Annexe_1_2.docx

7/28/2019 Annexe_1_2.docx

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Fred

Enclosure Methods for Multivariate Diff erentiable

Functions and Application to Global Optimization

eric Messine (Lab. LIA, Departement dInformatiqu e de lUniversit

[email protected])

Jean-Louis Lagouanelle (Lab. LIMA, Institut de Recherche en Informatique de Toulouse

[email protected])

Abstract: The efficiency of global optimization methods in connection with intervalarithmetic is no more to be demonstrated. They allow to determine the global optimum and the corresponding optimizers , with certainty and arbitrary accuracy. One of themain features of these algorithms is to deliver a function enclosure dened on a box (right parallelepiped). The studied method provides a lower bound (or upper bound ) of a function in that box throughout two diff erent strategies. As we shall see, these algorithms associated with various Branch and Bound methods lead to accelerated convergence and permit to avoid the cluster problem. Ke y Words: Globaloptimization,Intervalarithmetic,Multivariatefunctions,Branchand Bound algorithm, Taylor s expansion, Polyhedral cone.

1 Introduction

In this paper the methods which are proposed, concern the minimization problemin the context of Branch and Bound algorithms and interval arithmetic [6], [10]. We use a rs t order Taylor s expansion and the mean value theorem to builda ffi ne underestimations of the function over the box. The intersection of these underestimations is easily computed, then a lower bound of the function on thebox follows. This lower bound is often better than those which are issued from diff erent inclusion functions, in connection with interval arithmetic [9].

First, we shall deal with the building of the affine underestimations and theprinciple of the algorithm to get the lower bound. Then, its efficiency will be shown in section 5, when used within Branch and Bound algorithm, on simpleexamples of unconstrained minimization of multivariate functions.

Usually, the minimization problem is described by:

Minimize f ( x ), subject to x X , with X IR .

f is a diff erentiable n-variables function and X is the right parallelepiped in IR ,dened by X = { x = ( x , , x ) ; x x x } , i { 1 , 2 , , n } .

The principle used here, including the underlying mean value theorem, has already been studied in the case of univariate polynomial function by Jaumard, Hansen, and Xiong [5], by Alefeld [1], and by Visweswaran and Floudas [11].Mladineo gives, for functions that satisfy a Lipchitz condition, a method [8] based on the building of cones with spherical bases, where a mixed linear non-linear system must be solved.

e de Pau

n (1 )

n L

n i U i
mailto:[email protected]:[email protected]:InstitutdeRechercheenInformatiquedeToulouselagouane@irit.frmailto:InstitutdeRechercheenInformatiquedeToulouselagouane@irit.frmailto:InstitutdeRechercheenInformatiquedeToulouselagouane@irit.frmailto:InstitutdeRechercheenInformatiquedeToulouselagouane@irit.frmailto:InstitutdeRechercheenInformatiquedeToulouselagouane@irit.frmailto:InstitutdeRechercheenInformatiquedeToulouselagouane@irit.frmailto:InstitutdeRechercheenInformatiquedeToulouselagouane@irit.frmailto:InstitutdeRechercheenInformatiquedeToulouselagouane@irit.frmailto:InstitutdeRechercheenInformatiquedeToulouselagouane@irit.frmailto:InstitutdeRechercheenInformatiquedeToulouselagouane@irit.frmailto:InstitutdeRechercheenInformatiquedeToulouselagouane@irit.frmailto:InstitutdeRechercheenInformatiquedeToulouselagouane@irit.frmailto:InstitutdeRechercheenInformatiquedeToulouselagouane@irit.frmailto:InstitutdeRechercheenInformatiquedeToulouselagouane@irit.frmailto:InstitutdeRechercheenInformatiquedeToulouselagouane@irit.frmailto:InstitutdeRechercheenInformatiquedeToulouselagouane@irit.frmailto:InstitutdeRechercheenInformatiquedeToulouselagouane@irit.frmailto:InstitutdeRechercheenInformatiquedeToulouselagouane@irit.frmailto:InstitutdeRechercheenInformatiquedeToulouselagouane@irit.frmailto:InstitutdeRechercheenInformatiquedeToulouselagouane@irit.frmailto:InstitutdeRechercheenInformatiquedeToulouselagouane@irit.frmailto:InstitutdeRechercheenInformatiquedeToulouselagouane@irit.frmailto:InstitutdeRechercheenInformatiquedeToulouselagouane@irit.frmailto:InstitutdeRechercheenInformatiquedeToulouselagouane@irit.frmailto:[email protected]


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Lk L k

U k U k

i

In our method, we get polyhedral cones and we want to solve very simplelinear systems to keep the efficiency of the algorithms. The unique solution of each linear system corresponds to the vertex of a polyhedral cone that is meantfor a minimizer of f on a box X . This is reached by two suitable choices of vertices of the hypercube as explained in sections 3. Convergence and complexityproperties are given in section 4. Numerical experiments and applications to global optimization follow in section 5.

