119896 (12)

(119909119879

119861119894119909)12

= 119909119879

119861119894119908 (119909

119879

119862119894119909)12

= 119909119879

119862119894119911

119894 = 1 2 119896

(13)

119910 ge 0 (14)

V119894ge 0 119894 = 1 2 119896 (15)

Theorem 11 (sufficient optimality condition) Let 119909 isin 1198830be

a feasible solution of MFP1 and there exist 120582119894isin R+119908 119911 isin R119899

V119894isin R+ and 119910 isin R119898 satisfying the condition in Lemma 10 at

119909 Furthermore suppose that the following conditions hold

(i) 119875(119909) = sum119896

119894=1120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus

(119909119879

119862119894119909)12

] + sum119898

119895=1119910119895ℎ119895(119909) is second order 120588-

pseudounivex with respect to 120578 120595 and 119870 at 119909 isin 1198830

with (nabla2119875(119909))119901 = 0 where119891119894 119883 rarr R119892

119894 119883 rarr R

ℎ119895 119883 rarr R 119894 = 1 2 119896 119895 = 1 2 119898 119901 isin R119899

120578 119883times119883 rarr R119899119870 119883times119883 rarr R+ and 120595 R rarr R

satisfying 120595(119886) le 0 rArr 119886 le 0

(ii) 120588 ge 0

Then 119909 is an efficient solution of MFP1

Proof Suppose that the hypothesis holdsSince the conditions of Lemma 10 are satisfied from (9)

and (13) we have

nabla119875 (119909)

= nabla(

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minusV119894119892119894(119909) minus (119909

119879

119862119894119909)12

] +

119898

sum

119895=1

119910119895ℎ119895(119909))

4 International Scholarly Research Notices

= nabla(

119896

sum

119894=1

120582119894[119891119894(119909) + 119909

119879

119861119894119908 minus V119894119892119894(119909) minus 119909

119879

119862119894119911]

+

119898

sum

119895=1

119910119895ℎ119895(119909)) = 0

(16)

Also from hypothesis (i) we have nabla2119875(119909)119901 = 0So we can write nabla119875(119909) + nabla2119875(119909)119901 = 0Now for 120578(119909 119909) isin R119899 we can write 120578(119909 119909)119879(nabla119875(119909) +

nabla2

119875(119909)119901) = 0For 120588 ge 0 we have 120578(119909 119909)119879(nabla119875(119909) + nabla2119875(119909)119901) +

120588119909 minus 1199092

ge 0Since 119875(119909) is second order 120588-pseudounivex with respect

to 120578 120595 and 119870 at 119909 isin 1198830 we have 119870(119909 119909)120595119875(119909) minus 119875(119909) +

(12)119901119879

(nabla2

119875(119909))119901 ge 0 and using the properties of 119870 120595 itgives

119875 (119909) minus 119875 (119909) +1

2119901119879

nabla2

119875 (119909) 119901

ge 0 997904rArr 119875 (119909) ge 119875 (119909) minus1

2119901119879

nabla2

119875 (119909) 119901

(17)

Since nabla2119875(119909)119901 = 0 the above inequality implies 119875(119909) ge 119875(119909)

997904rArr

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minusV119894119892119894(119909) minus (119909

119879

119862119894119909)12

] +

119898

sum

119895=1

119910119895ℎ119895(119909)

ge

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

] +

119898

sum

119895=1

119910119895ℎ119895(119909)

(18)

Suppose that 119909 is not efficient solution of MFP1 then thereexist 119909 isin 119883

0such that

119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

le 119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

119894 = 1 2 119896

(19)

and 119891119905(119909) + (119909

119879

119861119905119909)12

minus V119905119892119905(119909) minus (119909

119879

119862119905119909)12

le 119891119905(119909) +

(119909119879

119861119905119909)12

minus V119905119892119905(119909)minus (119909

119879

119862119905119909)12

for some 119905 isin 1 2 119896

The above relation together with the relation 120582119894gt 0

implies that

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

]

lt

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

]

(20)

From the relations (5) (11) and (14) we get

119898

sum

119895=1

119910119895ℎ119895(119909) le

119898

sum

119895=1

119910119895ℎ119895(119909) (21)

Consequently (20) and (21) yield

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

]

+

119898

sum

119895=1

119910119895ℎ119895(119909)

lt

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

]

+

119898

sum

119895=1

119910119895ℎ119895(119909)

(22)

This contradicts (18) Hence 119909 is an efficient solution forMFP1

Theorem 12 (sufficient optimality condition) Let 119909 isin 1198830be

a feasible solution of MFP1 and there exist 120582119894isin R+119908 119911 isin R119899

V119894isin R+ and 119910 isin R119898 satisfying the condition in Lemma 10 at

119909 Furthermore suppose that the following conditions hold

(i) 119876(119909) = sum119896

119894=1120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus

(119909119879

119862119894119909)12

] is second order 120588-pseudounivex withrespect to 120578 120595

0 and 119870 at 119909 isin 119883

0and 119867(119909) =

sum119898

119895=1119910119895ℎ119895(119909) is second order 120590-quasiunivex with

respect to 120578 1205951 and 119870 at 119909 isin 119883

0with (nabla2119876(119909))119901 = 0

and (nabla2119867(119909))119901 = 0 where 119891119894 119883 rarr R 119892

119894 119883 rarr R

ℎ119895 119883 rarr R 119894 = 1 2 119896 119895 = 1 2 119898 119901 isin R119899

120578 119883 times 119883 rarr R119899 119870 119883 times 119883 rarr R+ and

1205950 1205951 R rarr R satisfying 120595

0(119886) ge 0 rArr 119886 ge 0 and

1205951(119886) le 0 rArr 119886 le 0

(ii) 120588 + 120590 ge 0

Then 119909 is an efficient solution of1198721198651198751

Proof Suppose hypothesis holds

International Scholarly Research Notices 5

From the relations (5) (11) and (14) we get119898

sum

119895=1

119910119895ℎ119895(119909)

le

119898

sum

119895=1

119910119895ℎ119895(119909) 997904rArr 119867(119909) le 119867 (119909) 997904rArr 119867(119909) minus 119867 (119909) le 0

(23)

Also from hypothesis (i) we get (nabla2119867(119909))119901 = 0So we have the following

119867(119909) minus 119867 (119909) +1

2119901119879

(nabla2

119867(119909)) 119901 le 0

997904rArr 119870 (119909 119909) 1205951119867 (119909) minus 119867 (119909) +

1

2119901119879

(nabla2

119867(119909)) 119901 le 0

(24)

Hence the 120590-quasiunivexity of 119867(119909) with respect to 120578 1205951

and119870 implies the following

120578(119909 119909)119879

nabla119867 (119909) + (nabla2

119867(119909)) 119901 + 120590119909 minus 1199092

le 0

997904rArr 120578(119909 119909)119879

nabla119867 (119909) + 120590119909 minus 1199092

le 0

(25)

From (9) we get

119896

sum

119894=1

120582119894[nabla119891119894(119909) + 119861

119894119908 minus V119894nabla119892119894(119909) minus 119862

119894119911]

+

119898

sum

119894=1

119910119894nablaℎ119894(119909) = 0

997904rArr nabla119876 (119909) + nabla119867 (119909) = 0

997904rArr 120578(119909 119909)119879

[nabla119876 (119909) + nabla119867 (119909)] = 0

997904rArr 120578(119909 119909)119879

nabla119876 (119909) + 120578(119909 119909)119879

nabla119867 (119909)

+ 120590119909 minus 1199092

minus 120590119909 minus 1199092

= 0

(26)

Using (25) in (26) we get

120578(119909 119909)119879

nabla119876 (119909) minus 120590119909 minus 1199092

ge 0 (27)

Since 120588 + 120590 ge 0 we get

120588119909 minus 1199092

ge minus120590119909 minus 1199092

(28)

So we have

120578(119909 119909)119879

nabla119876 (119909) + 120588119909 minus 1199092

ge 0

997904rArr 120578(119909 119909)119879

nabla119876 (119909) + nabla2

119876 (119909) 119901 + 120588119909 minus 1199092

ge 0

(29)

Since 119876(119909) is 120588-pseudounivex with respect to 120578 1205950 and 119870

we obtained

119870 (119909 119909) 1205950[119876 (119909) minus 119876 (119909) +

1

2119901119879

nabla2

119876 (119909) 119901] ge 0 (30)

Using the property of119870 and 1205950 we get

119876 (119909) minus 119876 (119909) +1

2119901119879

nabla2

119876 (119909) 119901 ge 0 997904rArr 119876 (119909) ge 119876 (119909)

997904rArr

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minusV119894119892119894(119909) minus (119909

119879

119862119894119909)12

]

ge

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

]

(31)

If 119909 were not an efficient solution to MFP1 then from (20)we have

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

]

lt

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minusV119894119892119894(119909) minus (119909

119879

119862119894119909)12

]

(32)

This contradicts (31)Therefore 119909 is an efficient solution for MFP1

3 Second Order Mixed Type MultiobjectiveFractional Duality

(i) MMFD0 Maximize

119871 (119906)

= (1198711(119906) minus

1

2119901119879

nabla2

1198711(119906) 119901 119871

2(119906)

minus1

2119901119879

nabla2

1198712(119906) 119901 119871

119896(119906) minus

1

2119901119879

nabla2

119871119896(119906) 119901)

(33)

where

119871119894(119906) =

119891119894(119906) + 119910

119879

1198691

ℎ1198691(119906) + 119906

119879

119861119894119908

119892119894(119906) minus 119906119879119862

119894119911

119894 = 1 2 119896 (34)

(ii) MMFD1 Maximize

119866 (119906)

= (1198661(119906) minus

1

2119901119879

nabla2

1198661(119906) 119901 119866

2(119906)

minus1

2119901119879

nabla2

1198662(119906) 119901 119866

119896(119906) minus

1

2119901119879

nabla2

119866119896(119906) 119901)

(35)

where 119866119894(119906) = 119891

119894(119906) + 119910

119879

1198691

ℎ1198691

(119906) + 119906119879

119861119894119908 minus ]119894119892119894(119906) minus

119906119879

119862119894119911 119894 = 1 2 119896 ]

119894are fixed parameters

6 International Scholarly Research Notices

(iii) 119872119872119865119863120582 Maximize 120582119866(119906) 120582 is 119896-dimensionalstrictly positive vector

all subject to same constraints

119896

sum

119894=1

120582119894[nabla119866119894(119906) + nabla

2

119866119894(119906) 119901] + 119910

119879

1198692

[nablaℎ1198692(119906) + nabla

2

nablaℎ1198692(119906) 119901]

= 0

(36)

119891119894(119906) + 119910

119879

1198691

ℎ1198691(119906) + 119906

119879

119861119894119908 minus V119894119892119894(119906) minus 119906

119879

119862119894119911 ge 0

for 119894 = 1 2 119896(37)

119910119879

1198692

ℎ1198692(119906) minus

1

2119901119879

nabla2

(119910119879

1198692

ℎ1198692(119906)) 119901 ge 0

1199101198692

isin R119898minus|1198691|

(38)

119908119879

119861119894119908 le 1 119911

119879

119862119894119911 le 1 119894 = 1 2 119896 (39)

119910 ge 0 V119894ge 0 119894 = 1 2 119896 (40)

where119891119894 119883 rarr R119892

119894 119883 rarr R ℎ

119895 119883 rarr R 119894 = 1 2 119896

119895 = 1 2 119898 are differentiable functions119908 119911 isin R119899119901 isin R119899119861119894 and 119862

119894 119894 = 1 2 119896 are positive semidefinite matrices of

order 119899For the following theorems we assume that 120578 119883 times119883 rarr

R119899 119870 119883 times 119883 rarr R+ and 120595

0 1205951 R rarr R satisfying

1205950(119886) ge 0 rArr 119886 ge 0 and 119887 le 0 rArr 120595

1(119887) le 0 and 120588 120590 isin R

Theorem 13 (weak duality) Let 119909 be a feasible solution for theprimalMFP120582 and let (119906 119910 V 119908) be feasible for dual SMMFD120582If

(i) sum119896119894=1120582119894119866119894(sdot) is second order 120588-pseudounivex with

respect to 120578 1205950 119870 and for 119910

1198692

isin R119898minus|1198691| 1199101198791198692

ℎ1198692

(sdot) issecond order 120590-quasiunivex with respect to 120578 120595

1 and

119870 along with

(ii) 120588 + 120590 ge 0 then Inf(120582119865(119909)) ge Sup(120582119866(119906))

Proof Now from the primal and dual constraints we have

ℎ (119909) le 0

119910119879

1198692

ℎ1198692(119906) minus

1

2119901119879

nabla2

(119910119879

1198692

ℎ1198692(119906)) 119901 ge 0

(41)

So

119910119879

1198692

ℎ1198692(119909) minus 119910

119879

1198692

ℎ1198692(119906) +

1

2119901119879

nabla2

(119910119879

1198692

ℎ1198692(119906)) 119901 le 0

997904rArr 119870 (119909 119906) 1205951[119910119879

1198692

ℎ1198692(119909) minus 119910

119879

1198692

ℎ1198692(119906)

+1

2119901119879

nabla2

(119910119879

1198692

ℎ1198692(119906)) 119901] le 0

(42)

Since 1199101198791198692

ℎ1198692

is second order 120590-quasiunivex with respect to120578 1205951 and119870 and in view of (42) for 119909 119906 isin R119899 we have

120578(119909 119906)119879

nabla [119910119879

1198692

ℎ1198692(119906)] + nabla

2

[119910119879

1198692

ℎ1198692(119906)] 119901 + 120590119909 minus 119906

2

le 0

(43)

Again from the dual constraint (36) we have119896

sum

119894=1

120582119894[nabla119866119894(119906) + nabla

2

119866119894(119906) 119901]

+ 119910119879

1198692

119879

[nablaℎ1198692(119906) + nabla

2

ℎ1198692(119906) 119901] = 0

(44)

Since 120578(119909 119906) isin R119899 we have

120578(119909 119906)119879

119896

sum

119894=1

120582119894[nabla119866119894(119906) + nabla

2

119866119894(119906) 119901]

+ 120578(119909 119906)119879

119910119879

1198692

[nablaℎ1198692(119906) + nabla

2

ℎ1198692(119906) 119901] = 0

997904rArr 120578(119909 119906)119879

119896

sum

119894=1

120582119894[nabla119866119894(119906) + nabla

2

119866119894(119906) 119901]

+ 120578(119909 119906)119879

119910119879

1198692

[nablaℎ1198692(119906) + nabla

2

ℎ1198692(119906) 119901]

+ 120590119909 minus 1199062

minus 120590119909 minus 1199062

= 0

(45)

Using (43) in above equation we get 120578(119909 119906)119879sum119896119894=1120582119894

[nabla119866119894(119906) + nabla

2

119866119894(119906)119901] minus 120590119909 minus 119906

2

ge 0Since 120588 + 120590 ge 0 we get 120588119909 minus 1199062 ge minus120590119909 minus 1199062So we have

120578(119909 119906)119879

119896

sum

119894=1

120582119894[nabla119866119894(119906) + nabla

2

119866119894(119906) 119901] + 120588119909 minus 119906

2

ge 0

(46)

Since sum119896

119894=1120582119894119866119894(119906) is second order 120588-pseudounivex

with respect to 120578 1205950 and 119870 by Definition 2 and

(46) we get 119870(119909 119906)1205950sum119896

119894=1120582119894119866119894(119909) minus sum

119896

119894=1120582119894119866119894(119906) +

(12)119901119879

sum119896

119894=1120582119894119866119894(119906)119901 ge 0

Using the property of 1205950and119870 we get

119896

sum

119894=1

120582119894119866119894(119909) minus

119896

sum

119894=1

120582119894119866119894(119906) +

1

2119901119879

119896

sum

119894=1

120582119894119866119894(119906) 119901 ge 0

997904rArr

119896

sum

119894=1

120582119894119866119894(119909) ge

119896

sum

119894=1

120582119894119866119894(119906)

997904rArr

119896

sum

119894=1

120582119894[119891119894(119909) + 119910

119879

1198691

ℎ1198691(119909)

+119909119879

119861119894119908 minus V119894119892119894(119909) minus 119909

119879

119862119894119911]

ge

119896

sum

119894=1

120582119894[119891119894(119906) + 119910

119879

1198691

ℎ1198691(119906)

+119906119879

119861119894119908 minus V119894119892119894(119906) minus 119909

119879

119862119894119911]

(47)

Equation (5) gives ℎ(119909) le 0 rArr 1199101198791198691

ℎ1198691

(119909) le 0 for 1199101198691

ge 0

International Scholarly Research Notices 7

So (47) implies that

119896

sum

119894=1

120582119894[119891119894(119909) + 119909

119879

119861119894119908 minus V119894119892119894(119909) minus 119909

119879

119862119894119911]

ge

119896

sum

119894=1

120582119894[119891119894(119906) + 119910

119879

1198691

ℎ1198691(119906) + 119906

119879

119861119894119908

minusV119894119892119894(119906) minus 119906

119879

119862119894119911]

(48)

Now by Schwarz Inequality and (39) we have

119909119879

119861119894119908 le (119909

119879

119861119894119909)12

(119908119879

119861119894119908)12

le (119909119879

119861119894119909)12

119909119879

119862119894119911 le (119909

119879

119862119894119909)12

(119911119879

119862119894119911)12

le (119909119879

119862119894119909)12

119894 = 1 2 119896

(49)

So both (48) and (49) imply that

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

]

ge

119896

sum

119894=1

120582119894[119891119894(119906) + 119910

119879

1198691

ℎ1198691(119906) + 119906

119879

119861119894119908

minus V119894119892119894(119906) minus 119906

119879

119862119894119911]

997904rArr Inf (120582119865 (119909)) ge Sup (120582119865 (119906))

(50)

Theorem 14 (strong duality) Let 119909 be optimal solution forMFP120582 and let 119882(119909) = 120601 Then there exist 120582

119894isin R+ 119908 119911 isin

R119899 V119894isin R+ and 119910 isin R119898 such that (119906 120582 119910 V 119908 119911 119901 =

0) is a feasible solution for dual and the objective values ofboth primal and dual are equal to zero Furthermore if (i)(119906 120582 119910 V 119908 119911) is feasible for dual (ii) sum119896

119894=1120582119894119866119894(sdot) is second

order 120588-pseudounivex with respect to 120578 1205950 and 119870 and for

1199101198692

isin R119898minus|1198691| 1199101198791198691

ℎ1198692

(sdot) is second order 120590-quasiunivex withrespect to 120578 120595

1 and 119870 along with (iii) 120588 + 120590 ge 0 then

(119906 120582 119910 V 119908 119911 119901 = 0) is properly efficient for SMMFD0

Proof Since 119909 is optimal solution for (MFP120582) by Lemma 10there exist 120582

119894isin R+119908 119911 isin R119899 V

119894isin R+ and 119910 isin R119898 such that

119896

sum

119894=1

120582119894[nabla119891119894(119909) + 119861

119894119908 minus V119894nabla119892119894(119909) minus 119862

119894119911] + 119910

119879

nablaℎ (119909) = 0

119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894(119892119894(119909) minus (119909

119879

119862119894119909)12

) = 0

119894 = 1 2 119896

119910119879

ℎ (119909) = 0

119908119879

119861119894119908 le 1 119911

119879

119862119894119911 le 1 119894 = 1 2 119896

(119909119879

119861119894119909)12

= 119909119879

119861119894119908 (119909

119879

119862119894119909)12

= 119909119879

119862119894119911

119894 = 1 2 119896

119910 ge 0

V119894ge 0 119894 = 1 2 119896

(51)

which can be written as119896

sum

119894=1

120582119894[nabla119891119894(119909) + 119910

119879

1198691

nablaℎ1198691(119909) + 119861

119894119908 minus V119894nabla119892119894(119909) minus 119862

119894119911]

+ 119910119879

1198691

nablaℎ1198692(119909) = 0

119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

+ 119910119879

1198691

ℎ1198691(119909) = 0

119910119879

1198691

ℎ1198692(119909) = 0

119908119879

119861119894119908 le 1 119911

119879

119862119894119911 le 1 119894 = 1 2 119896

(119909119879

119861119894119909)12

= 119909119879

119861119894119908 (119909

119879

119862119894119909)12

= 119909119879

119862119894119911

119894 = 1 2 119896

119910 ge 0

V119894ge 0 119894 = 1 2 119896

(52)

These are nothing but the dual constraintsSo (119906 120582 119910 V 119908 119911 119901 = 0) is feasible solution for dual

problemAnd the objective values of MFP120582 and SMMFD120582 are

equal to zero It follows fromTheorem 13 and for any feasiblesolution (119906 120582 119910 V 119908 119911 119901 = 0) of dual that 120582119866(119906) le 120582119866(119909)

So (119906 120582 119910 V 119908 119911 119901 = 0) is optimal solution ofSMMFD120582 Then applying Lemmas 8 and 9 we concludethat (119906 120582 119910 V 119908 119911 119901 = 0) is properly efficient for(SMMFD0)

4 Special Case

If 119901 = 0 119862119894= 0 119894 = 1 2 119896 then our dual programming

reduces to the dual programming proposed by Tripathy [14]

5 Conclusion

In this paper three approaches given by Dinklebaeh [28] andJagannathan [12] for both primal and second order mixedtype dual of a nondifferentiable multiobjective fractionalprogramming problem are introduced and the necessaryand sufficient optimality conditions are established and aparameterization technique is used to establish duality results

8 International Scholarly Research Notices

under generalized second order 120588-univexity assumptionTheresults developed in this paper can be further extendedto higher order mixed type fractional problem containingsquare root term Also the present work can be furtherextended to a class of nondifferentiable minimax mixedfractional programming problems

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] C R Bector ldquoDuality in nonlinear fractional programmingrdquoZeitschrift fur Operations Research vol 17 no 5 pp 183ndash1931973

[2] C R Bector and S Chandra ldquoFirst and Second order duality fora class of non differentiable fractional programmingrdquo Journal ofInformation amp Optimization Sciences vol 7 pp 335ndash348 1986

[3] B D Craven ldquoFractional programmingrdquo in Sig Ma Series inApplied Mathematics vol 4 pp 145ndash152 Heldermann BerlinGermany 1998

[4] B Mond and T Weir ldquoDuality for fractional programmingwith generalized convexity conditionrdquo Journal of Information ampOptimization Sciences vol 3 pp 105ndash124 1982

[5] I M Stancu-Minasian Fractional Programming TheoryMethod and Application Kluwer Academic PublishersDordrecht The Netherland 1997

[6] S Schaible and T Ibaraki ldquoFractional programmingrdquo EuropeanJournal of Operational Research vol 12 no 4 pp 325ndash338 1983

[7] I Ahmad S K Gupta N Kailey and R P Agarwal ldquoDualityin nondifferentiable minimax fractional programming with B-(pr)-invexityrdquo Journal of Inequalities and Applications vol 2011article 75 15 pages 2011

[8] I Ahmad and S Sharma ldquoSecond-order duality for nondif-ferentiable multiobjective programming problemsrdquo NumericalFunctional Analysis and Optimization vol 28 no 9-10 pp 975ndash988 2007

[9] T R Gulati I Ahmad and D Agarwal ldquoSufficiency andduality in multiobjective programming under generalized typei functionsrdquo Journal of Optimization Theory and Applicationsvol 135 no 3 pp 411ndash427 2007

[10] I Ahmad Z Husain and S Al-Homidan ldquoSecond-order dual-ity in nondifferentiable fractional programmingrdquo NonlinearAnalysis Real World Applications vol 12 no 2 pp 1103ndash11102011

[11] C R Bector S Chandra and I Husain ldquoOptimality conditionsand duality in subdifferentiable multiobjective fractional pro-grammingrdquo Journal of Optimization Theory and Applicationsvol 79 no 1 pp 105ndash125 1993

[12] R Jagannathan ldquoDuality for nonlinear fractional programsrdquoZeitschrift fur Operations Research vol 17 no 1 pp 1ndash3 1973

[13] R Osuna-Gomez A Ruız-Canales and P Rufian-Lizana ldquoMul-tiobjective fractional programmingwith generalized convexityrdquoTrabajos de Investigacion Operativa vol 8 no 1 pp 97ndash1102000

[14] A K Tripathy ldquoMixed type duality in multiobjective fractionalprogramming under generalized 120588-univex functionrdquo Journal ofMathematical Modelling and Algorithms in Operations Research2013

[15] S K Mishra S Wang and K K Lai ldquoRole of 120572-pseudo-univex functions in vector variational-like inequality problemsrdquoJournal of Systems Science and Complexity vol 20 no 3 pp344ndash349 2007

[16] S Schaible ldquoFractional programmingrdquo in Handbook of GlobalOptimization R Horst and P M Pardalos Eds pp 495ndash608 KluwerAcademic Publishers DordrechtTheNetherlands1995

[17] Z A Liang H X Huang and P M Pardalos ldquoOptimalityconditions and duality for a class of nonlinear fractionalprogramming problemsrdquo Journal of Optimization Theory andApplications vol 110 no 3 pp 611ndash619 2001

[18] Z-A Liang H-X Huang and P M Pardalos ldquoEfficiencyconditions and duality for a class of multiobjective fractionalprogramming problemsrdquo Journal of Global Optimization vol27 no 4 pp 447ndash471 2003

[19] L B Santos R Osuna-Gomez and A Rojas-Medar ldquoNons-moothmultiobjective fractional programmingwith generalizedconvexityrdquo Revista Integracion vol 26 no 1 pp 1ndash12 2008

[20] C R Bector S Chandra and A Abha ldquoOn mixed duality inmathematical programmingrdquo Journal of Mathematical Analysisand Applications vol 259 no 1 pp 346ndash356 2001

[21] Z Xu ldquoMixed type duality in multiobjective programmingproblemsrdquo Journal of Mathematical Analysis and Applicationsvol 198 no 3 pp 621ndash635 1996

[22] H Zhou and Y Wang ldquoOptimality and mixed duality for non-smooth multi-objective fractional programmingrdquo Pure Mathe-matics and Applications vol 14 no 3 pp 263ndash274 2003

[23] I Ahmad andZHusain ldquoDuality in nondifferentiableminimaxfractional programming with generalized convexityrdquo AppliedMathematics and Computation vol 176 no 2 pp 545ndash5512006

[24] I Ahmad and Z Husain ldquoOptimality conditions and dualityin nondifferentiable minimax fractional programming withgeneralized convexityrdquo Journal of Optimization Theory andApplications vol 129 no 2 pp 255ndash275 2006

[25] B Mond ldquoA class of nondifferentiable fractional program-ming problemrdquo Zeitschrift fur Angewandte Mathematik undMechanik vol 58 no 8 pp 337ndash341 1978

[26] D S Kim S J Kim and M H Kim ldquoOptimality andduality for a class of nondifferentiable multiobjective fractionalprogramming problemsrdquo Journal of Optimization Theory andApplications vol 129 no 1 pp 131ndash146 2006

[27] D S Kim Y J Lee and K D Bae ldquoDuality in nondifferentiablemultiobjective fractional programs involving conesrdquo TaiwaneseJournal of Mathematics vol 13 no 6 A pp 1811ndash1821 2009

[28] W Dinklebaeh ldquoOn nonlinear fractional programmingrdquoMan-agement Science vol 13 no 7 pp 492ndash498 1967

[29] M A Hanson ldquoOn sufficiency of the Kuhn-Tucker conditionsrdquoJournal of Mathematical Analysis and Applications vol 80 no2 pp 545ndash550 1981

[30] C R Bector S K Suneja and S Gupta ldquoUnivex functionsand univex nonlinear programmingrdquo in Proceeding of theAdministrative Science Association of Canada pp 115ndash124 1992

[31] S K Mishra ldquoOn multiple-objective optimization with gener-alized univexityrdquo Journal of Mathematical Analysis and Applica-tions vol 224 no 1 pp 131ndash148 1998

[32] A Jayswal ldquoNon-differentiable minimax fractional program-ming with generalized 120572-univexityrdquo Journal of Computationaland Applied Mathematics vol 214 no 1 pp 121ndash135 2008

[33] A M Geoffrion ldquoProper efficiency and the theory of vectormaximizationrdquo Journal of Mathematical Analysis and Applica-tions vol 22 no 3 pp 618ndash630 1968

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 3: Research Article Second Order Duality in Multiobjective Fractional Programming …downloads.hindawi.com/archive/2014/541524.pdf · 2019-07-31 · Research Article Second Order Duality

International Scholarly Research Notices 3

Definition 3 The real differentiable function 119891119894is said to be

second order 120588-quasiunivex at 119909 isin 119883 with respect to 120578 120595and119870 if

119870 (119909 119909) 120595 [119891119894(119909) minus 119891

119894(119909) +

1

2119901119879

(nabla2

119891119894(119909) 119901)] le 0

997904rArr 120578(119909 119909)119879

[nabla119891119894(119909) + nabla

2

119891119894(119909) 119901] + 120588119909 minus 119909

2

le 0

forall119909 isin 119883

(8)

Remark 4 If 119901 = 0 the above definitions reduce to thedefinitions of 120588-univex 120588-pseudounivex and 120588-quasiunivexas introduced in [14]

Definition 5 A feasible point119909 is said to be efficient forMFP1if there exists no other feasible point 119909 in MFP1 such that119865119894(119909) le 119865

119894(119909) 119894 = 1 2 119896 and 119865

119903(119909) lt 119865

119903(119909) for some

119903 isin 1 2 119896

Definition 6 (see [33]) A feasible point119909 is said to be properlyefficient for MFP1 if it is efficient and there exist 119872 gt 0

such that for each 119894 isin 1 2 119896 and for all feasible point119909 in MFP1 satisfying 119865

119894(119909) lt 119865

119894(119909) we have 119865

119894(119909) minus 119865

119894(119909) le

119872(119865119903(119909) minus 119865

119903(119909)) for some 119903 isin 1 2 119896 such that 119865

119903(119909) gt

119865119903(119909)

We assume that 119891119894(119909) + (119909

119879

119861119894119909)12

ge 0 119892119894(119909) minus

(119909119879

119862119894119909)12

gt 0 119894 = 1 2 119896 for all 119909 isin 119883

Definition 7 (generalized Schwarz Inequality) Let 119861 be apositive semidefinite matrix of order 119899 Then for all 119909 119908 isinR119899 119909119879119861119908 le (119909119879119861119909)12(119908119879119861119908)12

The equality holds if 119861119909 = 120582119861119908 for some 120582 ge 0Let 1198691(119909) = 119895 isin 119872 = 1 2 119898 ℎ

119895(119909) = 0 and

V119894= (119891119894(119909) + (119909

119879

119861119894119909)12

)(119892119894(119909) minus (119909

119879

119862119894119909)12

)Then define the set119882(119909) = 119908 isin R119899 119908119879nablaℎ

119895(119909) le 0 119895 isin

119869(119909) satisfying any one of the following conditions

(a) 119909119879119861119894119909 gt 0 119909119879119862

119894119909 = 0 rArr 119908

119879

(nabla119891119894(119909) + 119861

119894119909

(119909119879

119861119894119909)12

minus V119894nabla119892119894(119909)) + (119908

119879

(V2119894119862119894)119908)12

ge 0 119908 isin119882(119909) 119894 = 1 2 119896

(b) 119909119879119861119894119909 = 0 119909119879119862

119894119909 gt 0 rArr 119908

119879

(nabla119891119894(119909) minus V

119894nabla119892119894(119909) minus

119862119894119909(119909119879

119862119894119909)12

) + (119908119879

119861119894119908)12

ge 0 119908 isin 119882(119909) 119894 =1 2 119896

(c) 119909119879119861119894119909 = 0 119909119879119862

119894119909 = 0 rArr 119908

119879

(nabla119891119894(119909) minus V

119894nabla119892119894(119909)) +

(119908119879

119861119894119908)12

+ (119908119879

(V2119894119862119894)119908)12

ge 0 119908 isin 119882(119909) 119894 =1 2 119896

(d) 119909119879119861119894119909 gt 0 119909119879119862

119894119909 gt 0 rArr 119908

119879

(nabla119891119894(119909) +

119861119894119909(119909119879

119861119894119909)12

minus V119894nabla119892119894(119909) minus 119862

119894119909(119909119879

119862119894119909)12

) ge 0119908 isin 119882(119909) 119894 = 1 2 119896

Lemma 8 (see [33]) If 1199090 isin 1198830is an optimal solution of

119872119865119875120582 then 1199090 is properly efficient for MFP1

Lemma 9 (see [12]) 1199090 isin 1198830is an efficient solution for MFP0

if and only if it is an efficient solution of MFP1 with 119865(1199090) = 0

Lemma 10 (see [10] necessary optimality condition) If 119909 isin1198830is an optimal solution of (119872119865119875120582) such that119882(119909) = 120601 then

there exist V119894isin R+ 119908 119911 isin R119899 and 119910 isin R119898 such that

nabla120582119865 (119909) + 119910119879

nablaℎ (119909)

=

119896

sum

119894=1

120582119894[nabla119891119894(119909) + 119861

119894119908 minus V119894nabla119892119894(119909) minus 119862

119894119911]

+

119898

sum

119895=1

119910119895nablaℎ119895(119909) = 0

(9)

119865119894(119909) = 119891

119894(119909) + (119909

119879

119861119894119909)12

minus V119894(119892119894(119909) minus (119909

119879

119862119894119909)12

) = 0

119894 = 1 2 119896

(10)

119910119879

ℎ (119909) = 0 (11)

119908119879

119861119894119908 le 1 119911

119879

119862119894119911 le 1 119894 = 1 2 119896 (12)

(119909119879

119861119894119909)12

= 119909119879

119861119894119908 (119909

119879

119862119894119909)12

= 119909119879

119862119894119911

119894 = 1 2 119896

(13)

119910 ge 0 (14)

V119894ge 0 119894 = 1 2 119896 (15)

Theorem 11 (sufficient optimality condition) Let 119909 isin 1198830be

a feasible solution of MFP1 and there exist 120582119894isin R+119908 119911 isin R119899

V119894isin R+ and 119910 isin R119898 satisfying the condition in Lemma 10 at

119909 Furthermore suppose that the following conditions hold

(i) 119875(119909) = sum119896

119894=1120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus

(119909119879

119862119894119909)12

] + sum119898

119895=1119910119895ℎ119895(119909) is second order 120588-

pseudounivex with respect to 120578 120595 and 119870 at 119909 isin 1198830

with (nabla2119875(119909))119901 = 0 where119891119894 119883 rarr R119892

119894 119883 rarr R

ℎ119895 119883 rarr R 119894 = 1 2 119896 119895 = 1 2 119898 119901 isin R119899

120578 119883times119883 rarr R119899119870 119883times119883 rarr R+ and 120595 R rarr R

satisfying 120595(119886) le 0 rArr 119886 le 0

(ii) 120588 ge 0

Then 119909 is an efficient solution of MFP1

Proof Suppose that the hypothesis holdsSince the conditions of Lemma 10 are satisfied from (9)

and (13) we have

nabla119875 (119909)

= nabla(

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minusV119894119892119894(119909) minus (119909

119879

119862119894119909)12

] +

119898

sum

119895=1

119910119895ℎ119895(119909))

4 International Scholarly Research Notices

= nabla(

119896

sum

119894=1

120582119894[119891119894(119909) + 119909

119879

119861119894119908 minus V119894119892119894(119909) minus 119909

119879

119862119894119911]

+

119898

sum

119895=1

119910119895ℎ119895(119909)) = 0

(16)

Also from hypothesis (i) we have nabla2119875(119909)119901 = 0So we can write nabla119875(119909) + nabla2119875(119909)119901 = 0Now for 120578(119909 119909) isin R119899 we can write 120578(119909 119909)119879(nabla119875(119909) +

nabla2

119875(119909)119901) = 0For 120588 ge 0 we have 120578(119909 119909)119879(nabla119875(119909) + nabla2119875(119909)119901) +

120588119909 minus 1199092

ge 0Since 119875(119909) is second order 120588-pseudounivex with respect

to 120578 120595 and 119870 at 119909 isin 1198830 we have 119870(119909 119909)120595119875(119909) minus 119875(119909) +

(12)119901119879

(nabla2

119875(119909))119901 ge 0 and using the properties of 119870 120595 itgives

119875 (119909) minus 119875 (119909) +1

2119901119879

nabla2

119875 (119909) 119901

ge 0 997904rArr 119875 (119909) ge 119875 (119909) minus1

2119901119879

nabla2

119875 (119909) 119901

(17)

Since nabla2119875(119909)119901 = 0 the above inequality implies 119875(119909) ge 119875(119909)

997904rArr

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minusV119894119892119894(119909) minus (119909

119879

119862119894119909)12

] +

119898

sum

119895=1

119910119895ℎ119895(119909)

ge

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

] +

119898

sum

119895=1

119910119895ℎ119895(119909)

(18)

Suppose that 119909 is not efficient solution of MFP1 then thereexist 119909 isin 119883

0such that

119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

le 119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

119894 = 1 2 119896

(19)

and 119891119905(119909) + (119909

119879

119861119905119909)12

minus V119905119892119905(119909) minus (119909

119879

119862119905119909)12

le 119891119905(119909) +

(119909119879

119861119905119909)12

minus V119905119892119905(119909)minus (119909

119879

119862119905119909)12

for some 119905 isin 1 2 119896

The above relation together with the relation 120582119894gt 0

implies that

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

]

lt

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

]

(20)

From the relations (5) (11) and (14) we get

119898

sum

119895=1

119910119895ℎ119895(119909) le

119898

sum

119895=1

119910119895ℎ119895(119909) (21)

Consequently (20) and (21) yield

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

]

+

119898

sum

119895=1

119910119895ℎ119895(119909)

lt

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

]

+

119898

sum

119895=1

119910119895ℎ119895(119909)

(22)

This contradicts (18) Hence 119909 is an efficient solution forMFP1

Theorem 12 (sufficient optimality condition) Let 119909 isin 1198830be

a feasible solution of MFP1 and there exist 120582119894isin R+119908 119911 isin R119899