The following section gives the basic tools for the construction of the poly-hedral cones.

2 Underestimating a ffi ne functions

According to Taylor s expansion of a function f , we get some inclusion functionof f in the box X [9]. This inclusion function is called the Taylor form:

( x, y ) X , f ( y ) f ( x ) + ( X x ) g ( X ) ,

Where g ( X ) is an interval vector of the gradient inclusion of f ; g will be auto-matically calculated by automatic di ff erentiation [3], [6].

Let X = [ x , x ] and G ( X ) = [g ( X ) , g ( X ) ] for i = 1 , 2 , ,n .

So, for every x X , ( x ) G ( X ) .

We assume, for the following, that g ( X ) .g ( X ) < 0. Let s = (s , , s ) be the coordinates of a vertex S of the box X and

g ( X ) = (g ( X ) , , g ( X ) )

g

( X ) =

g

( X ) +

g

( X ) ,

k

{ 1 , 2 , , n }

Then, from the following trivial inequalities:

(y x )g ( X ) (y x ) (y x )g ( X ) and (y x )g ( X ) (y x ) (y x )g ( X )

for all i = 1 , 2 , , n and any given G ( X ) , we can deduce for every vertexS of the box X , an affine underestimation of a function f over X , that is

f ( y ) f ( s ) + ( y s ) g ( X ), y X

So we can obtain 2 affine underestimations of the function f for the box X . Each one of these hyperplanes is a maximum supporting hyperplane at the

vertex S , for the box X .

2 T

L U i i Li U i

f x i

L i U

i

1 n S S

1 S

n T

where

S k

s x k x x Lk k

L s x k x x U k k

U (2 )

i L L

i i i Li i L U

i i U U i i U i U

L i i

i

T S (3 )

n


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Conditions (4 ) and (5 ) follow directly from the fact that is a maximum k

3 Global minimum enclosure methods

The rs t aim of the following algorithm is to get a lower bound of a functionf on a box X . This will be achieved by the intersection of n + 1 hyperplanes chosen amongst the 2 hyperplanes that we can build with the properties givenin section 2; but that does no t give automatically a lower bound of f on X ; except in the case of an univariate function [5], [11].

For k = 1 , , 2 le t be a maximum supporting hyperplane, u it s affinerepresentation and E the restriction to the box X of the associated half-space dened by

E = { ( x , z ) IR I R ; z u ( x ), x X }

classically we have the

Property 1 I f E is a polyhedra l cone with vertex C which contains the

graph of the function f then fo r | K | = n + 1 , { C } = exists and is unique.

Remark. Thechoiceof |K| = n +1isminimal,itispossibletoset |K | > n +1 wi than analogous convexity property for a polyhedrical set but then the computation of the lower bound becomes too expensive.

One way to obtain existence and uniqueness of such a point C is given bythe following choice of the hyperplanes .

First of al l the n + 1 vertices S selected are no t contained in the same

u (s ) = f ( s )

u

( x )

f ( x ) ,

x

X

u (s ) u ( x ), x X

supporting hyperplane. Condition (6 ) supposes, here, that according to section 2, g ( X )g ( X ) < 0 for al l i ; this is obtained by application of a monotonicity test to f on X .