V119894isin R+ and 119910 isin R119898 satisfying the condition in Lemma 10 at

119909 Furthermore suppose that the following conditions hold

(i) 119876(119909) = sum119896

119894=1120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus

(119909119879

119862119894119909)12

] is second order 120588-pseudounivex withrespect to 120578 120595

0 and 119870 at 119909 isin 119883

0and 119867(119909) =

sum119898

119895=1119910119895ℎ119895(119909) is second order 120590-quasiunivex with

respect to 120578 1205951 and 119870 at 119909 isin 119883

0with (nabla2119876(119909))119901 = 0

and (nabla2119867(119909))119901 = 0 where 119891119894 119883 rarr R 119892

119894 119883 rarr R

ℎ119895 119883 rarr R 119894 = 1 2 119896 119895 = 1 2 119898 119901 isin R119899

120578 119883 times 119883 rarr R119899 119870 119883 times 119883 rarr R+ and

1205950 1205951 R rarr R satisfying 120595

0(119886) ge 0 rArr 119886 ge 0 and

1205951(119886) le 0 rArr 119886 le 0

(ii) 120588 + 120590 ge 0

Then 119909 is an efficient solution of1198721198651198751

Proof Suppose hypothesis holds

International Scholarly Research Notices 5

From the relations (5) (11) and (14) we get119898

sum

119895=1

119910119895ℎ119895(119909)

le

119898

sum

119895=1

119910119895ℎ119895(119909) 997904rArr 119867(119909) le 119867 (119909) 997904rArr 119867(119909) minus 119867 (119909) le 0

(23)

Also from hypothesis (i) we get (nabla2119867(119909))119901 = 0So we have the following

119867(119909) minus 119867 (119909) +1

2119901119879

(nabla2

119867(119909)) 119901 le 0

997904rArr 119870 (119909 119909) 1205951119867 (119909) minus 119867 (119909) +

1

2119901119879

(nabla2

119867(119909)) 119901 le 0

(24)

Hence the 120590-quasiunivexity of 119867(119909) with respect to 120578 1205951

and119870 implies the following

120578(119909 119909)119879

nabla119867 (119909) + (nabla2

119867(119909)) 119901 + 120590119909 minus 1199092

le 0

997904rArr 120578(119909 119909)119879

nabla119867 (119909) + 120590119909 minus 1199092

le 0

(25)

From (9) we get

119896

sum

119894=1

120582119894[nabla119891119894(119909) + 119861

119894119908 minus V119894nabla119892119894(119909) minus 119862

119894119911]

+

119898

sum

119894=1

119910119894nablaℎ119894(119909) = 0

997904rArr nabla119876 (119909) + nabla119867 (119909) = 0

997904rArr 120578(119909 119909)119879

[nabla119876 (119909) + nabla119867 (119909)] = 0

997904rArr 120578(119909 119909)119879

nabla119876 (119909) + 120578(119909 119909)119879

nabla119867 (119909)

+ 120590119909 minus 1199092

minus 120590119909 minus 1199092

= 0

(26)

Using (25) in (26) we get

120578(119909 119909)119879

nabla119876 (119909) minus 120590119909 minus 1199092

ge 0 (27)

Since 120588 + 120590 ge 0 we get

120588119909 minus 1199092

ge minus120590119909 minus 1199092

(28)

So we have

120578(119909 119909)119879

nabla119876 (119909) + 120588119909 minus 1199092

ge 0

997904rArr 120578(119909 119909)119879

nabla119876 (119909) + nabla2

119876 (119909) 119901 + 120588119909 minus 1199092

ge 0

(29)

Since 119876(119909) is 120588-pseudounivex with respect to 120578 1205950 and 119870

we obtained

119870 (119909 119909) 1205950[119876 (119909) minus 119876 (119909) +

1

2119901119879

nabla2

119876 (119909) 119901] ge 0 (30)

Using the property of119870 and 1205950 we get

119876 (119909) minus 119876 (119909) +1

2119901119879

nabla2

119876 (119909) 119901 ge 0 997904rArr 119876 (119909) ge 119876 (119909)

997904rArr

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minusV119894119892119894(119909) minus (119909

119879

119862119894119909)12

]

ge

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

]

(31)

If 119909 were not an efficient solution to MFP1 then from (20)we have

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

]

lt

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minusV119894119892119894(119909) minus (119909

119879

119862119894119909)12

]

(32)

This contradicts (31)Therefore 119909 is an efficient solution for MFP1

3 Second Order Mixed Type MultiobjectiveFractional Duality

(i) MMFD0 Maximize

119871 (119906)

= (1198711(119906) minus

1

2119901119879

nabla2

1198711(119906) 119901 119871

2(119906)

minus1

2119901119879

nabla2

1198712(119906) 119901 119871

119896(119906) minus

1

2119901119879

nabla2

119871119896(119906) 119901)

(33)

where

119871119894(119906) =

119891119894(119906) + 119910

119879

1198691

ℎ1198691(119906) + 119906

119879

119861119894119908

119892119894(119906) minus 119906119879119862

119894119911

119894 = 1 2 119896 (34)

(ii) MMFD1 Maximize

119866 (119906)

= (1198661(119906) minus

1

2119901119879

nabla2

1198661(119906) 119901 119866

2(119906)

minus1

2119901119879

nabla2

1198662(119906) 119901 119866

119896(119906) minus

1

2119901119879

nabla2

119866119896(119906) 119901)

(35)

where 119866119894(119906) = 119891

119894(119906) + 119910

119879

1198691

ℎ1198691

(119906) + 119906119879

119861119894119908 minus ]119894119892119894(119906) minus

119906119879

119862119894119911 119894 = 1 2 119896 ]

119894are fixed parameters

6 International Scholarly Research Notices

(iii) 119872119872119865119863120582 Maximize 120582119866(119906) 120582 is 119896-dimensionalstrictly positive vector

all subject to same constraints

119896

sum

119894=1

120582119894[nabla119866119894(119906) + nabla

2

119866119894(119906) 119901] + 119910

119879

1198692

[nablaℎ1198692(119906) + nabla

2

nablaℎ1198692(119906) 119901]

= 0

(36)

119891119894(119906) + 119910

119879

1198691

ℎ1198691(119906) + 119906

119879

119861119894119908 minus V119894119892119894(119906) minus 119906

119879

119862119894119911 ge 0

for 119894 = 1 2 119896(37)

119910119879

1198692

ℎ1198692(119906) minus

1

2119901119879

nabla2

(119910119879

1198692

ℎ1198692(119906)) 119901 ge 0

1199101198692

isin R119898minus|1198691|

(38)

119908119879

119861119894119908 le 1 119911

119879

119862119894119911 le 1 119894 = 1 2 119896 (39)

119910 ge 0 V119894ge 0 119894 = 1 2 119896 (40)

where119891119894 119883 rarr R119892

119894 119883 rarr R ℎ

119895 119883 rarr R 119894 = 1 2 119896

119895 = 1 2 119898 are differentiable functions119908 119911 isin R119899119901 isin R119899119861119894 and 119862

119894 119894 = 1 2 119896 are positive semidefinite matrices of

order 119899For the following theorems we assume that 120578 119883 times119883 rarr

R119899 119870 119883 times 119883 rarr R+ and 120595

0 1205951 R rarr R satisfying

1205950(119886) ge 0 rArr 119886 ge 0 and 119887 le 0 rArr 120595

1(119887) le 0 and 120588 120590 isin R

Theorem 13 (weak duality) Let 119909 be a feasible solution for theprimalMFP120582 and let (119906 119910 V 119908) be feasible for dual SMMFD120582If

(i) sum119896119894=1120582119894119866119894(sdot) is second order 120588-pseudounivex with

respect to 120578 1205950 119870 and for 119910

1198692

isin R119898minus|1198691| 1199101198791198692

ℎ1198692

(sdot) issecond order 120590-quasiunivex with respect to 120578 120595

1 and

119870 along with

(ii) 120588 + 120590 ge 0 then Inf(120582119865(119909)) ge Sup(120582119866(119906))

Proof Now from the primal and dual constraints we have

ℎ (119909) le 0

119910119879

1198692

ℎ1198692(119906) minus

1

2119901119879

nabla2

(119910119879

1198692

ℎ1198692(119906)) 119901 ge 0

(41)

So

119910119879

1198692

ℎ1198692(119909) minus 119910

119879

1198692

ℎ1198692(119906) +

1

2119901119879

nabla2

(119910119879

1198692

ℎ1198692(119906)) 119901 le 0

997904rArr 119870 (119909 119906) 1205951[119910119879

1198692

ℎ1198692(119909) minus 119910

119879

1198692

ℎ1198692(119906)

+1

2119901119879

nabla2

(119910119879

1198692

ℎ1198692(119906)) 119901] le 0

(42)

Since 1199101198791198692

ℎ1198692

is second order 120590-quasiunivex with respect to120578 1205951 and119870 and in view of (42) for 119909 119906 isin R119899 we have

120578(119909 119906)119879

nabla [119910119879

1198692

ℎ1198692(119906)] + nabla

2

[119910119879

1198692

ℎ1198692(119906)] 119901 + 120590119909 minus 119906

2

le 0

(43)

Again from the dual constraint (36) we have119896

sum

119894=1

120582119894[nabla119866119894(119906) + nabla

2

119866119894(119906) 119901]

+ 119910119879

1198692

119879

[nablaℎ1198692(119906) + nabla

2

ℎ1198692(119906) 119901] = 0

(44)

Since 120578(119909 119906) isin R119899 we have

120578(119909 119906)119879

119896

sum

119894=1

120582119894[nabla119866119894(119906) + nabla

2

119866119894(119906) 119901]

+ 120578(119909 119906)119879

119910119879

1198692

[nablaℎ1198692(119906) + nabla

2

ℎ1198692(119906) 119901] = 0

997904rArr 120578(119909 119906)119879

119896

sum

119894=1

120582119894[nabla119866119894(119906) + nabla

2

119866119894(119906) 119901]

+ 120578(119909 119906)119879

119910119879

1198692

[nablaℎ1198692(119906) + nabla

2

ℎ1198692(119906) 119901]

+ 120590119909 minus 1199062

minus 120590119909 minus 1199062

= 0

(45)

Using (43) in above equation we get 120578(119909 119906)119879sum119896119894=1120582119894

[nabla119866119894(119906) + nabla

2

119866119894(119906)119901] minus 120590119909 minus 119906

2

ge 0Since 120588 + 120590 ge 0 we get 120588119909 minus 1199062 ge minus120590119909 minus 1199062So we have

120578(119909 119906)119879

119896

sum

119894=1

120582119894[nabla119866119894(119906) + nabla

2

119866119894(119906) 119901] + 120588119909 minus 119906

2

ge 0

(46)

Since sum119896

119894=1120582119894119866119894(119906) is second order 120588-pseudounivex

with respect to 120578 1205950 and 119870 by Definition 2 and

(46) we get 119870(119909 119906)1205950sum119896

119894=1120582119894119866119894(119909) minus sum

119896

119894=1120582119894119866119894(119906) +

(12)119901119879

sum119896

119894=1120582119894119866119894(119906)119901 ge 0

Using the property of 1205950and119870 we get

119896

sum

119894=1

120582119894119866119894(119909) minus

119896

sum

119894=1

120582119894119866119894(119906) +

1

2119901119879

119896

sum

119894=1

120582119894119866119894(119906) 119901 ge 0

997904rArr

119896

sum

119894=1

120582119894119866119894(119909) ge

119896

sum

119894=1

120582119894119866119894(119906)

997904rArr

119896

sum

119894=1

120582119894[119891119894(119909) + 119910

119879

1198691

ℎ1198691(119909)

+119909119879

119861119894119908 minus V119894119892119894(119909) minus 119909

119879

119862119894119911]

ge

119896

sum

119894=1

120582119894[119891119894(119906) + 119910

119879

1198691

ℎ1198691(119906)

+119906119879

119861119894119908 minus V119894119892119894(119906) minus 119909

119879

119862119894119911]

(47)

Equation (5) gives ℎ(119909) le 0 rArr 1199101198791198691

ℎ1198691

(119909) le 0 for 1199101198691

ge 0

International Scholarly Research Notices 7

So (47) implies that

119896

sum

119894=1

120582119894[119891119894(119909) + 119909

119879

119861119894119908 minus V119894119892119894(119909) minus 119909

119879

119862119894119911]

ge

119896

sum

119894=1

120582119894[119891119894(119906) + 119910

119879

1198691

ℎ1198691(119906) + 119906

119879

119861119894119908

minusV119894119892119894(119906) minus 119906

119879

119862119894119911]

(48)

Now by Schwarz Inequality and (39) we have

119909119879

119861119894119908 le (119909

119879

119861119894119909)12

(119908119879

119861119894119908)12

le (119909119879

119861119894119909)12

119909119879

119862119894119911 le (119909

119879

119862119894119909)12

(119911119879

119862119894119911)12

le (119909119879

119862119894119909)12

119894 = 1 2 119896

(49)

So both (48) and (49) imply that

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

]

ge

119896

sum

119894=1

120582119894[119891119894(119906) + 119910

119879

1198691

ℎ1198691(119906) + 119906

119879

119861119894119908

minus V119894119892119894(119906) minus 119906

119879

119862119894119911]

997904rArr Inf (120582119865 (119909)) ge Sup (120582119865 (119906))

(50)

Theorem 14 (strong duality) Let 119909 be optimal solution forMFP120582 and let 119882(119909) = 120601 Then there exist 120582

119894isin R+ 119908 119911 isin

R119899 V119894isin R+ and 119910 isin R119898 such that (119906 120582 119910 V 119908 119911 119901 =

0) is a feasible solution for dual and the objective values ofboth primal and dual are equal to zero Furthermore if (i)(119906 120582 119910 V 119908 119911) is feasible for dual (ii) sum119896

119894=1120582119894119866119894(sdot) is second

order 120588-pseudounivex with respect to 120578 1205950 and 119870 and for

1199101198692

isin R119898minus|1198691| 1199101198791198691

ℎ1198692

(sdot) is second order 120590-quasiunivex withrespect to 120578 120595

1 and 119870 along with (iii) 120588 + 120590 ge 0 then

(119906 120582 119910 V 119908 119911 119901 = 0) is properly efficient for SMMFD0

Proof Since 119909 is optimal solution for (MFP120582) by Lemma 10there exist 120582

119894isin R+119908 119911 isin R119899 V

119894isin R+ and 119910 isin R119898 such that

119896

sum

119894=1

120582119894[nabla119891119894(119909) + 119861

119894119908 minus V119894nabla119892119894(119909) minus 119862

119894119911] + 119910

119879

nablaℎ (119909) = 0

119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894(119892119894(119909) minus (119909

119879

119862119894119909)12

) = 0

119894 = 1 2 119896

119910119879

ℎ (119909) = 0

119908119879

119861119894119908 le 1 119911

119879

119862119894119911 le 1 119894 = 1 2 119896

(119909119879

119861119894119909)12

= 119909119879

119861119894119908 (119909

119879

119862119894119909)12

= 119909119879

119862119894119911

119894 = 1 2 119896

119910 ge 0

V119894ge 0 119894 = 1 2 119896

(51)

which can be written as119896

sum

119894=1

120582119894[nabla119891119894(119909) + 119910

119879

1198691

nablaℎ1198691(119909) + 119861

119894119908 minus V119894nabla119892119894(119909) minus 119862

119894119911]

+ 119910119879

1198691

nablaℎ1198692(119909) = 0

119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

+ 119910119879

1198691

ℎ1198691(119909) = 0

119910119879

1198691

ℎ1198692(119909) = 0

119908119879

119861119894119908 le 1 119911

119879

119862119894119911 le 1 119894 = 1 2 119896

(119909119879

119861119894119909)12

= 119909119879

119861119894119908 (119909

119879

119862119894119909)12

= 119909119879

119862119894119911

119894 = 1 2 119896

119910 ge 0

V119894ge 0 119894 = 1 2 119896

(52)

These are nothing but the dual constraintsSo (119906 120582 119910 V 119908 119911 119901 = 0) is feasible solution for dual

problemAnd the objective values of MFP120582 and SMMFD120582 are

equal to zero It follows fromTheorem 13 and for any feasiblesolution (119906 120582 119910 V 119908 119911 119901 = 0) of dual that 120582119866(119906) le 120582119866(119909)

So (119906 120582 119910 V 119908 119911 119901 = 0) is optimal solution ofSMMFD120582 Then applying Lemmas 8 and 9 we concludethat (119906 120582 119910 V 119908 119911 119901 = 0) is properly efficient for(SMMFD0)

4 Special Case

If 119901 = 0 119862119894= 0 119894 = 1 2 119896 then our dual programming

reduces to the dual programming proposed by Tripathy [14]

5 Conclusion

In this paper three approaches given by Dinklebaeh [28] andJagannathan [12] for both primal and second order mixedtype dual of a nondifferentiable multiobjective fractionalprogramming problem are introduced and the necessaryand sufficient optimality conditions are established and aparameterization technique is used to establish duality results

8 International Scholarly Research Notices

under generalized second order 120588-univexity assumptionTheresults developed in this paper can be further extendedto higher order mixed type fractional problem containingsquare root term Also the present work can be furtherextended to a class of nondifferentiable minimax mixedfractional programming problems

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] C R Bector ldquoDuality in nonlinear fractional programmingrdquoZeitschrift fur Operations Research vol 17 no 5 pp 183ndash1931973

[2] C R Bector and S Chandra ldquoFirst and Second order duality fora class of non differentiable fractional programmingrdquo Journal ofInformation amp Optimization Sciences vol 7 pp 335ndash348 1986

[3] B D Craven ldquoFractional programmingrdquo in Sig Ma Series inApplied Mathematics vol 4 pp 145ndash152 Heldermann BerlinGermany 1998

[4] B Mond and T Weir ldquoDuality for fractional programmingwith generalized convexity conditionrdquo Journal of Information ampOptimization Sciences vol 3 pp 105ndash124 1982

[5] I M Stancu-Minasian Fractional Programming TheoryMethod and Application Kluwer Academic PublishersDordrecht The Netherland 1997

[6] S Schaible and T Ibaraki ldquoFractional programmingrdquo EuropeanJournal of Operational Research vol 12 no 4 pp 325ndash338 1983

[7] I Ahmad S K Gupta N Kailey and R P Agarwal ldquoDualityin nondifferentiable minimax fractional programming with B-(pr)-invexityrdquo Journal of Inequalities and Applications vol 2011article 75 15 pages 2011

[8] I Ahmad and S Sharma ldquoSecond-order duality for nondif-ferentiable multiobjective programming problemsrdquo NumericalFunctional Analysis and Optimization vol 28 no 9-10 pp 975ndash988 2007

[9] T R Gulati I Ahmad and D Agarwal ldquoSufficiency andduality in multiobjective programming under generalized typei functionsrdquo Journal of Optimization Theory and Applicationsvol 135 no 3 pp 411ndash427 2007

[10] I Ahmad Z Husain and S Al-Homidan ldquoSecond-order dual-ity in nondifferentiable fractional programmingrdquo NonlinearAnalysis Real World Applications vol 12 no 2 pp 1103ndash11102011

[11] C R Bector S Chandra and I Husain ldquoOptimality conditionsand duality in subdifferentiable multiobjective fractional pro-grammingrdquo Journal of Optimization Theory and Applicationsvol 79 no 1 pp 105ndash125 1993