Let ( x , z ) IR IR the coordinates of the point C and u ( x ) = f ( s ) +

Deni t ion1 . A set of n + 1 vertices of X is said admissible i f the vertex C of the polyhedral cone dened by n + 1 hyperplanes, built as above, satise s z f ( x ) , x X and x X . For commodity the simplex, convex hull of the

Such a point C gives a lower bound z of f and x is destined to be a currentminimizer for problem (1). Let us see on a short example that any arbitrary set of n + 1 vertices of a box X in IR may no t be admissible.

k

c c

c c

hyperplane of IR . Then each affine function u satise s the relations

( x s ) g ( X ) , we introduce now the

vertices S S S is said admissible.

n

n +k

+k

n k

k K +k

k K k

k

n

k

k k k (4 )

k (5 )

(6 )

i i

n k k

k T S k

0 1 n

c c

n


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Example1.

f : X IR IR

( x , x ) f ( x , x ) = x 2 x x + 3 x 5 x [ 5 , 5] [ 15 , 10]

then G ( X ) = ([27 ,43],[15 ,5] ) Let s = ( 5 , 15) , s = (5 , 15 ) , s = (5 ,10) . For these vertices and

u ( x , x ) = f ( 5 , 1 5 ) 2 7 ( x + 5 ) 1 5 ( x + 1 5 ) u ( x , x ) = f ( 5 , 1 5 ) + 4 3 ( x 5 ) 1 5 ( x + 1 5 ) u ( x , x ) = f ( 5 , 1 0 ) 2 7 ( x + 5 ) + 5 ( x 1 0 )

we get ( x , z ) = (3 .57 , 15 , 103 .6) but f (5 , 10) = 110, hence z is no t a

In sections 3.1 and 3.2 we deal with two kinds of admissible sets of vertices; insection 3.2 we show that such sets always exist and can be easily identied .

3.1 Admissible r ight simplexes

The rs t idea to get straightforward formulae from linear systems is to consider the n + 1 vertices of a right simplex generated by a reference vertex S and by

of the others S , only one variable is modied.

Figure 1: Right Simplex

The following result gives a su ffi cient condition for a right simplex to beadmissible.

1 2

c c

0

u , u , u the corresponding hyperplanes

lower bound; { s , s , s } is no t admissible.

the n adjacent vertices of a box X . So doing the linear system is easily solved because from the vertex S to one

2

1 2 1 2 21 1 2 1 2 X =

T T T

3 T

1 2 3

1 1 2 1 2 2 1 2 1 2 3 1 2 1 2

c 1 2 3

0

S 2

S

S S 0

3

1


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g ( X ) g ( X )

I f x , x , , x are the basis variables and e = 0 ,k = 0 , 1 , ,n , x =

e is a slack variable.

First, note that S diff ers from S only by one coordinate, supposed to be x .

Proof. Let s = (s , s , , s ) be the coordinates of the vertex S for al l k

k

n T 1

2

k

0 k

k k

Theo re m2. Let S be a vertex of a box X i n IR ,let S , S , , S be the ver-tices adjacent to S . I f g ( X ) is dene d by (2) fo r al l k . Then the corresponding right simplex T is admissible i f and only i f

g ( X ) g ( X )

where g ( X ) = g ( X ) i f (s ) = x and g ( X ) = g ( X ) i f (s ) = x .

and u the affine function

u ( x ) = f ( s ) + ( x s ) g ( X )

Consider the auxiliary problem

Minimize z subject to ( x , z ) E , k K

I f x is a solution of this linear program, then ( x , z ) satise s property 1; z isa lower bound of f on the box X and x X .

Standard form of (8 ) is

Minimize z subject to z = e + f ( s ) + ( x s ) g

0 e , k = 0 , 1 , 2 , , n ( x , z ) E , k K

The optimal solution of (9 ) is reached at the unique extremal point C , vertexof the polyhedral cone:

C =

( x , x , , x ) is solution of the linear system

z = f ( s ) + ( x s ) g ( X ) , k = 0 , 1 , 2 , , n

Moreover S is adjacent to S for al l k and g is built following (2). Then bysubtraction of equation (k) from equation (0 ) one gets directly

f ( s ) f ( s ) s g ( X ) s g ( X ) g ( X ) g ( X )

k = 1 , 2 , , n

x X that is proved using (3 ) and the eventually optimal value z for theproblem (8 ) is

z = f ( s ) + ( x s ) g ( X )

0 n

1 2 n 0 S

0 n

i = 1 g ( X ) S 0 i

S i

S 0 i

1 (7 )

S k i L

i k i Li

S k i U

i k i U i

k k T n k

k k k T S k

k

+ k

x X (8 )

k k T S ( X ) , k +

k

x X (9 )

n

k = 0

k

1 2 n

k k T S k

k 0 S k

x

=

k

0 k S k S 0 k +

k 0 k S

k k S

k S k k S 0 k (10)