[12] R Jagannathan ldquoDuality for nonlinear fractional programsrdquoZeitschrift fur Operations Research vol 17 no 1 pp 1ndash3 1973

[13] R Osuna-Gomez A Ruız-Canales and P Rufian-Lizana ldquoMul-tiobjective fractional programmingwith generalized convexityrdquoTrabajos de Investigacion Operativa vol 8 no 1 pp 97ndash1102000

[14] A K Tripathy ldquoMixed type duality in multiobjective fractionalprogramming under generalized 120588-univex functionrdquo Journal ofMathematical Modelling and Algorithms in Operations Research2013

[15] S K Mishra S Wang and K K Lai ldquoRole of 120572-pseudo-univex functions in vector variational-like inequality problemsrdquoJournal of Systems Science and Complexity vol 20 no 3 pp344ndash349 2007

[16] S Schaible ldquoFractional programmingrdquo in Handbook of GlobalOptimization R Horst and P M Pardalos Eds pp 495ndash608 KluwerAcademic Publishers DordrechtTheNetherlands1995

[17] Z A Liang H X Huang and P M Pardalos ldquoOptimalityconditions and duality for a class of nonlinear fractionalprogramming problemsrdquo Journal of Optimization Theory andApplications vol 110 no 3 pp 611ndash619 2001

[18] Z-A Liang H-X Huang and P M Pardalos ldquoEfficiencyconditions and duality for a class of multiobjective fractionalprogramming problemsrdquo Journal of Global Optimization vol27 no 4 pp 447ndash471 2003

[19] L B Santos R Osuna-Gomez and A Rojas-Medar ldquoNons-moothmultiobjective fractional programmingwith generalizedconvexityrdquo Revista Integracion vol 26 no 1 pp 1ndash12 2008

[20] C R Bector S Chandra and A Abha ldquoOn mixed duality inmathematical programmingrdquo Journal of Mathematical Analysisand Applications vol 259 no 1 pp 346ndash356 2001

[21] Z Xu ldquoMixed type duality in multiobjective programmingproblemsrdquo Journal of Mathematical Analysis and Applicationsvol 198 no 3 pp 621ndash635 1996

[22] H Zhou and Y Wang ldquoOptimality and mixed duality for non-smooth multi-objective fractional programmingrdquo Pure Mathe-matics and Applications vol 14 no 3 pp 263ndash274 2003

[23] I Ahmad andZHusain ldquoDuality in nondifferentiableminimaxfractional programming with generalized convexityrdquo AppliedMathematics and Computation vol 176 no 2 pp 545ndash5512006

[24] I Ahmad and Z Husain ldquoOptimality conditions and dualityin nondifferentiable minimax fractional programming withgeneralized convexityrdquo Journal of Optimization Theory andApplications vol 129 no 2 pp 255ndash275 2006

[25] B Mond ldquoA class of nondifferentiable fractional program-ming problemrdquo Zeitschrift fur Angewandte Mathematik undMechanik vol 58 no 8 pp 337ndash341 1978

[26] D S Kim S J Kim and M H Kim ldquoOptimality andduality for a class of nondifferentiable multiobjective fractionalprogramming problemsrdquo Journal of Optimization Theory andApplications vol 129 no 1 pp 131ndash146 2006

[27] D S Kim Y J Lee and K D Bae ldquoDuality in nondifferentiablemultiobjective fractional programs involving conesrdquo TaiwaneseJournal of Mathematics vol 13 no 6 A pp 1811ndash1821 2009

[28] W Dinklebaeh ldquoOn nonlinear fractional programmingrdquoMan-agement Science vol 13 no 7 pp 492ndash498 1967

[29] M A Hanson ldquoOn sufficiency of the Kuhn-Tucker conditionsrdquoJournal of Mathematical Analysis and Applications vol 80 no2 pp 545ndash550 1981

[30] C R Bector S K Suneja and S Gupta ldquoUnivex functionsand univex nonlinear programmingrdquo in Proceeding of theAdministrative Science Association of Canada pp 115ndash124 1992

[31] S K Mishra ldquoOn multiple-objective optimization with gener-alized univexityrdquo Journal of Mathematical Analysis and Applica-tions vol 224 no 1 pp 131ndash148 1998

[32] A Jayswal ldquoNon-differentiable minimax fractional program-ming with generalized 120572-univexityrdquo Journal of Computationaland Applied Mathematics vol 214 no 1 pp 121ndash135 2008

[33] A M Geoffrion ldquoProper efficiency and the theory of vectormaximizationrdquo Journal of Mathematical Analysis and Applica-tions vol 22 no 3 pp 618ndash630 1968

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 4: Research Article Second Order Duality in Multiobjective Fractional Programming …downloads.hindawi.com/archive/2014/541524.pdf · 2019-07-31 · Research Article Second Order Duality

4 International Scholarly Research Notices

= nabla(

119896

sum

119894=1

120582119894[119891119894(119909) + 119909

119879

119861119894119908 minus V119894119892119894(119909) minus 119909

119879

119862119894119911]

+

119898

sum

119895=1

119910119895ℎ119895(119909)) = 0

(16)

Also from hypothesis (i) we have nabla2119875(119909)119901 = 0So we can write nabla119875(119909) + nabla2119875(119909)119901 = 0Now for 120578(119909 119909) isin R119899 we can write 120578(119909 119909)119879(nabla119875(119909) +

nabla2

119875(119909)119901) = 0For 120588 ge 0 we have 120578(119909 119909)119879(nabla119875(119909) + nabla2119875(119909)119901) +

120588119909 minus 1199092

ge 0Since 119875(119909) is second order 120588-pseudounivex with respect

to 120578 120595 and 119870 at 119909 isin 1198830 we have 119870(119909 119909)120595119875(119909) minus 119875(119909) +

(12)119901119879

(nabla2

119875(119909))119901 ge 0 and using the properties of 119870 120595 itgives

119875 (119909) minus 119875 (119909) +1

2119901119879

nabla2

119875 (119909) 119901

ge 0 997904rArr 119875 (119909) ge 119875 (119909) minus1

2119901119879

nabla2

119875 (119909) 119901

(17)

Since nabla2119875(119909)119901 = 0 the above inequality implies 119875(119909) ge 119875(119909)

997904rArr

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minusV119894119892119894(119909) minus (119909

119879

119862119894119909)12

] +

119898

sum

119895=1

119910119895ℎ119895(119909)

ge

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

] +

119898

sum

119895=1

119910119895ℎ119895(119909)

(18)

Suppose that 119909 is not efficient solution of MFP1 then thereexist 119909 isin 119883

0such that

119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

le 119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

119894 = 1 2 119896

(19)

and 119891119905(119909) + (119909

119879

119861119905119909)12

minus V119905119892119905(119909) minus (119909

119879

119862119905119909)12

le 119891119905(119909) +

(119909119879

119861119905119909)12

minus V119905119892119905(119909)minus (119909

119879

119862119905119909)12

for some 119905 isin 1 2 119896

The above relation together with the relation 120582119894gt 0

implies that

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

]

lt

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

]

(20)

From the relations (5) (11) and (14) we get

119898

sum

119895=1

119910119895ℎ119895(119909) le

119898

sum

119895=1

119910119895ℎ119895(119909) (21)

Consequently (20) and (21) yield

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

]

+

119898

sum

119895=1

119910119895ℎ119895(119909)

lt

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

]

+

119898

sum

119895=1

119910119895ℎ119895(119909)

(22)

This contradicts (18) Hence 119909 is an efficient solution forMFP1

Theorem 12 (sufficient optimality condition) Let 119909 isin 1198830be

a feasible solution of MFP1 and there exist 120582119894isin R+119908 119911 isin R119899

V119894isin R+ and 119910 isin R119898 satisfying the condition in Lemma 10 at

119909 Furthermore suppose that the following conditions hold

(i) 119876(119909) = sum119896

119894=1120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus

(119909119879

119862119894119909)12

] is second order 120588-pseudounivex withrespect to 120578 120595

0 and 119870 at 119909 isin 119883

0and 119867(119909) =

sum119898

119895=1119910119895ℎ119895(119909) is second order 120590-quasiunivex with

respect to 120578 1205951 and 119870 at 119909 isin 119883

0with (nabla2119876(119909))119901 = 0

and (nabla2119867(119909))119901 = 0 where 119891119894 119883 rarr R 119892

119894 119883 rarr R

ℎ119895 119883 rarr R 119894 = 1 2 119896 119895 = 1 2 119898 119901 isin R119899

120578 119883 times 119883 rarr R119899 119870 119883 times 119883 rarr R+ and

1205950 1205951 R rarr R satisfying 120595

0(119886) ge 0 rArr 119886 ge 0 and

1205951(119886) le 0 rArr 119886 le 0

(ii) 120588 + 120590 ge 0

Then 119909 is an efficient solution of1198721198651198751

Proof Suppose hypothesis holds

International Scholarly Research Notices 5

From the relations (5) (11) and (14) we get119898

sum

119895=1

119910119895ℎ119895(119909)

le

119898

sum

119895=1

119910119895ℎ119895(119909) 997904rArr 119867(119909) le 119867 (119909) 997904rArr 119867(119909) minus 119867 (119909) le 0

(23)

Also from hypothesis (i) we get (nabla2119867(119909))119901 = 0So we have the following

119867(119909) minus 119867 (119909) +1

2119901119879

(nabla2

119867(119909)) 119901 le 0

997904rArr 119870 (119909 119909) 1205951119867 (119909) minus 119867 (119909) +

1

2119901119879

(nabla2

119867(119909)) 119901 le 0

(24)

Hence the 120590-quasiunivexity of 119867(119909) with respect to 120578 1205951

and119870 implies the following

120578(119909 119909)119879

nabla119867 (119909) + (nabla2

119867(119909)) 119901 + 120590119909 minus 1199092

le 0

997904rArr 120578(119909 119909)119879

nabla119867 (119909) + 120590119909 minus 1199092

le 0

(25)

From (9) we get

119896

sum

119894=1

120582119894[nabla119891119894(119909) + 119861

119894119908 minus V119894nabla119892119894(119909) minus 119862

119894119911]

+

119898

sum

119894=1

119910119894nablaℎ119894(119909) = 0

997904rArr nabla119876 (119909) + nabla119867 (119909) = 0

997904rArr 120578(119909 119909)119879

[nabla119876 (119909) + nabla119867 (119909)] = 0

997904rArr 120578(119909 119909)119879

nabla119876 (119909) + 120578(119909 119909)119879

nabla119867 (119909)

+ 120590119909 minus 1199092

minus 120590119909 minus 1199092

= 0

(26)

Using (25) in (26) we get

120578(119909 119909)119879

nabla119876 (119909) minus 120590119909 minus 1199092

ge 0 (27)

Since 120588 + 120590 ge 0 we get

120588119909 minus 1199092

ge minus120590119909 minus 1199092

(28)

So we have

120578(119909 119909)119879

nabla119876 (119909) + 120588119909 minus 1199092

ge 0

997904rArr 120578(119909 119909)119879

nabla119876 (119909) + nabla2

119876 (119909) 119901 + 120588119909 minus 1199092

ge 0

(29)

Since 119876(119909) is 120588-pseudounivex with respect to 120578 1205950 and 119870

we obtained

119870 (119909 119909) 1205950[119876 (119909) minus 119876 (119909) +

1

2119901119879

nabla2

119876 (119909) 119901] ge 0 (30)

Using the property of119870 and 1205950 we get

119876 (119909) minus 119876 (119909) +1

2119901119879

nabla2

119876 (119909) 119901 ge 0 997904rArr 119876 (119909) ge 119876 (119909)

997904rArr

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minusV119894119892119894(119909) minus (119909

119879

119862119894119909)12

]

ge

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

]

(31)

If 119909 were not an efficient solution to MFP1 then from (20)we have

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

]

lt

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minusV119894119892119894(119909) minus (119909

119879

119862119894119909)12

]

(32)

This contradicts (31)Therefore 119909 is an efficient solution for MFP1

3 Second Order Mixed Type MultiobjectiveFractional Duality

(i) MMFD0 Maximize

119871 (119906)

= (1198711(119906) minus

1

2119901119879

nabla2

1198711(119906) 119901 119871

2(119906)

minus1

2119901119879

nabla2

1198712(119906) 119901 119871

119896(119906) minus

1

2119901119879

nabla2

119871119896(119906) 119901)

(33)

where

119871119894(119906) =

119891119894(119906) + 119910

119879

1198691

ℎ1198691(119906) + 119906

119879

119861119894119908

119892119894(119906) minus 119906119879119862

119894119911

119894 = 1 2 119896 (34)

(ii) MMFD1 Maximize

119866 (119906)

= (1198661(119906) minus

1

2119901119879

nabla2

1198661(119906) 119901 119866

2(119906)

minus1

2119901119879

nabla2

1198662(119906) 119901 119866

119896(119906) minus

1

2119901119879

nabla2

119866119896(119906) 119901)

(35)

where 119866119894(119906) = 119891

119894(119906) + 119910

119879

1198691

ℎ1198691

(119906) + 119906119879

119861119894119908 minus ]119894119892119894(119906) minus

119906119879

119862119894119911 119894 = 1 2 119896 ]

119894are fixed parameters

6 International Scholarly Research Notices

(iii) 119872119872119865119863120582 Maximize 120582119866(119906) 120582 is 119896-dimensionalstrictly positive vector

all subject to same constraints

119896

sum

119894=1

120582119894[nabla119866119894(119906) + nabla

2

119866119894(119906) 119901] + 119910

119879

1198692

[nablaℎ1198692(119906) + nabla

2

nablaℎ1198692(119906) 119901]

= 0

(36)

119891119894(119906) + 119910

119879

1198691

ℎ1198691(119906) + 119906

119879

119861119894119908 minus V119894119892119894(119906) minus 119906

119879

119862119894119911 ge 0

for 119894 = 1 2 119896(37)

119910119879

1198692

ℎ1198692(119906) minus

1

2119901119879

nabla2

(119910119879

1198692

ℎ1198692(119906)) 119901 ge 0

1199101198692

isin R119898minus|1198691|

(38)

119908119879

119861119894119908 le 1 119911

119879

119862119894119911 le 1 119894 = 1 2 119896 (39)

119910 ge 0 V119894ge 0 119894 = 1 2 119896 (40)

where119891119894 119883 rarr R119892

119894 119883 rarr R ℎ

119895 119883 rarr R 119894 = 1 2 119896

119895 = 1 2 119898 are differentiable functions119908 119911 isin R119899119901 isin R119899119861119894 and 119862

119894 119894 = 1 2 119896 are positive semidefinite matrices of

order 119899For the following theorems we assume that 120578 119883 times119883 rarr

R119899 119870 119883 times 119883 rarr R+ and 120595

0 1205951 R rarr R satisfying

1205950(119886) ge 0 rArr 119886 ge 0 and 119887 le 0 rArr 120595

1(119887) le 0 and 120588 120590 isin R

Theorem 13 (weak duality) Let 119909 be a feasible solution for theprimalMFP120582 and let (119906 119910 V 119908) be feasible for dual SMMFD120582If

(i) sum119896119894=1120582119894119866119894(sdot) is second order 120588-pseudounivex with

respect to 120578 1205950 119870 and for 119910

1198692

isin R119898minus|1198691| 1199101198791198692

ℎ1198692

(sdot) issecond order 120590-quasiunivex with respect to 120578 120595

1 and

119870 along with

(ii) 120588 + 120590 ge 0 then Inf(120582119865(119909)) ge Sup(120582119866(119906))

Proof Now from the primal and dual constraints we have

ℎ (119909) le 0

119910119879

1198692

ℎ1198692(119906) minus

1

2119901119879

nabla2

(119910119879

1198692

ℎ1198692(119906)) 119901 ge 0

(41)

So

119910119879

1198692

ℎ1198692(119909) minus 119910

119879

1198692

ℎ1198692(119906) +

1

2119901119879

nabla2

(119910119879

1198692

ℎ1198692(119906)) 119901 le 0

997904rArr 119870 (119909 119906) 1205951[119910119879

1198692

ℎ1198692(119909) minus 119910

119879

1198692

ℎ1198692(119906)

+1

2119901119879

nabla2

(119910119879

1198692

ℎ1198692(119906)) 119901] le 0

(42)

Since 1199101198791198692

ℎ1198692

is second order 120590-quasiunivex with respect to120578 1205951 and119870 and in view of (42) for 119909 119906 isin R119899 we have

120578(119909 119906)119879

nabla [119910119879

1198692

ℎ1198692(119906)] + nabla

2

[119910119879

1198692

ℎ1198692(119906)] 119901 + 120590119909 minus 119906

2

le 0

(43)

Again from the dual constraint (36) we have119896

sum

119894=1

120582119894[nabla119866119894(119906) + nabla

2

119866119894(119906) 119901]

+ 119910119879

1198692

119879

[nablaℎ1198692(119906) + nabla

2

ℎ1198692(119906) 119901] = 0

(44)

Since 120578(119909 119906) isin R119899 we have

120578(119909 119906)119879

119896

sum

119894=1

120582119894[nabla119866119894(119906) + nabla

2

119866119894(119906) 119901]

+ 120578(119909 119906)119879

119910119879

1198692

[nablaℎ1198692(119906) + nabla

2

ℎ1198692(119906) 119901] = 0

997904rArr 120578(119909 119906)119879

119896

sum

119894=1

120582119894[nabla119866119894(119906) + nabla

2

119866119894(119906) 119901]

+ 120578(119909 119906)119879

119910119879

1198692

[nablaℎ1198692(119906) + nabla

2

ℎ1198692(119906) 119901]

+ 120590119909 minus 1199062

minus 120590119909 minus 1199062

= 0

(45)

Using (43) in above equation we get 120578(119909 119906)119879sum119896119894=1120582119894

[nabla119866119894(119906) + nabla

2

119866119894(119906)119901] minus 120590119909 minus 119906

2

ge 0Since 120588 + 120590 ge 0 we get 120588119909 minus 1199062 ge minus120590119909 minus 1199062So we have

120578(119909 119906)119879

119896

sum

119894=1

120582119894[nabla119866119894(119906) + nabla

2

119866119894(119906) 119901] + 120588119909 minus 119906

2

ge 0

(46)

Since sum119896

119894=1120582119894119866119894(119906) is second order 120588-pseudounivex

with respect to 120578 1205950 and 119870 by Definition 2 and

(46) we get 119870(119909 119906)1205950sum119896

119894=1120582119894119866119894(119909) minus sum

119896

119894=1120582119894119866119894(119906) +

(12)119901119879

sum119896

119894=1120582119894119866119894(119906)119901 ge 0

Using the property of 1205950and119870 we get

119896

sum

119894=1

120582119894119866119894(119909) minus

119896

sum

119894=1

120582119894119866119894(119906) +

1

2119901119879

119896

sum

119894=1

120582119894119866119894(119906) 119901 ge 0

997904rArr

119896

sum

119894=1

120582119894119866119894(119909) ge

119896

sum

119894=1

120582119894119866119894(119906)

997904rArr

119896

sum

119894=1

120582119894[119891119894(119909) + 119910

119879

1198691

ℎ1198691(119909)