0 T S 0 (11)


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Relaxation: one moves some hyperplanes until the condition (7 ) is satis-

but the assumption g ( X )g ( X ) < 0 induces that cm > 0 for any k = 0 and

k

U k L

k

But this solution is optimal i f and only i f the marginal costs are non negative.From the standard form (9 ) we get

z = z + cm e

with cm = 1 + g ( X ) g ( X )

and cm = g ( X ) g ( X )

, k = 1 , , n

optimality of ( x , z ) implies that cm > 0. Let w (G ( X ) ) : = g ( X ) g ( X ) and < G ( X ) > : = min{g ( X ) , | g ( X ) | }

be respectively the width and the mignitude of the interval G ( X ) , for i =

1 , 2 , , n .

The following condition of existence of an admissible right simplex is deduced from the worst case in the previous inequality (7).

Theo re m3. For n 3 admissible right simplex exists on the box X i f and only i f

< G ( X ) > w (G ( X ) )

Remark. As i t can be seen in Algo ri th m 1 when no admissible right simplex canbe found on the box, this condition is weakened anda lower bound will be found again, but with a loss of accuracy.

For univariate or bivariate functions, inequality (12) is always satised anda right simplex can be found.

This leads to theAlgorithm 1-

Applicati on of a monotony test (elimination of components). Choose a right simplex satisfying (12); determination of the set K . For n 3 i f no such

simplex exists then apply one of these local strategies :

- relaxation of the gradient; then modify until the condition is satised,

- Return the intersection of the polyhedral cone with the unbounded cylin-der which has for base the box X : return the lowe r point so obtained.

Compute x , z by (10) and (11).

Proof. i f the optimality condition (7 ) is satised then a lower bound is di - rectly obtained by computation of { C } = ,

ed, Intersection is equivalent to compute directly the rs t order Taylor inclusion

function [10].

n

k = 0

0 n

k = 1 g ( X ) S 0 k

Sk k

0 k

k g ( X ) S 0 k

k Sk 0

0

i U i Li i U i Li i

n

i = 1 i

i 1 (12)

k

k K k


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U when s = x + x L

This algorithm runs only for right simplexes; the aim of the next section isto answer positively the question : for a given box X IR , is i t always possible to nd an admissible set of n + 1 vertices?

3.2 Construction of admissible simplexes

Let x = ( x , x , , x ) and x = ( x , x , , x ) be the vertices of X withextremal coordinates. S is called the opposite of a given vertex S on the box X

s . We naturally associate to the box X , the symmetric directed graph con-

structed from its vertices and edges. Then the following result shows that, for any given vertex S and the opposite one to nd a path of length n : SS S S such that the corresponding simplex is admissible.

Figure 2: Admissible Simplex

g ( X ) being dened by (2), we denote

w (G ( X ) ) , i = { 1 , , n }

Theo re m4. Let S be an arbitrary vertex of the box X , S it s opposite. Let the real numbers K be sorted i n increasing order so that

1 < K K K < 0

Then the simplex dene d by vertices SS S S ) is admissible when the path from S to S is dene d fo r j = 0 , 1 , 2 , , n 1 by

: = s , i { 1 , 2 , , n } \ k

S on the box X , i t is always possible

n

1 2 1 2 n n

k 1 k n 1

..

S

..

S ....

2 .. ....

.... .... ..

. .

....

. .

...

............

... ....... . . . . . . . . .. . . . . . .

.... .. .. .. .... . .. .. .. ..

.. .. ..

........

.. .. .. . . . ..... ... .... ... . . . . . . . . ..

.. .. .. .. .. .. . .. .. .. .. ... .. .. .. .. .. .. . .. .. .. .. ...

.. .. .. .. .. .. .. .. .... . .... .... .... ...

.... .. .. .. .. .. .... . .... .... ....

...

S 3 S =S

0 00

0

1 11

S 1

2 2

S =S 3 3 33

S i

K = i |g ( X ) | S i

k 1 k 2 n k 1 k n 1 n S ( = k

s k j + 1i k j + 1k j + 1

k i + 1

s : = x Lk j + 1 + x U k j + 1 s k k j + 1

(13)


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the hypothesis g g < 0 and (3). one must show that x X and that z

L U k k

where S = S and S = S. This path is unique fo r strict inequalities.