+119909119879

119861119894119908 minus V119894119892119894(119909) minus 119909

119879

119862119894119911]

ge

119896

sum

119894=1

120582119894[119891119894(119906) + 119910

119879

1198691

ℎ1198691(119906)

+119906119879

119861119894119908 minus V119894119892119894(119906) minus 119909

119879

119862119894119911]

(47)

Equation (5) gives ℎ(119909) le 0 rArr 1199101198791198691

ℎ1198691

(119909) le 0 for 1199101198691

ge 0

International Scholarly Research Notices 7

So (47) implies that

119896

sum

119894=1

120582119894[119891119894(119909) + 119909

119879

119861119894119908 minus V119894119892119894(119909) minus 119909

119879

119862119894119911]

ge

119896

sum

119894=1

120582119894[119891119894(119906) + 119910

119879

1198691

ℎ1198691(119906) + 119906

119879

119861119894119908

minusV119894119892119894(119906) minus 119906

119879

119862119894119911]

(48)

Now by Schwarz Inequality and (39) we have

119909119879

119861119894119908 le (119909

119879

119861119894119909)12

(119908119879

119861119894119908)12

le (119909119879

119861119894119909)12

119909119879

119862119894119911 le (119909

119879

119862119894119909)12

(119911119879

119862119894119911)12

le (119909119879

119862119894119909)12

119894 = 1 2 119896

(49)

So both (48) and (49) imply that

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

]

ge

119896

sum

119894=1

120582119894[119891119894(119906) + 119910

119879

1198691

ℎ1198691(119906) + 119906

119879

119861119894119908

minus V119894119892119894(119906) minus 119906

119879

119862119894119911]

997904rArr Inf (120582119865 (119909)) ge Sup (120582119865 (119906))

(50)

Theorem 14 (strong duality) Let 119909 be optimal solution forMFP120582 and let 119882(119909) = 120601 Then there exist 120582

119894isin R+ 119908 119911 isin

R119899 V119894isin R+ and 119910 isin R119898 such that (119906 120582 119910 V 119908 119911 119901 =

0) is a feasible solution for dual and the objective values ofboth primal and dual are equal to zero Furthermore if (i)(119906 120582 119910 V 119908 119911) is feasible for dual (ii) sum119896

119894=1120582119894119866119894(sdot) is second

order 120588-pseudounivex with respect to 120578 1205950 and 119870 and for

1199101198692

isin R119898minus|1198691| 1199101198791198691

ℎ1198692

(sdot) is second order 120590-quasiunivex withrespect to 120578 120595

1 and 119870 along with (iii) 120588 + 120590 ge 0 then

(119906 120582 119910 V 119908 119911 119901 = 0) is properly efficient for SMMFD0

Proof Since 119909 is optimal solution for (MFP120582) by Lemma 10there exist 120582

119894isin R+119908 119911 isin R119899 V

119894isin R+ and 119910 isin R119898 such that

119896

sum

119894=1

120582119894[nabla119891119894(119909) + 119861

119894119908 minus V119894nabla119892119894(119909) minus 119862

119894119911] + 119910

119879

nablaℎ (119909) = 0

119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894(119892119894(119909) minus (119909

119879

119862119894119909)12

) = 0

119894 = 1 2 119896

119910119879

ℎ (119909) = 0

119908119879

119861119894119908 le 1 119911

119879

119862119894119911 le 1 119894 = 1 2 119896

(119909119879

119861119894119909)12

= 119909119879

119861119894119908 (119909

119879

119862119894119909)12

= 119909119879

119862119894119911

119894 = 1 2 119896

119910 ge 0

V119894ge 0 119894 = 1 2 119896

(51)

which can be written as119896

sum

119894=1

120582119894[nabla119891119894(119909) + 119910

119879

1198691

nablaℎ1198691(119909) + 119861

119894119908 minus V119894nabla119892119894(119909) minus 119862

119894119911]

+ 119910119879

1198691

nablaℎ1198692(119909) = 0

119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

+ 119910119879

1198691

ℎ1198691(119909) = 0

119910119879

1198691

ℎ1198692(119909) = 0

119908119879

119861119894119908 le 1 119911

119879

119862119894119911 le 1 119894 = 1 2 119896

(119909119879

119861119894119909)12

= 119909119879

119861119894119908 (119909

119879

119862119894119909)12

= 119909119879

119862119894119911

119894 = 1 2 119896

119910 ge 0

V119894ge 0 119894 = 1 2 119896

(52)

These are nothing but the dual constraintsSo (119906 120582 119910 V 119908 119911 119901 = 0) is feasible solution for dual

problemAnd the objective values of MFP120582 and SMMFD120582 are

equal to zero It follows fromTheorem 13 and for any feasiblesolution (119906 120582 119910 V 119908 119911 119901 = 0) of dual that 120582119866(119906) le 120582119866(119909)

So (119906 120582 119910 V 119908 119911 119901 = 0) is optimal solution ofSMMFD120582 Then applying Lemmas 8 and 9 we concludethat (119906 120582 119910 V 119908 119911 119901 = 0) is properly efficient for(SMMFD0)

4 Special Case

If 119901 = 0 119862119894= 0 119894 = 1 2 119896 then our dual programming

reduces to the dual programming proposed by Tripathy [14]

5 Conclusion

In this paper three approaches given by Dinklebaeh [28] andJagannathan [12] for both primal and second order mixedtype dual of a nondifferentiable multiobjective fractionalprogramming problem are introduced and the necessaryand sufficient optimality conditions are established and aparameterization technique is used to establish duality results

8 International Scholarly Research Notices

under generalized second order 120588-univexity assumptionTheresults developed in this paper can be further extendedto higher order mixed type fractional problem containingsquare root term Also the present work can be furtherextended to a class of nondifferentiable minimax mixedfractional programming problems

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] C R Bector ldquoDuality in nonlinear fractional programmingrdquoZeitschrift fur Operations Research vol 17 no 5 pp 183ndash1931973

[2] C R Bector and S Chandra ldquoFirst and Second order duality fora class of non differentiable fractional programmingrdquo Journal ofInformation amp Optimization Sciences vol 7 pp 335ndash348 1986

[3] B D Craven ldquoFractional programmingrdquo in Sig Ma Series inApplied Mathematics vol 4 pp 145ndash152 Heldermann BerlinGermany 1998

[4] B Mond and T Weir ldquoDuality for fractional programmingwith generalized convexity conditionrdquo Journal of Information ampOptimization Sciences vol 3 pp 105ndash124 1982

[5] I M Stancu-Minasian Fractional Programming TheoryMethod and Application Kluwer Academic PublishersDordrecht The Netherland 1997

[6] S Schaible and T Ibaraki ldquoFractional programmingrdquo EuropeanJournal of Operational Research vol 12 no 4 pp 325ndash338 1983

[7] I Ahmad S K Gupta N Kailey and R P Agarwal ldquoDualityin nondifferentiable minimax fractional programming with B-(pr)-invexityrdquo Journal of Inequalities and Applications vol 2011article 75 15 pages 2011

[8] I Ahmad and S Sharma ldquoSecond-order duality for nondif-ferentiable multiobjective programming problemsrdquo NumericalFunctional Analysis and Optimization vol 28 no 9-10 pp 975ndash988 2007

[9] T R Gulati I Ahmad and D Agarwal ldquoSufficiency andduality in multiobjective programming under generalized typei functionsrdquo Journal of Optimization Theory and Applicationsvol 135 no 3 pp 411ndash427 2007

[10] I Ahmad Z Husain and S Al-Homidan ldquoSecond-order dual-ity in nondifferentiable fractional programmingrdquo NonlinearAnalysis Real World Applications vol 12 no 2 pp 1103ndash11102011

[11] C R Bector S Chandra and I Husain ldquoOptimality conditionsand duality in subdifferentiable multiobjective fractional pro-grammingrdquo Journal of Optimization Theory and Applicationsvol 79 no 1 pp 105ndash125 1993

[12] R Jagannathan ldquoDuality for nonlinear fractional programsrdquoZeitschrift fur Operations Research vol 17 no 1 pp 1ndash3 1973

[13] R Osuna-Gomez A Ruız-Canales and P Rufian-Lizana ldquoMul-tiobjective fractional programmingwith generalized convexityrdquoTrabajos de Investigacion Operativa vol 8 no 1 pp 97ndash1102000

[14] A K Tripathy ldquoMixed type duality in multiobjective fractionalprogramming under generalized 120588-univex functionrdquo Journal ofMathematical Modelling and Algorithms in Operations Research2013

[15] S K Mishra S Wang and K K Lai ldquoRole of 120572-pseudo-univex functions in vector variational-like inequality problemsrdquoJournal of Systems Science and Complexity vol 20 no 3 pp344ndash349 2007

[16] S Schaible ldquoFractional programmingrdquo in Handbook of GlobalOptimization R Horst and P M Pardalos Eds pp 495ndash608 KluwerAcademic Publishers DordrechtTheNetherlands1995

[17] Z A Liang H X Huang and P M Pardalos ldquoOptimalityconditions and duality for a class of nonlinear fractionalprogramming problemsrdquo Journal of Optimization Theory andApplications vol 110 no 3 pp 611ndash619 2001

[18] Z-A Liang H-X Huang and P M Pardalos ldquoEfficiencyconditions and duality for a class of multiobjective fractionalprogramming problemsrdquo Journal of Global Optimization vol27 no 4 pp 447ndash471 2003

[19] L B Santos R Osuna-Gomez and A Rojas-Medar ldquoNons-moothmultiobjective fractional programmingwith generalizedconvexityrdquo Revista Integracion vol 26 no 1 pp 1ndash12 2008

[20] C R Bector S Chandra and A Abha ldquoOn mixed duality inmathematical programmingrdquo Journal of Mathematical Analysisand Applications vol 259 no 1 pp 346ndash356 2001

[21] Z Xu ldquoMixed type duality in multiobjective programmingproblemsrdquo Journal of Mathematical Analysis and Applicationsvol 198 no 3 pp 621ndash635 1996

[22] H Zhou and Y Wang ldquoOptimality and mixed duality for non-smooth multi-objective fractional programmingrdquo Pure Mathe-matics and Applications vol 14 no 3 pp 263ndash274 2003

[23] I Ahmad andZHusain ldquoDuality in nondifferentiableminimaxfractional programming with generalized convexityrdquo AppliedMathematics and Computation vol 176 no 2 pp 545ndash5512006

[24] I Ahmad and Z Husain ldquoOptimality conditions and dualityin nondifferentiable minimax fractional programming withgeneralized convexityrdquo Journal of Optimization Theory andApplications vol 129 no 2 pp 255ndash275 2006

[25] B Mond ldquoA class of nondifferentiable fractional program-ming problemrdquo Zeitschrift fur Angewandte Mathematik undMechanik vol 58 no 8 pp 337ndash341 1978

[26] D S Kim S J Kim and M H Kim ldquoOptimality andduality for a class of nondifferentiable multiobjective fractionalprogramming problemsrdquo Journal of Optimization Theory andApplications vol 129 no 1 pp 131ndash146 2006

[27] D S Kim Y J Lee and K D Bae ldquoDuality in nondifferentiablemultiobjective fractional programs involving conesrdquo TaiwaneseJournal of Mathematics vol 13 no 6 A pp 1811ndash1821 2009

[28] W Dinklebaeh ldquoOn nonlinear fractional programmingrdquoMan-agement Science vol 13 no 7 pp 492ndash498 1967

[29] M A Hanson ldquoOn sufficiency of the Kuhn-Tucker conditionsrdquoJournal of Mathematical Analysis and Applications vol 80 no2 pp 545ndash550 1981

[30] C R Bector S K Suneja and S Gupta ldquoUnivex functionsand univex nonlinear programmingrdquo in Proceeding of theAdministrative Science Association of Canada pp 115ndash124 1992

[31] S K Mishra ldquoOn multiple-objective optimization with gener-alized univexityrdquo Journal of Mathematical Analysis and Applica-tions vol 224 no 1 pp 131ndash148 1998

[32] A Jayswal ldquoNon-differentiable minimax fractional program-ming with generalized 120572-univexityrdquo Journal of Computationaland Applied Mathematics vol 214 no 1 pp 121ndash135 2008

[33] A M Geoffrion ldquoProper efficiency and the theory of vectormaximizationrdquo Journal of Mathematical Analysis and Applica-tions vol 22 no 3 pp 618ndash630 1968

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 5: Research Article Second Order Duality in Multiobjective Fractional Programming …downloads.hindawi.com/archive/2014/541524.pdf · 2019-07-31 · Research Article Second Order Duality

International Scholarly Research Notices 5

From the relations (5) (11) and (14) we get119898

sum

119895=1

119910119895ℎ119895(119909)

le

119898

sum

119895=1

119910119895ℎ119895(119909) 997904rArr 119867(119909) le 119867 (119909) 997904rArr 119867(119909) minus 119867 (119909) le 0

(23)

Also from hypothesis (i) we get (nabla2119867(119909))119901 = 0So we have the following

119867(119909) minus 119867 (119909) +1

2119901119879

(nabla2

119867(119909)) 119901 le 0

997904rArr 119870 (119909 119909) 1205951119867 (119909) minus 119867 (119909) +

1

2119901119879

(nabla2

119867(119909)) 119901 le 0

(24)

Hence the 120590-quasiunivexity of 119867(119909) with respect to 120578 1205951

and119870 implies the following

120578(119909 119909)119879

nabla119867 (119909) + (nabla2

119867(119909)) 119901 + 120590119909 minus 1199092

le 0

997904rArr 120578(119909 119909)119879

nabla119867 (119909) + 120590119909 minus 1199092

le 0

(25)

From (9) we get

119896

sum

119894=1

120582119894[nabla119891119894(119909) + 119861

119894119908 minus V119894nabla119892119894(119909) minus 119862

119894119911]

+

119898

sum

119894=1

119910119894nablaℎ119894(119909) = 0

997904rArr nabla119876 (119909) + nabla119867 (119909) = 0

997904rArr 120578(119909 119909)119879

[nabla119876 (119909) + nabla119867 (119909)] = 0

997904rArr 120578(119909 119909)119879

nabla119876 (119909) + 120578(119909 119909)119879

nabla119867 (119909)

+ 120590119909 minus 1199092

minus 120590119909 minus 1199092

= 0

(26)

Using (25) in (26) we get

120578(119909 119909)119879

nabla119876 (119909) minus 120590119909 minus 1199092

ge 0 (27)

Since 120588 + 120590 ge 0 we get

120588119909 minus 1199092

ge minus120590119909 minus 1199092

(28)

So we have

120578(119909 119909)119879

nabla119876 (119909) + 120588119909 minus 1199092

ge 0

997904rArr 120578(119909 119909)119879

nabla119876 (119909) + nabla2

119876 (119909) 119901 + 120588119909 minus 1199092

ge 0

(29)

Since 119876(119909) is 120588-pseudounivex with respect to 120578 1205950 and 119870

we obtained

119870 (119909 119909) 1205950[119876 (119909) minus 119876 (119909) +

1

2119901119879

nabla2

119876 (119909) 119901] ge 0 (30)

Using the property of119870 and 1205950 we get

119876 (119909) minus 119876 (119909) +1

2119901119879

nabla2

119876 (119909) 119901 ge 0 997904rArr 119876 (119909) ge 119876 (119909)

997904rArr

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minusV119894119892119894(119909) minus (119909

119879

119862119894119909)12

]

ge

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

]

(31)

If 119909 were not an efficient solution to MFP1 then from (20)we have

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

]

lt

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minusV119894119892119894(119909) minus (119909

119879

119862119894119909)12

]

(32)

This contradicts (31)Therefore 119909 is an efficient solution for MFP1

3 Second Order Mixed Type MultiobjectiveFractional Duality

(i) MMFD0 Maximize

119871 (119906)

= (1198711(119906) minus

1

2119901119879

nabla2

1198711(119906) 119901 119871

2(119906)

minus1

2119901119879

nabla2

1198712(119906) 119901 119871

119896(119906) minus

1

2119901119879

nabla2

119871119896(119906) 119901)

(33)

where

119871119894(119906) =

119891119894(119906) + 119910

119879

1198691

ℎ1198691(119906) + 119906

119879

119861119894119908

119892119894(119906) minus 119906119879119862

119894119911

119894 = 1 2 119896 (34)

(ii) MMFD1 Maximize

119866 (119906)

= (1198661(119906) minus

1

2119901119879

nabla2

1198661(119906) 119901 119866

2(119906)

minus1

2119901119879

nabla2

1198662(119906) 119901 119866

119896(119906) minus

1

2119901119879

nabla2

119866119896(119906) 119901)

(35)

where 119866119894(119906) = 119891

119894(119906) + 119910

119879

1198691

ℎ1198691

(119906) + 119906119879

119861119894119908 minus ]119894119892119894(119906) minus

119906119879

119862119894119911 119894 = 1 2 119896 ]

119894are fixed parameters

6 International Scholarly Research Notices

(iii) 119872119872119865119863120582 Maximize 120582119866(119906) 120582 is 119896-dimensionalstrictly positive vector

all subject to same constraints

119896

sum

119894=1

120582119894[nabla119866119894(119906) + nabla

2

119866119894(119906) 119901] + 119910

119879

1198692

[nablaℎ1198692(119906) + nabla

2

nablaℎ1198692(119906) 119901]

= 0

(36)

119891119894(119906) + 119910

119879

1198691

ℎ1198691(119906) + 119906

119879

119861119894119908 minus V119894119892119894(119906) minus 119906

119879

119862119894119911 ge 0

for 119894 = 1 2 119896(37)

119910119879

1198692

ℎ1198692(119906) minus

1

2119901119879

nabla2

(119910119879

1198692

ℎ1198692(119906)) 119901 ge 0

1199101198692

isin R119898minus|1198691|

(38)

119908119879

119861119894119908 le 1 119911

119879

119862119894119911 le 1 119894 = 1 2 119896 (39)

119910 ge 0 V119894ge 0 119894 = 1 2 119896 (40)

where119891119894 119883 rarr R119892

119894 119883 rarr R ℎ

119895 119883 rarr R 119894 = 1 2 119896

119895 = 1 2 119898 are differentiable functions119908 119911 isin R119899119901 isin R119899119861119894 and 119862

119894 119894 = 1 2 119896 are positive semidefinite matrices of

order 119899For the following theorems we assume that 120578 119883 times119883 rarr

R119899 119870 119883 times 119883 rarr R+ and 120595

0 1205951 R rarr R satisfying

1205950(119886) ge 0 rArr 119886 ge 0 and 119887 le 0 rArr 120595

1(119887) le 0 and 120588 120590 isin R

Theorem 13 (weak duality) Let 119909 be a feasible solution for theprimalMFP120582 and let (119906 119910 V 119908) be feasible for dual SMMFD120582If

(i) sum119896119894=1120582119894119866119894(sdot) is second order 120588-pseudounivex with

respect to 120578 1205950 119870 and for 119910

1198692

isin R119898minus|1198691| 1199101198791198692

ℎ1198692

(sdot) issecond order 120590-quasiunivex with respect to 120578 120595

1 and

119870 along with

(ii) 120588 + 120590 ge 0 then Inf(120582119865(119909)) ge Sup(120582119866(119906))

Proof Now from the primal and dual constraints we have

ℎ (119909) le 0

119910119879

1198692

ℎ1198692(119906) minus

1

2119901119879

nabla2

(119910119879

1198692

ℎ1198692(119906)) 119901 ge 0

(41)

So

119910119879

1198692

ℎ1198692(119909) minus 119910

119879

1198692

ℎ1198692(119906) +

1

2119901119879

nabla2

(119910119879

1198692

ℎ1198692(119906)) 119901 le 0

997904rArr 119870 (119909 119906) 1205951[119910119879

1198692

ℎ1198692(119909) minus 119910

119879

1198692

ℎ1198692(119906)

+1

2119901119879

nabla2

(119910119879

1198692

ℎ1198692(119906)) 119901] le 0

(42)

Since 1199101198791198692

ℎ1198692

is second order 120590-quasiunivex with respect to120578 1205951 and119870 and in view of (42) for 119909 119906 isin R119899 we have

120578(119909 119906)119879

nabla [119910119879

1198692

ℎ1198692(119906)] + nabla

2

[119910119879

1198692

ℎ1198692(119906)] 119901 + 120590119909 minus 119906

2

le 0

(43)

Again from the dual constraint (36) we have119896

sum

119894=1

120582119894[nabla119866119894(119906) + nabla

2

119866119894(119906) 119901]

+ 119910119879

1198692

119879

[nablaℎ1198692(119906) + nabla

2

ℎ1198692(119906) 119901] = 0

(44)

Since 120578(119909 119906) isin R119899 we have

120578(119909 119906)119879

119896

sum

119894=1

120582119894[nabla119866119894(119906) + nabla

2

119866119894(119906) 119901]

+ 120578(119909 119906)119879

119910119879

1198692

[nablaℎ1198692(119906) + nabla

2

ℎ1198692(119906) 119901] = 0

997904rArr 120578(119909 119906)119879

119896

sum

119894=1

120582119894[nabla119866119894(119906) + nabla

2

119866119894(119906) 119901]

+ 120578(119909 119906)119879

119910119879

1198692

[nablaℎ1198692(119906) + nabla

2

ℎ1198692(119906) 119901]

+ 120590119909 minus 1199062

minus 120590119909 minus 1199062

= 0

(45)

Using (43) in above equation we get 120578(119909 119906)119879sum119896119894=1120582119894

[nabla119866119894(119906) + nabla

2

119866119894(119906)119901] minus 120590119909 minus 119906

2

ge 0Since 120588 + 120590 ge 0 we get 120588119909 minus 1199062 ge minus120590119909 minus 1199062So we have

120578(119909 119906)119879

119896

sum

119894=1

120582119894[nabla119866119894(119906) + nabla

2

119866119894(119906) 119901] + 120588119909 minus 119906

2

ge 0

(46)

Since sum119896

119894=1120582119894119866119894(119906) is second order 120588-pseudounivex

with respect to 120578 1205950 and 119870 by Definition 2 and

(46) we get 119870(119909 119906)1205950sum119896

119894=1120582119894119866119894(119909) minus sum

119896

119894=1120582119894119866119894(119906) +

(12)119901119879

sum119896

119894=1120582119894119866119894(119906)119901 ge 0

Using the property of 1205950and119870 we get

119896

sum

119894=1

120582119894119866119894(119909) minus

119896

sum

119894=1

120582119894119866119894(119906) +

1

2119901119879

119896

sum

119894=1

120582119894119866119894(119906) 119901 ge 0

997904rArr

119896

sum

119894=1

120582119894119866119894(119909) ge

119896

sum

119894=1

120582119894119866119894(119906)

997904rArr

119896

sum

119894=1

120582119894[119891119894(119909) + 119910

119879

1198691

ℎ1198691(119909)

+119909119879

119861119894119908 minus V119894119892119894(119909) minus 119909

119879

119862119894119911]

ge

119896

sum

119894=1

120582119894[119891119894(119906) + 119910

119879

1198691

ℎ1198691(119906)

+119906119879

119861119894119908 minus V119894119892119894(119906) minus 119909

119879

119862119894119911]

(47)

Equation (5) gives ℎ(119909) le 0 rArr 1199101198791198691

ℎ1198691

(119909) le 0 for 1199101198691

ge 0

International Scholarly Research Notices 7

So (47) implies that

119896

sum

119894=1

120582119894[119891119894(119909) + 119909

119879

119861119894119908 minus V119894119892119894(119909) minus 119909

119879

119862119894119911]

ge

119896

sum

119894=1

120582119894[119891119894(119906) + 119910

119879

1198691

ℎ1198691(119906) + 119906

119879

119861119894119908

minusV119894119892119894(119906) minus 119906

119879

119862119894119911]

(48)

Now by Schwarz Inequality and (39) we have

119909119879

119861119894119908 le (119909

119879

119861119894119909)12

(119908119879

119861119894119908)12

le (119909119879

119861119894119909)12

119909119879

119862119894119911 le (119909

119879

119862119894119909)12

(119911119879

119862119894119911)12

le (119909119879

119862119894119909)12

119894 = 1 2 119896

(49)

So both (48) and (49) imply that

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

]

ge

119896

sum

119894=1

120582119894[119891119894(119906) + 119910

119879

1198691

ℎ1198691(119906) + 119906

119879

119861119894119908

minus V119894119892119894(119906) minus 119906

119879

119862119894119911]

997904rArr Inf (120582119865 (119909)) ge Sup (120582119865 (119906))

(50)

Theorem 14 (strong duality) Let 119909 be optimal solution forMFP120582 and let 119882(119909) = 120601 Then there exist 120582

119894isin R+ 119908 119911 isin

R119899 V119894isin R+ and 119910 isin R119898 such that (119906 120582 119910 V 119908 119911 119901 =

0) is a feasible solution for dual and the objective values ofboth primal and dual are equal to zero Furthermore if (i)(119906 120582 119910 V 119908 119911) is feasible for dual (ii) sum119896

119894=1120582119894119866119894(sdot) is second

order 120588-pseudounivex with respect to 120578 1205950 and 119870 and for

1199101198692

isin R119898minus|1198691| 1199101198791198691

ℎ1198692

(sdot) is second order 120590-quasiunivex withrespect to 120578 120595

1 and 119870 along with (iii) 120588 + 120590 ge 0 then

(119906 120582 119910 V 119908 119911 119901 = 0) is properly efficient for SMMFD0

Proof Since 119909 is optimal solution for (MFP120582) by Lemma 10there exist 120582

119894isin R+119908 119911 isin R119899 V

119894isin R+ and 119910 isin R119898 such that

119896

sum

119894=1

120582119894[nabla119891119894(119909) + 119861

119894119908 minus V119894nabla119892119894(119909) minus 119862

119894119911] + 119910

119879

nablaℎ (119909) = 0

119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894(119892119894(119909) minus (119909

119879

119862119894119909)12

) = 0

119894 = 1 2 119896

119910119879

ℎ (119909) = 0

119908119879

119861119894119908 le 1 119911

119879

119862119894119911 le 1 119894 = 1 2 119896

(119909119879

119861119894119909)12

= 119909119879

119861119894119908 (119909

119879

119862119894119909)12

= 119909119879

119862119894119911

119894 = 1 2 119896

119910 ge 0

V119894ge 0 119894 = 1 2 119896

(51)

which can be written as119896

sum

119894=1

120582119894[nabla119891119894(119909) + 119910

119879

1198691

nablaℎ1198691(119909) + 119861

119894119908 minus V119894nabla119892119894(119909) minus 119862

119894119911]

+ 119910119879

1198691

nablaℎ1198692(119909) = 0

119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

+ 119910119879

1198691

ℎ1198691(119909) = 0

119910119879

1198691

ℎ1198692(119909) = 0

119908119879

119861119894119908 le 1 119911

119879

119862119894119911 le 1 119894 = 1 2 119896

(119909119879

119861119894119909)12

= 119909119879

119861119894119908 (119909

119879

119862119894119909)12

= 119909119879

119862119894119911

119894 = 1 2 119896

119910 ge 0

V119894ge 0 119894 = 1 2 119896

(52)

These are nothing but the dual constraintsSo (119906 120582 119910 V 119908 119911 119901 = 0) is feasible solution for dual

problemAnd the objective values of MFP120582 and SMMFD120582 are

equal to zero It follows fromTheorem 13 and for any feasiblesolution (119906 120582 119910 V 119908 119911 119901 = 0) of dual that 120582119866(119906) le 120582119866(119909)

So (119906 120582 119910 V 119908 119911 119901 = 0) is optimal solution ofSMMFD120582 Then applying Lemmas 8 and 9 we concludethat (119906 120582 119910 V 119908 119911 119901 = 0) is properly efficient for(SMMFD0)

4 Special Case

If 119901 = 0 119862119894= 0 119894 = 1 2 119896 then our dual programming

reduces to the dual programming proposed by Tripathy [14]

5 Conclusion

In this paper three approaches given by Dinklebaeh [28] andJagannathan [12] for both primal and second order mixedtype dual of a nondifferentiable multiobjective fractionalprogramming problem are introduced and the necessaryand sufficient optimality conditions are established and aparameterization technique is used to establish duality results

8 International Scholarly Research Notices

under generalized second order 120588-univexity assumptionTheresults developed in this paper can be further extendedto higher order mixed type fractional problem containingsquare root term Also the present work can be furtherextended to a class of nondifferentiable minimax mixedfractional programming problems

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] C R Bector ldquoDuality in nonlinear fractional programmingrdquoZeitschrift fur Operations Research vol 17 no 5 pp 183ndash1931973

[2] C R Bector and S Chandra ldquoFirst and Second order duality fora class of non differentiable fractional programmingrdquo Journal ofInformation amp Optimization Sciences vol 7 pp 335ndash348 1986

[3] B D Craven ldquoFractional programmingrdquo in Sig Ma Series inApplied Mathematics vol 4 pp 145ndash152 Heldermann BerlinGermany 1998

[4] B Mond and T Weir ldquoDuality for fractional programmingwith generalized convexity conditionrdquo Journal of Information ampOptimization Sciences vol 3 pp 105ndash124 1982

[5] I M Stancu-Minasian Fractional Programming TheoryMethod and Application Kluwer Academic PublishersDordrecht The Netherland 1997

[6] S Schaible and T Ibaraki ldquoFractional programmingrdquo EuropeanJournal of Operational Research vol 12 no 4 pp 325ndash338 1983

[7] I Ahmad S K Gupta N Kailey and R P Agarwal ldquoDualityin nondifferentiable minimax fractional programming with B-(pr)-invexityrdquo Journal of Inequalities and Applications vol 2011article 75 15 pages 2011

[8] I Ahmad and S Sharma ldquoSecond-order duality for nondif-ferentiable multiobjective programming problemsrdquo NumericalFunctional Analysis and Optimization vol 28 no 9-10 pp 975ndash988 2007

[9] T R Gulati I Ahmad and D Agarwal ldquoSufficiency andduality in multiobjective programming under generalized typei functionsrdquo Journal of Optimization Theory and Applicationsvol 135 no 3 pp 411ndash427 2007

[10] I Ahmad Z Husain and S Al-Homidan ldquoSecond-order dual-ity in nondifferentiable fractional programmingrdquo NonlinearAnalysis Real World Applications vol 12 no 2 pp 1103ndash11102011

[11] C R Bector S Chandra and I Husain ldquoOptimality conditionsand duality in subdifferentiable multiobjective fractional pro-grammingrdquo Journal of Optimization Theory and Applicationsvol 79 no 1 pp 105ndash125 1993

[12] R Jagannathan ldquoDuality for nonlinear fractional programsrdquoZeitschrift fur Operations Research vol 17 no 1 pp 1ndash3 1973

[13] R Osuna-Gomez A Ruız-Canales and P Rufian-Lizana ldquoMul-tiobjective fractional programmingwith generalized convexityrdquoTrabajos de Investigacion Operativa vol 8 no 1 pp 97ndash1102000

[14] A K Tripathy ldquoMixed type duality in multiobjective fractionalprogramming under generalized 120588-univex functionrdquo Journal ofMathematical Modelling and Algorithms in Operations Research2013

[15] S K Mishra S Wang and K K Lai ldquoRole of 120572-pseudo-univex functions in vector variational-like inequality problemsrdquoJournal of Systems Science and Complexity vol 20 no 3 pp344ndash349 2007

[16] S Schaible ldquoFractional programmingrdquo in Handbook of GlobalOptimization R Horst and P M Pardalos Eds pp 495ndash608 KluwerAcademic Publishers DordrechtTheNetherlands1995

[17] Z A Liang H X Huang and P M Pardalos ldquoOptimalityconditions and duality for a class of nonlinear fractionalprogramming problemsrdquo Journal of Optimization Theory andApplications vol 110 no 3 pp 611ndash619 2001

[18] Z-A Liang H-X Huang and P M Pardalos ldquoEfficiencyconditions and duality for a class of multiobjective fractionalprogramming problemsrdquo Journal of Global Optimization vol27 no 4 pp 447ndash471 2003

[19] L B Santos R Osuna-Gomez and A Rojas-Medar ldquoNons-moothmultiobjective fractional programmingwith generalizedconvexityrdquo Revista Integracion vol 26 no 1 pp 1ndash12 2008

[20] C R Bector S Chandra and A Abha ldquoOn mixed duality inmathematical programmingrdquo Journal of Mathematical Analysisand Applications vol 259 no 1 pp 346ndash356 2001

[21] Z Xu ldquoMixed type duality in multiobjective programmingproblemsrdquo Journal of Mathematical Analysis and Applicationsvol 198 no 3 pp 621ndash635 1996

[22] H Zhou and Y Wang ldquoOptimality and mixed duality for non-smooth multi-objective fractional programmingrdquo Pure Mathe-matics and Applications vol 14 no 3 pp 263ndash274 2003

[23] I Ahmad andZHusain ldquoDuality in nondifferentiableminimaxfractional programming with generalized convexityrdquo AppliedMathematics and Computation vol 176 no 2 pp 545ndash5512006

[24] I Ahmad and Z Husain ldquoOptimality conditions and dualityin nondifferentiable minimax fractional programming withgeneralized convexityrdquo Journal of Optimization Theory andApplications vol 129 no 2 pp 255ndash275 2006

[25] B Mond ldquoA class of nondifferentiable fractional program-ming problemrdquo Zeitschrift fur Angewandte Mathematik undMechanik vol 58 no 8 pp 337ndash341 1978

[26] D S Kim S J Kim and M H Kim ldquoOptimality andduality for a class of nondifferentiable multiobjective fractionalprogramming problemsrdquo Journal of Optimization Theory andApplications vol 129 no 1 pp 131ndash146 2006

[27] D S Kim Y J Lee and K D Bae ldquoDuality in nondifferentiablemultiobjective fractional programs involving conesrdquo TaiwaneseJournal of Mathematics vol 13 no 6 A pp 1811ndash1821 2009

[28] W Dinklebaeh ldquoOn nonlinear fractional programmingrdquoMan-agement Science vol 13 no 7 pp 492ndash498 1967

[29] M A Hanson ldquoOn sufficiency of the Kuhn-Tucker conditionsrdquoJournal of Mathematical Analysis and Applications vol 80 no2 pp 545ndash550 1981

[30] C R Bector S K Suneja and S Gupta ldquoUnivex functionsand univex nonlinear programmingrdquo in Proceeding of theAdministrative Science Association of Canada pp 115ndash124 1992

[31] S K Mishra ldquoOn multiple-objective optimization with gener-alized univexityrdquo Journal of Mathematical Analysis and Applica-tions vol 224 no 1 pp 131ndash148 1998

[32] A Jayswal ldquoNon-differentiable minimax fractional program-ming with generalized 120572-univexityrdquo Journal of Computationaland Applied Mathematics vol 214 no 1 pp 121ndash135 2008

[33] A M Geoffrion ldquoProper efficiency and the theory of vectormaximizationrdquo Journal of Mathematical Analysis and Applica-tions vol 22 no 3 pp 618ndash630 1968

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 6: Research Article Second Order Duality in Multiobjective Fractional Programming …downloads.hindawi.com/archive/2014/541524.pdf · 2019-07-31 · Research Article Second Order Duality

6 International Scholarly Research Notices

(iii) 119872119872119865119863120582 Maximize 120582119866(119906) 120582 is 119896-dimensionalstrictly positive vector

all subject to same constraints

119896

sum

119894=1

120582119894[nabla119866119894(119906) + nabla

2

119866119894(119906) 119901] + 119910

119879

1198692

[nablaℎ1198692(119906) + nabla

2

nablaℎ1198692(119906) 119901]

= 0

(36)

119891119894(119906) + 119910

119879

1198691

ℎ1198691(119906) + 119906

119879

119861119894119908 minus V119894119892119894(119906) minus 119906

119879

119862119894119911 ge 0

for 119894 = 1 2 119896(37)

119910119879

1198692

ℎ1198692(119906) minus

1

2119901119879

nabla2

(119910119879

1198692

ℎ1198692(119906)) 119901 ge 0

1199101198692

isin R119898minus|1198691|

(38)

119908119879

119861119894119908 le 1 119911

119879

119862119894119911 le 1 119894 = 1 2 119896 (39)

119910 ge 0 V119894ge 0 119894 = 1 2 119896 (40)

where119891119894 119883 rarr R119892

119894 119883 rarr R ℎ

119895 119883 rarr R 119894 = 1 2 119896

119895 = 1 2 119898 are differentiable functions119908 119911 isin R119899119901 isin R119899119861119894 and 119862

119894 119894 = 1 2 119896 are positive semidefinite matrices of

order 119899For the following theorems we assume that 120578 119883 times119883 rarr

R119899 119870 119883 times 119883 rarr R+ and 120595

0 1205951 R rarr R satisfying

1205950(119886) ge 0 rArr 119886 ge 0 and 119887 le 0 rArr 120595

1(119887) le 0 and 120588 120590 isin R

Theorem 13 (weak duality) Let 119909 be a feasible solution for theprimalMFP120582 and let (119906 119910 V 119908) be feasible for dual SMMFD120582If

(i) sum119896119894=1120582119894119866119894(sdot) is second order 120588-pseudounivex with

respect to 120578 1205950 119870 and for 119910

1198692

isin R119898minus|1198691| 1199101198791198692

ℎ1198692

(sdot) issecond order 120590-quasiunivex with respect to 120578 120595

1 and

119870 along with

(ii) 120588 + 120590 ge 0 then Inf(120582119865(119909)) ge Sup(120582119866(119906))

Proof Now from the primal and dual constraints we have

ℎ (119909) le 0

119910119879

1198692

ℎ1198692(119906) minus

1

2119901119879

nabla2

(119910119879

1198692

ℎ1198692(119906)) 119901 ge 0

(41)

So

119910119879

1198692

ℎ1198692(119909) minus 119910

119879

1198692

ℎ1198692(119906) +

1

2119901119879

nabla2

(119910119879

1198692

ℎ1198692(119906)) 119901 le 0

997904rArr 119870 (119909 119906) 1205951[119910119879

1198692

ℎ1198692(119909) minus 119910

119879

1198692

ℎ1198692(119906)

+1

2119901119879

nabla2

(119910119879

1198692

ℎ1198692(119906)) 119901] le 0

(42)

Since 1199101198791198692

ℎ1198692

is second order 120590-quasiunivex with respect to120578 1205951 and119870 and in view of (42) for 119909 119906 isin R119899 we have

120578(119909 119906)119879

nabla [119910119879

1198692

ℎ1198692(119906)] + nabla

2

[119910119879

1198692

ℎ1198692(119906)] 119901 + 120590119909 minus 119906

2

le 0

(43)

Again from the dual constraint (36) we have119896

sum

119894=1

120582119894[nabla119866119894(119906) + nabla

2

119866119894(119906) 119901]

+ 119910119879

1198692

119879

[nablaℎ1198692(119906) + nabla

2

ℎ1198692(119906) 119901] = 0

(44)

Since 120578(119909 119906) isin R119899 we have

120578(119909 119906)119879

119896

sum

119894=1

120582119894[nabla119866119894(119906) + nabla

2

119866119894(119906) 119901]

+ 120578(119909 119906)119879

119910119879

1198692

[nablaℎ1198692(119906) + nabla

2

ℎ1198692(119906) 119901] = 0

997904rArr 120578(119909 119906)119879

119896

sum

119894=1

120582119894[nabla119866119894(119906) + nabla

2

119866119894(119906) 119901]

+ 120578(119909 119906)119879

119910119879

1198692

[nablaℎ1198692(119906) + nabla

2

ℎ1198692(119906) 119901]

+ 120590119909 minus 1199062

minus 120590119909 minus 1199062

= 0

(45)

Using (43) in above equation we get 120578(119909 119906)119879sum119896119894=1120582119894

[nabla119866119894(119906) + nabla

2

119866119894(119906)119901] minus 120590119909 minus 119906

2

ge 0Since 120588 + 120590 ge 0 we get 120588119909 minus 1199062 ge minus120590119909 minus 1199062So we have

120578(119909 119906)119879

119896

sum

119894=1

120582119894[nabla119866119894(119906) + nabla

2

119866119894(119906) 119901] + 120588119909 minus 119906

2

ge 0

(46)

Since sum119896

119894=1120582119894119866119894(119906) is second order 120588-pseudounivex

with respect to 120578 1205950 and 119870 by Definition 2 and

(46) we get 119870(119909 119906)1205950sum119896

119894=1120582119894119866119894(119909) minus sum

119896

119894=1120582119894119866119894(119906) +

(12)119901119879

sum119896

119894=1120582119894119866119894(119906)119901 ge 0

Using the property of 1205950and119870 we get

119896

sum

119894=1

120582119894119866119894(119909) minus

119896

sum

119894=1

120582119894119866119894(119906) +

1

2119901119879

119896

sum

119894=1

120582119894119866119894(119906) 119901 ge 0

997904rArr

119896

sum

119894=1

120582119894119866119894(119909) ge

119896

sum

119894=1

120582119894119866119894(119906)

997904rArr

119896

sum

119894=1

120582119894[119891119894(119909) + 119910

119879

1198691

ℎ1198691(119909)

+119909119879

119861119894119908 minus V119894119892119894(119909) minus 119909

119879

119862119894119911]

ge

119896

sum

119894=1

120582119894[119891119894(119906) + 119910

119879

1198691

ℎ1198691(119906)

+119906119879

119861119894119908 minus V119894119892119894(119906) minus 119909

119879

119862119894119911]

(47)

Equation (5) gives ℎ(119909) le 0 rArr 1199101198791198691

ℎ1198691

(119909) le 0 for 1199101198691

ge 0

International Scholarly Research Notices 7

So (47) implies that

119896

sum

119894=1

120582119894[119891119894(119909) + 119909

119879

119861119894119908 minus V119894119892119894(119909) minus 119909

119879

119862119894119911]

ge

119896

sum

119894=1

120582119894[119891119894(119906) + 119910

119879

1198691

ℎ1198691(119906) + 119906

119879

119861119894119908

minusV119894119892119894(119906) minus 119906

119879

119862119894119911]

(48)

Now by Schwarz Inequality and (39) we have

119909119879

119861119894119908 le (119909

119879

119861119894119909)12

(119908119879

119861119894119908)12

le (119909119879

119861119894119909)12

119909119879

119862119894119911 le (119909

119879

119862119894119909)12

(119911119879

119862119894119911)12

le (119909119879

119862119894119909)12

119894 = 1 2 119896

(49)

So both (48) and (49) imply that

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

]

ge

119896

sum

119894=1

120582119894[119891119894(119906) + 119910

119879

1198691

ℎ1198691(119906) + 119906

119879

119861119894119908

minus V119894119892119894(119906) minus 119906

119879

119862119894119911]

997904rArr Inf (120582119865 (119909)) ge Sup (120582119865 (119906))

(50)

Theorem 14 (strong duality) Let 119909 be optimal solution forMFP120582 and let 119882(119909) = 120601 Then there exist 120582

119894isin R+ 119908 119911 isin

R119899 V119894isin R+ and 119910 isin R119898 such that (119906 120582 119910 V 119908 119911 119901 =

0) is a feasible solution for dual and the objective values ofboth primal and dual are equal to zero Furthermore if (i)(119906 120582 119910 V 119908 119911) is feasible for dual (ii) sum119896

119894=1120582119894119866119894(sdot) is second

order 120588-pseudounivex with respect to 120578 1205950 and 119870 and for

1199101198692

isin R119898minus|1198691| 1199101198791198691

ℎ1198692

(sdot) is second order 120590-quasiunivex withrespect to 120578 120595

1 and 119870 along with (iii) 120588 + 120590 ge 0 then

(119906 120582 119910 V 119908 119911 119901 = 0) is properly efficient for SMMFD0

Proof Since 119909 is optimal solution for (MFP120582) by Lemma 10there exist 120582

119894isin R+119908 119911 isin R119899 V

119894isin R+ and 119910 isin R119898 such that

119896

sum

119894=1

120582119894[nabla119891119894(119909) + 119861

119894119908 minus V119894nabla119892119894(119909) minus 119862

119894119911] + 119910

119879

nablaℎ (119909) = 0

119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894(119892119894(119909) minus (119909

119879

119862119894119909)12

) = 0

119894 = 1 2 119896

119910119879

ℎ (119909) = 0

119908119879

119861119894119908 le 1 119911

119879

119862119894119911 le 1 119894 = 1 2 119896

(119909119879

119861119894119909)12

= 119909119879

119861119894119908 (119909

119879

119862119894119909)12

= 119909119879

119862119894119911

119894 = 1 2 119896

119910 ge 0

V119894ge 0 119894 = 1 2 119896

(51)

which can be written as119896

sum

119894=1

120582119894[nabla119891119894(119909) + 119910

119879

1198691

nablaℎ1198691(119909) + 119861

119894119908 minus V119894nabla119892119894(119909) minus 119862

119894119911]

+ 119910119879

1198691

nablaℎ1198692(119909) = 0

119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

+ 119910119879

1198691

ℎ1198691(119909) = 0

119910119879

1198691

ℎ1198692(119909) = 0

119908119879

119861119894119908 le 1 119911

119879

119862119894119911 le 1 119894 = 1 2 119896

(119909119879

119861119894119909)12

= 119909119879

119861119894119908 (119909

119879

119862119894119909)12

= 119909119879

119862119894119911

119894 = 1 2 119896

119910 ge 0

V119894ge 0 119894 = 1 2 119896

(52)

These are nothing but the dual constraintsSo (119906 120582 119910 V 119908 119911 119901 = 0) is feasible solution for dual

problemAnd the objective values of MFP120582 and SMMFD120582 are

equal to zero It follows fromTheorem 13 and for any feasiblesolution (119906 120582 119910 V 119908 119911 119901 = 0) of dual that 120582119866(119906) le 120582119866(119909)

So (119906 120582 119910 V 119908 119911 119901 = 0) is optimal solution ofSMMFD120582 Then applying Lemmas 8 and 9 we concludethat (119906 120582 119910 V 119908 119911 119901 = 0) is properly efficient for(SMMFD0)

4 Special Case

If 119901 = 0 119862119894= 0 119894 = 1 2 119896 then our dual programming

reduces to the dual programming proposed by Tripathy [14]

5 Conclusion

In this paper three approaches given by Dinklebaeh [28] andJagannathan [12] for both primal and second order mixedtype dual of a nondifferentiable multiobjective fractionalprogramming problem are introduced and the necessaryand sufficient optimality conditions are established and aparameterization technique is used to establish duality results

8 International Scholarly Research Notices

under generalized second order 120588-univexity assumptionTheresults developed in this paper can be further extendedto higher order mixed type fractional problem containingsquare root term Also the present work can be furtherextended to a class of nondifferentiable minimax mixedfractional programming problems

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] C R Bector ldquoDuality in nonlinear fractional programmingrdquoZeitschrift fur Operations Research vol 17 no 5 pp 183ndash1931973

[2] C R Bector and S Chandra ldquoFirst and Second order duality fora class of non differentiable fractional programmingrdquo Journal ofInformation amp Optimization Sciences vol 7 pp 335ndash348 1986

[3] B D Craven ldquoFractional programmingrdquo in Sig Ma Series inApplied Mathematics vol 4 pp 145ndash152 Heldermann BerlinGermany 1998

[4] B Mond and T Weir ldquoDuality for fractional programmingwith generalized convexity conditionrdquo Journal of Information ampOptimization Sciences vol 3 pp 105ndash124 1982

[5] I M Stancu-Minasian Fractional Programming TheoryMethod and Application Kluwer Academic PublishersDordrecht The Netherland 1997

[6] S Schaible and T Ibaraki ldquoFractional programmingrdquo EuropeanJournal of Operational Research vol 12 no 4 pp 325ndash338 1983

[7] I Ahmad S K Gupta N Kailey and R P Agarwal ldquoDualityin nondifferentiable minimax fractional programming with B-(pr)-invexityrdquo Journal of Inequalities and Applications vol 2011article 75 15 pages 2011

[8] I Ahmad and S Sharma ldquoSecond-order duality for nondif-ferentiable multiobjective programming problemsrdquo NumericalFunctional Analysis and Optimization vol 28 no 9-10 pp 975ndash988 2007

[9] T R Gulati I Ahmad and D Agarwal ldquoSufficiency andduality in multiobjective programming under generalized typei functionsrdquo Journal of Optimization Theory and Applicationsvol 135 no 3 pp 411ndash427 2007

[10] I Ahmad Z Husain and S Al-Homidan ldquoSecond-order dual-ity in nondifferentiable fractional programmingrdquo NonlinearAnalysis Real World Applications vol 12 no 2 pp 1103ndash11102011

[11] C R Bector S Chandra and I Husain ldquoOptimality conditionsand duality in subdifferentiable multiobjective fractional pro-grammingrdquo Journal of Optimization Theory and Applicationsvol 79 no 1 pp 105ndash125 1993

[12] R Jagannathan ldquoDuality for nonlinear fractional programsrdquoZeitschrift fur Operations Research vol 17 no 1 pp 1ndash3 1973

[13] R Osuna-Gomez A Ruız-Canales and P Rufian-Lizana ldquoMul-tiobjective fractional programmingwith generalized convexityrdquoTrabajos de Investigacion Operativa vol 8 no 1 pp 97ndash1102000

[14] A K Tripathy ldquoMixed type duality in multiobjective fractionalprogramming under generalized 120588-univex functionrdquo Journal ofMathematical Modelling and Algorithms in Operations Research2013

[15] S K Mishra S Wang and K K Lai ldquoRole of 120572-pseudo-univex functions in vector variational-like inequality problemsrdquoJournal of Systems Science and Complexity vol 20 no 3 pp344ndash349 2007

[16] S Schaible ldquoFractional programmingrdquo in Handbook of GlobalOptimization R Horst and P M Pardalos Eds pp 495ndash608 KluwerAcademic Publishers DordrechtTheNetherlands1995

[17] Z A Liang H X Huang and P M Pardalos ldquoOptimalityconditions and duality for a class of nonlinear fractionalprogramming problemsrdquo Journal of Optimization Theory andApplications vol 110 no 3 pp 611ndash619 2001

[18] Z-A Liang H-X Huang and P M Pardalos ldquoEfficiencyconditions and duality for a class of multiobjective fractionalprogramming problemsrdquo Journal of Global Optimization vol27 no 4 pp 447ndash471 2003

[19] L B Santos R Osuna-Gomez and A Rojas-Medar ldquoNons-moothmultiobjective fractional programmingwith generalizedconvexityrdquo Revista Integracion vol 26 no 1 pp 1ndash12 2008

[20] C R Bector S Chandra and A Abha ldquoOn mixed duality inmathematical programmingrdquo Journal of Mathematical Analysisand Applications vol 259 no 1 pp 346ndash356 2001

[21] Z Xu ldquoMixed type duality in multiobjective programmingproblemsrdquo Journal of Mathematical Analysis and Applicationsvol 198 no 3 pp 621ndash635 1996

[22] H Zhou and Y Wang ldquoOptimality and mixed duality for non-smooth multi-objective fractional programmingrdquo Pure Mathe-matics and Applications vol 14 no 3 pp 263ndash274 2003

[23] I Ahmad andZHusain ldquoDuality in nondifferentiableminimaxfractional programming with generalized convexityrdquo AppliedMathematics and Computation vol 176 no 2 pp 545ndash5512006

[24] I Ahmad and Z Husain ldquoOptimality conditions and dualityin nondifferentiable minimax fractional programming withgeneralized convexityrdquo Journal of Optimization Theory andApplications vol 129 no 2 pp 255ndash275 2006

[25] B Mond ldquoA class of nondifferentiable fractional program-ming problemrdquo Zeitschrift fur Angewandte Mathematik undMechanik vol 58 no 8 pp 337ndash341 1978

[26] D S Kim S J Kim and M H Kim ldquoOptimality andduality for a class of nondifferentiable multiobjective fractionalprogramming problemsrdquo Journal of Optimization Theory andApplications vol 129 no 1 pp 131ndash146 2006

[27] D S Kim Y J Lee and K D Bae ldquoDuality in nondifferentiablemultiobjective fractional programs involving conesrdquo TaiwaneseJournal of Mathematics vol 13 no 6 A pp 1811ndash1821 2009

[28] W Dinklebaeh ldquoOn nonlinear fractional programmingrdquoMan-agement Science vol 13 no 7 pp 492ndash498 1967

[29] M A Hanson ldquoOn sufficiency of the Kuhn-Tucker conditionsrdquoJournal of Mathematical Analysis and Applications vol 80 no2 pp 545ndash550 1981

[30] C R Bector S K Suneja and S Gupta ldquoUnivex functionsand univex nonlinear programmingrdquo in Proceeding of theAdministrative Science Association of Canada pp 115ndash124 1992

[31] S K Mishra ldquoOn multiple-objective optimization with gener-alized univexityrdquo Journal of Mathematical Analysis and Applica-tions vol 224 no 1 pp 131ndash148 1998

[32] A Jayswal ldquoNon-differentiable minimax fractional program-ming with generalized 120572-univexityrdquo Journal of Computationaland Applied Mathematics vol 214 no 1 pp 121ndash135 2008

[33] A M Geoffrion ldquoProper efficiency and the theory of vectormaximizationrdquo Journal of Mathematical Analysis and Applica-tions vol 22 no 3 pp 618ndash630 1968

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 7: Research Article Second Order Duality in Multiobjective Fractional Programming …downloads.hindawi.com/archive/2014/541524.pdf · 2019-07-31 · Research Article Second Order Duality

International Scholarly Research Notices 7

So (47) implies that

119896

sum

119894=1

120582119894[119891119894(119909) + 119909

119879

119861119894119908 minus V119894119892119894(119909) minus 119909

119879

119862119894119911]

ge

119896

sum

119894=1

120582119894[119891119894(119906) + 119910

119879

1198691

ℎ1198691(119906) + 119906

119879

119861119894119908

minusV119894119892119894(119906) minus 119906

119879

119862119894119911]

(48)

Now by Schwarz Inequality and (39) we have

119909119879

119861119894119908 le (119909

119879

119861119894119909)12

(119908119879

119861119894119908)12

le (119909119879

119861119894119909)12

119909119879

119862119894119911 le (119909

119879

119862119894119909)12

(119911119879

119862119894119911)12

le (119909119879

119862119894119909)12

119894 = 1 2 119896

(49)

So both (48) and (49) imply that

119896

sum

119894=1

120582119894[119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

]

ge

119896

sum

119894=1

120582119894[119891119894(119906) + 119910

119879

1198691

ℎ1198691(119906) + 119906

119879

119861119894119908

minus V119894119892119894(119906) minus 119906

119879

119862119894119911]

997904rArr Inf (120582119865 (119909)) ge Sup (120582119865 (119906))

(50)

Theorem 14 (strong duality) Let 119909 be optimal solution forMFP120582 and let 119882(119909) = 120601 Then there exist 120582

119894isin R+ 119908 119911 isin

R119899 V119894isin R+ and 119910 isin R119898 such that (119906 120582 119910 V 119908 119911 119901 =

0) is a feasible solution for dual and the objective values ofboth primal and dual are equal to zero Furthermore if (i)(119906 120582 119910 V 119908 119911) is feasible for dual (ii) sum119896

119894=1120582119894119866119894(sdot) is second

order 120588-pseudounivex with respect to 120578 1205950 and 119870 and for

1199101198692

isin R119898minus|1198691| 1199101198791198691

ℎ1198692

(sdot) is second order 120590-quasiunivex withrespect to 120578 120595

1 and 119870 along with (iii) 120588 + 120590 ge 0 then

(119906 120582 119910 V 119908 119911 119901 = 0) is properly efficient for SMMFD0

Proof Since 119909 is optimal solution for (MFP120582) by Lemma 10there exist 120582

119894isin R+119908 119911 isin R119899 V

119894isin R+ and 119910 isin R119898 such that

119896

sum

119894=1

120582119894[nabla119891119894(119909) + 119861

119894119908 minus V119894nabla119892119894(119909) minus 119862

119894119911] + 119910

119879

nablaℎ (119909) = 0

119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894(119892119894(119909) minus (119909

119879

119862119894119909)12

) = 0

119894 = 1 2 119896

119910119879

ℎ (119909) = 0

119908119879

119861119894119908 le 1 119911

119879

119862119894119911 le 1 119894 = 1 2 119896

(119909119879

119861119894119909)12

= 119909119879

119861119894119908 (119909

119879

119862119894119909)12

= 119909119879

119862119894119911

119894 = 1 2 119896

119910 ge 0

V119894ge 0 119894 = 1 2 119896

(51)

which can be written as119896

sum

119894=1

120582119894[nabla119891119894(119909) + 119910

119879

1198691

nablaℎ1198691(119909) + 119861

119894119908 minus V119894nabla119892119894(119909) minus 119862

119894119911]

+ 119910119879

1198691

nablaℎ1198692(119909) = 0

119891119894(119909) + (119909

119879

119861119894119909)12

minus V119894119892119894(119909) minus (119909

119879

119862119894119909)12

+ 119910119879

1198691

ℎ1198691(119909) = 0

119910119879

1198691

ℎ1198692(119909) = 0

119908119879

119861119894119908 le 1 119911

119879

119862119894119911 le 1 119894 = 1 2 119896

(119909119879

119861119894119909)12

= 119909119879

119861119894119908 (119909

119879

119862119894119909)12

= 119909119879

119862119894119911

119894 = 1 2 119896

119910 ge 0

V119894ge 0 119894 = 1 2 119896

(52)

These are nothing but the dual constraintsSo (119906 120582 119910 V 119908 119911 119901 = 0) is feasible solution for dual

problemAnd the objective values of MFP120582 and SMMFD120582 are

equal to zero It follows fromTheorem 13 and for any feasiblesolution (119906 120582 119910 V 119908 119911 119901 = 0) of dual that 120582119866(119906) le 120582119866(119909)

So (119906 120582 119910 V 119908 119911 119901 = 0) is optimal solution ofSMMFD120582 Then applying Lemmas 8 and 9 we concludethat (119906 120582 119910 V 119908 119911 119901 = 0) is properly efficient for(SMMFD0)

4 Special Case

If 119901 = 0 119862119894= 0 119894 = 1 2 119896 then our dual programming

reduces to the dual programming proposed by Tripathy [14]

5 Conclusion

In this paper three approaches given by Dinklebaeh [28] andJagannathan [12] for both primal and second order mixedtype dual of a nondifferentiable multiobjective fractionalprogramming problem are introduced and the necessaryand sufficient optimality conditions are established and aparameterization technique is used to establish duality results

8 International Scholarly Research Notices

under generalized second order 120588-univexity assumptionTheresults developed in this paper can be further extendedto higher order mixed type fractional problem containingsquare root term Also the present work can be furtherextended to a class of nondifferentiable minimax mixedfractional programming problems

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] C R Bector ldquoDuality in nonlinear fractional programmingrdquoZeitschrift fur Operations Research vol 17 no 5 pp 183ndash1931973

[2] C R Bector and S Chandra ldquoFirst and Second order duality fora class of non differentiable fractional programmingrdquo Journal ofInformation amp Optimization Sciences vol 7 pp 335ndash348 1986

[3] B D Craven ldquoFractional programmingrdquo in Sig Ma Series inApplied Mathematics vol 4 pp 145ndash152 Heldermann BerlinGermany 1998

[4] B Mond and T Weir ldquoDuality for fractional programmingwith generalized convexity conditionrdquo Journal of Information ampOptimization Sciences vol 3 pp 105ndash124 1982

[5] I M Stancu-Minasian Fractional Programming TheoryMethod and Application Kluwer Academic PublishersDordrecht The Netherland 1997

[6] S Schaible and T Ibaraki ldquoFractional programmingrdquo EuropeanJournal of Operational Research vol 12 no 4 pp 325ndash338 1983

[7] I Ahmad S K Gupta N Kailey and R P Agarwal ldquoDualityin nondifferentiable minimax fractional programming with B-(pr)-invexityrdquo Journal of Inequalities and Applications vol 2011article 75 15 pages 2011

[8] I Ahmad and S Sharma ldquoSecond-order duality for nondif-ferentiable multiobjective programming problemsrdquo NumericalFunctional Analysis and Optimization vol 28 no 9-10 pp 975ndash988 2007

[9] T R Gulati I Ahmad and D Agarwal ldquoSufficiency andduality in multiobjective programming under generalized typei functionsrdquo Journal of Optimization Theory and Applicationsvol 135 no 3 pp 411ndash427 2007

[10] I Ahmad Z Husain and S Al-Homidan ldquoSecond-order dual-ity in nondifferentiable fractional programmingrdquo NonlinearAnalysis Real World Applications vol 12 no 2 pp 1103ndash11102011

[11] C R Bector S Chandra and I Husain ldquoOptimality conditionsand duality in subdifferentiable multiobjective fractional pro-grammingrdquo Journal of Optimization Theory and Applicationsvol 79 no 1 pp 105ndash125 1993

[12] R Jagannathan ldquoDuality for nonlinear fractional programsrdquoZeitschrift fur Operations Research vol 17 no 1 pp 1ndash3 1973

[13] R Osuna-Gomez A Ruız-Canales and P Rufian-Lizana ldquoMul-tiobjective fractional programmingwith generalized convexityrdquoTrabajos de Investigacion Operativa vol 8 no 1 pp 97ndash1102000

[14] A K Tripathy ldquoMixed type duality in multiobjective fractionalprogramming under generalized 120588-univex functionrdquo Journal ofMathematical Modelling and Algorithms in Operations Research2013

[15] S K Mishra S Wang and K K Lai ldquoRole of 120572-pseudo-univex functions in vector variational-like inequality problemsrdquoJournal of Systems Science and Complexity vol 20 no 3 pp344ndash349 2007

[16] S Schaible ldquoFractional programmingrdquo in Handbook of GlobalOptimization R Horst and P M Pardalos Eds pp 495ndash608 KluwerAcademic Publishers DordrechtTheNetherlands1995

[17] Z A Liang H X Huang and P M Pardalos ldquoOptimalityconditions and duality for a class of nonlinear fractionalprogramming problemsrdquo Journal of Optimization Theory andApplications vol 110 no 3 pp 611ndash619 2001

[18] Z-A Liang H-X Huang and P M Pardalos ldquoEfficiencyconditions and duality for a class of multiobjective fractionalprogramming problemsrdquo Journal of Global Optimization vol27 no 4 pp 447ndash471 2003

[19] L B Santos R Osuna-Gomez and A Rojas-Medar ldquoNons-moothmultiobjective fractional programmingwith generalizedconvexityrdquo Revista Integracion vol 26 no 1 pp 1ndash12 2008

[20] C R Bector S Chandra and A Abha ldquoOn mixed duality inmathematical programmingrdquo Journal of Mathematical Analysisand Applications vol 259 no 1 pp 346ndash356 2001

[21] Z Xu ldquoMixed type duality in multiobjective programmingproblemsrdquo Journal of Mathematical Analysis and Applicationsvol 198 no 3 pp 621ndash635 1996

[22] H Zhou and Y Wang ldquoOptimality and mixed duality for non-smooth multi-objective fractional programmingrdquo Pure Mathe-matics and Applications vol 14 no 3 pp 263ndash274 2003

[23] I Ahmad andZHusain ldquoDuality in nondifferentiableminimaxfractional programming with generalized convexityrdquo AppliedMathematics and Computation vol 176 no 2 pp 545ndash5512006

[24] I Ahmad and Z Husain ldquoOptimality conditions and dualityin nondifferentiable minimax fractional programming withgeneralized convexityrdquo Journal of Optimization Theory andApplications vol 129 no 2 pp 255ndash275 2006

[25] B Mond ldquoA class of nondifferentiable fractional program-ming problemrdquo Zeitschrift fur Angewandte Mathematik undMechanik vol 58 no 8 pp 337ndash341 1978

[26] D S Kim S J Kim and M H Kim ldquoOptimality andduality for a class of nondifferentiable multiobjective fractionalprogramming problemsrdquo Journal of Optimization Theory andApplications vol 129 no 1 pp 131ndash146 2006

[27] D S Kim Y J Lee and K D Bae ldquoDuality in nondifferentiablemultiobjective fractional programs involving conesrdquo TaiwaneseJournal of Mathematics vol 13 no 6 A pp 1811ndash1821 2009

[28] W Dinklebaeh ldquoOn nonlinear fractional programmingrdquoMan-agement Science vol 13 no 7 pp 492ndash498 1967

[29] M A Hanson ldquoOn sufficiency of the Kuhn-Tucker conditionsrdquoJournal of Mathematical Analysis and Applications vol 80 no2 pp 545ndash550 1981

[30] C R Bector S K Suneja and S Gupta ldquoUnivex functionsand univex nonlinear programmingrdquo in Proceeding of theAdministrative Science Association of Canada pp 115ndash124 1992

[31] S K Mishra ldquoOn multiple-objective optimization with gener-alized univexityrdquo Journal of Mathematical Analysis and Applica-tions vol 224 no 1 pp 131ndash148 1998

[32] A Jayswal ldquoNon-differentiable minimax fractional program-ming with generalized 120572-univexityrdquo Journal of Computationaland Applied Mathematics vol 214 no 1 pp 121ndash135 2008

[33] A M Geoffrion ldquoProper efficiency and the theory of vectormaximizationrdquo Journal of Mathematical Analysis and Applica-tions vol 22 no 3 pp 618ndash630 1968

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 8: Research Article Second Order Duality in Multiobjective Fractional Programming …downloads.hindawi.com/archive/2014/541524.pdf · 2019-07-31 · Research Article Second Order Duality

8 International Scholarly Research Notices

under generalized second order 120588-univexity assumptionTheresults developed in this paper can be further extendedto higher order mixed type fractional problem containingsquare root term Also the present work can be furtherextended to a class of nondifferentiable minimax mixedfractional programming problems

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] C R Bector ldquoDuality in nonlinear fractional programmingrdquoZeitschrift fur Operations Research vol 17 no 5 pp 183ndash1931973

[2] C R Bector and S Chandra ldquoFirst and Second order duality fora class of non differentiable fractional programmingrdquo Journal ofInformation amp Optimization Sciences vol 7 pp 335ndash348 1986

[3] B D Craven ldquoFractional programmingrdquo in Sig Ma Series inApplied Mathematics vol 4 pp 145ndash152 Heldermann BerlinGermany 1998

[4] B Mond and T Weir ldquoDuality for fractional programmingwith generalized convexity conditionrdquo Journal of Information ampOptimization Sciences vol 3 pp 105ndash124 1982

[5] I M Stancu-Minasian Fractional Programming TheoryMethod and Application Kluwer Academic PublishersDordrecht The Netherland 1997

[6] S Schaible and T Ibaraki ldquoFractional programmingrdquo EuropeanJournal of Operational Research vol 12 no 4 pp 325ndash338 1983

[7] I Ahmad S K Gupta N Kailey and R P Agarwal ldquoDualityin nondifferentiable minimax fractional programming with B-(pr)-invexityrdquo Journal of Inequalities and Applications vol 2011article 75 15 pages 2011

[8] I Ahmad and S Sharma ldquoSecond-order duality for nondif-ferentiable multiobjective programming problemsrdquo NumericalFunctional Analysis and Optimization vol 28 no 9-10 pp 975ndash988 2007

[9] T R Gulati I Ahmad and D Agarwal ldquoSufficiency andduality in multiobjective programming under generalized typei functionsrdquo Journal of Optimization Theory and Applicationsvol 135 no 3 pp 411ndash427 2007

[10] I Ahmad Z Husain and S Al-Homidan ldquoSecond-order dual-ity in nondifferentiable fractional programmingrdquo NonlinearAnalysis Real World Applications vol 12 no 2 pp 1103ndash11102011

[11] C R Bector S Chandra and I Husain ldquoOptimality conditionsand duality in subdifferentiable multiobjective fractional pro-grammingrdquo Journal of Optimization Theory and Applicationsvol 79 no 1 pp 105ndash125 1993

[12] R Jagannathan ldquoDuality for nonlinear fractional programsrdquoZeitschrift fur Operations Research vol 17 no 1 pp 1ndash3 1973

[13] R Osuna-Gomez A Ruız-Canales and P Rufian-Lizana ldquoMul-tiobjective fractional programmingwith generalized convexityrdquoTrabajos de Investigacion Operativa vol 8 no 1 pp 97ndash1102000

[14] A K Tripathy ldquoMixed type duality in multiobjective fractionalprogramming under generalized 120588-univex functionrdquo Journal ofMathematical Modelling and Algorithms in Operations Research2013

[15] S K Mishra S Wang and K K Lai ldquoRole of 120572-pseudo-univex functions in vector variational-like inequality problemsrdquoJournal of Systems Science and Complexity vol 20 no 3 pp344ndash349 2007

[16] S Schaible ldquoFractional programmingrdquo in Handbook of GlobalOptimization R Horst and P M Pardalos Eds pp 495ndash608 KluwerAcademic Publishers DordrechtTheNetherlands1995

[17] Z A Liang H X Huang and P M Pardalos ldquoOptimalityconditions and duality for a class of nonlinear fractionalprogramming problemsrdquo Journal of Optimization Theory andApplications vol 110 no 3 pp 611ndash619 2001

[18] Z-A Liang H-X Huang and P M Pardalos ldquoEfficiencyconditions and duality for a class of multiobjective fractionalprogramming problemsrdquo Journal of Global Optimization vol27 no 4 pp 447ndash471 2003

[19] L B Santos R Osuna-Gomez and A Rojas-Medar ldquoNons-moothmultiobjective fractional programmingwith generalizedconvexityrdquo Revista Integracion vol 26 no 1 pp 1ndash12 2008

[20] C R Bector S Chandra and A Abha ldquoOn mixed duality inmathematical programmingrdquo Journal of Mathematical Analysisand Applications vol 259 no 1 pp 346ndash356 2001

[21] Z Xu ldquoMixed type duality in multiobjective programmingproblemsrdquo Journal of Mathematical Analysis and Applicationsvol 198 no 3 pp 621ndash635 1996

[22] H Zhou and Y Wang ldquoOptimality and mixed duality for non-smooth multi-objective fractional programmingrdquo Pure Mathe-matics and Applications vol 14 no 3 pp 263ndash274 2003

[23] I Ahmad andZHusain ldquoDuality in nondifferentiableminimaxfractional programming with generalized convexityrdquo AppliedMathematics and Computation vol 176 no 2 pp 545ndash5512006

[24] I Ahmad and Z Husain ldquoOptimality conditions and dualityin nondifferentiable minimax fractional programming withgeneralized convexityrdquo Journal of Optimization Theory andApplications vol 129 no 2 pp 255ndash275 2006

[25] B Mond ldquoA class of nondifferentiable fractional program-ming problemrdquo Zeitschrift fur Angewandte Mathematik undMechanik vol 58 no 8 pp 337ndash341 1978

[26] D S Kim S J Kim and M H Kim ldquoOptimality andduality for a class of nondifferentiable multiobjective fractionalprogramming problemsrdquo Journal of Optimization Theory andApplications vol 129 no 1 pp 131ndash146 2006

[27] D S Kim Y J Lee and K D Bae ldquoDuality in nondifferentiablemultiobjective fractional programs involving conesrdquo TaiwaneseJournal of Mathematics vol 13 no 6 A pp 1811ndash1821 2009

[28] W Dinklebaeh ldquoOn nonlinear fractional programmingrdquoMan-agement Science vol 13 no 7 pp 492ndash498 1967

[29] M A Hanson ldquoOn sufficiency of the Kuhn-Tucker conditionsrdquoJournal of Mathematical Analysis and Applications vol 80 no2 pp 545ndash550 1981

[30] C R Bector S K Suneja and S Gupta ldquoUnivex functionsand univex nonlinear programmingrdquo in Proceeding of theAdministrative Science Association of Canada pp 115ndash124 1992

[31] S K Mishra ldquoOn multiple-objective optimization with gener-alized univexityrdquo Journal of Mathematical Analysis and Applica-tions vol 224 no 1 pp 131ndash148 1998

[32] A Jayswal ldquoNon-differentiable minimax fractional program-ming with generalized 120572-univexityrdquo Journal of Computationaland Applied Mathematics vol 214 no 1 pp 121ndash135 2008

[33] A M Geoffrion ldquoProper efficiency and the theory of vectormaximizationrdquo Journal of Mathematical Analysis and Applica-tions vol 22 no 3 pp 618ndash630 1968

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 9: Research Article Second Order Duality in Multiobjective Fractional Programming …downloads.hindawi.com/archive/2014/541524.pdf · 2019-07-31 · Research Article Second Order Duality

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Structural Duality I

Structural Duality I

Duality in Condensed Matter Physics and Quantum Field Theoryeduardo.physics.illinois.edu/homepage/quantum... · Duality of Forms • Geometric duality • p forms in D dimensions

Duality in Condensed Matter Physics and Quantum Field Theoryeduardo.physics.illinois.edu/homepage/quantum... · Duality of Forms • Geometric duality • p forms in D dimensions

Duality Opt

Duality Opt

A Second Course in String Theory - SepnetA Second Course in String Theory Conformal Field Theory, Superstrings, Duality and Twistors Edward Hughes, Queen Mary University of London

A Second Course in String Theory - SepnetA Second Course in String Theory Conformal Field Theory, Superstrings, Duality and Twistors Edward Hughes, Queen Mary University of London

holographic duality

holographic duality

A Short Course on Duality, Adjoint Operators, Green’s Functions, …estep/assets/adjointcourse.pdf · Duality, Adjoint Operators, and Green’s Functions Duality, Adjoints, Green’s

A Short Course on Duality, Adjoint Operators, Green’s Functions, …estep/assets/adjointcourse.pdf · Duality, Adjoint Operators, and Green’s Functions Duality, Adjoints, Green’s

Duality Sensitivity Analysis - TRANSP-OR€¦ · Duality Sensitivity Analysis ... Decision Aid Methodologies In Transportation Lecture 3: Duality and sensitivity analysis. Motivation

Duality Sensitivity Analysis - TRANSP-OR€¦ · Duality Sensitivity Analysis ... Decision Aid Methodologies In Transportation Lecture 3: Duality and sensitivity analysis. Motivation

Duality methods for pricing contingent claimsbcc.impan.pl/.../PricingConjDualityV2013.pdf · duality and mathemati-cal finance Introduction Duality theory Pricing Final comments

Duality methods for pricing contingent claimsbcc.impan.pl/.../PricingConjDualityV2013.pdf · duality and mathemati-cal finance Introduction Duality theory Pricing Final comments

The wave-particle duality of photons - Prime Theory · The wave-particle duality of photons . On realism, causality and locality . Laurențiu Mihăescu . Bucharest, Romania . Second

The wave-particle duality of photons - Prime Theory · The wave-particle duality of photons . On realism, causality and locality . Laurențiu Mihăescu . Bucharest, Romania . Second

Article sur second life

Article sur second life

Duality (I)

Duality (I)

The duality of duality and non-duality

The duality of duality and non-duality

Duality for topological abelian group stacks and T-duality · Duality for topological abelian group stacks and T-duality U. Bunke, Th. Schick, M. Spitzweck, and A. Thom January 26,

Duality for topological abelian group stacks and T-duality · Duality for topological abelian group stacks and T-duality U. Bunke, Th. Schick, M. Spitzweck, and A. Thom January 26,

Principal of duality

Principal of duality

Review article: Three new complementing second language ...

Review article: Three new complementing second language ...

LP Duality

LP Duality

From Schur-Weyl duality to quantum symmetric pairs · Schur-Weyl duality. . . . . . . Quantization. . . . . . . . . q-Schur duality BLM construction. . . . . . . Quantum symmetric

From Schur-Weyl duality to quantum symmetric pairs · Schur-Weyl duality. . . . . . . Quantization. . . . . . . . . q-Schur duality BLM construction. . . . . . . Quantum symmetric

Simplex and Duality

Simplex and Duality

Seiberg Duality

Seiberg Duality

Duality in Society: Duality in Teacher Education ..."Duality" 1. Summary of the Program. 17 The project "Duality in Society; Duality in Teacher. Education" was initiated Winter, 1970

Duality in Society: Duality in Teacher Education ..."Duality" 1. Summary of the Program. 17 The project "Duality in Society; Duality in Teacher. Education" was initiated Winter, 1970