Proof. One proceeds analogously to the previous theorem. The only diff erence lies on the fact that in this case, two consecutive vertices have n 1 identical coordinates. Consequently x optimal solution of the associated linear programis obtained directly by subtraction of two consecutive equations; and then, we get

) f ( s ) + g ( X ) s g

g ( X ) g

for k = 1 , 2 , , n

z = f ( s ) + ( x s ) g ( X )

is a lower bound of f . x X is straightforward when one writes x x x , for al l k under

Otherwise z is the minimal value of auxiliary problem (8 ) i f and only i f themarginal costs are non-negative because

z = z + e + g ( X ) g

which implies that

g ( X ) g

g ( X ) g

for j = 1 , 2 , , n 1 But considering (13) and (2 ) we get

( X ) = = g

g ( X ) =

g

( X ) g

( X ) g

( X ) + g ( X ) g

g

( X ) g

( X )

for j = 1 , 2 , , n . Therefore, in any case,

g ( X ) g w (G ( X ) )

k 0 k n

x = k f ( s k 1 k S

k k k

S k 1 k ( X ) s

1 k

S k

S 1 k ( X )

(14)

and T S (15)

L k

k U k

k 0

n

k = 1 e k 1 e k

S k

S 1 k ( X )

g ( X ) S k 0

k

0 < 1 + g ( X ) S

k 0

k 1 S k 1 k 1

S k 0 k 1 ( X )

and g ( X ) S

k 0

k S k k

S k 1 k ( X )

g S

k 0

k j + 1 ( X )

g S k + 1 k j + 1 ( X ) g

S k k + 1 ( X )

g S 1 k ( X ) = g

S 2 k

S k 0 k ( X )

and

S k k

g S k 1 k ( X ) g L

k ( X )

U L L k

U k

S k 1 k ( X )

U L g ( X ) U k

g ( X ) S k 0

k S k

S 1 k ( X )

= |g S

k 0

k ( X ) |

k


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i U i

7 7

then an admissible path from s = (5 , 15) to

because g ( X ) < 0 < g ( X ) , for j = 1 , 2 , , n . Hence for, any initial vertex S , at least one admissible simplex, can be found

applying the rule given in theorem(4) that gives a path from S to S .

I f the initial vertex is S instead of S , i t is easily seen that one nds the same path in reverse order towards S . Then, for a box X , i t exists at least 2admissible simplexes and as much lower bounds for f . We do no t study here the best choice amongst al l these possibilities, a choice that must take into accountthe values of f and it s derivative.

For illustration, we apply this method to example 1

s = (5 , 15 ) is the initial vertex, g = (4 3 , 15 ) and K =

which gives 1 < K < K < 0

s = ( 5 ,10) is obtained by

modifying rs t

the

coordinate

x

which

leads

to

the

vertex

(5 ,10)

and

secondly

the coordinate x which leads to (5 ,10) the terminal vertex.

One gets with the corresponding hyperplanes x = ( , 5 ) and z = 315 .9. Now we see why the right simplex { s , s , s } was no t admissible in this exam-

ple; generally for a bivariate function on a box X , two simplexes are admissible; which, in this case, coincides with the right ones.

To get a lower bound of a function f from theorem (4), we applyAlgorithm 2-

Choose a vertex S ; determine g ( X ) . Applicati on of monotony test (elimination of components). Computation of

g ( X ) g ( X ) , k = { 1 , , n }

Classify the associated numbers K in increasing order. Determination of an admissible simplex following theorem (4 ) Compute { x , z } with (14) and (15).

Remark. Several very efficient improvements may be used to get tighter lower bounds: Slope matrices will advantageously take place of derivatives when their computation is possible or narrower intervals for the components of g may also result from Taylor expansion of E. Hansen [3], where some interval arguments are replaced by real quantities.

I f an

upper

bound

is

required

instead

of

a lower

bound,

Algo ri th m

2 is

mod-ie d as follows

For a given vertex S of the box X , let H = | K |, for i = 1 , 2 , , n .

g ( X ) = g ( X ) +

L U

n 1

T S T 1

43 70 ,

K = 2

15

20 2 1

2

1 8 2211

1 2 3

K = k |g ( X ) | Sk

U k Lk

k

i i

S i

s x i x x Li i

U s x i x x U i

i L i i

g ( X ) L (16)


10/16

f + While the minimizers are no t found with a given tolerance Do

- Extract the rs t element from L , write i t as follows: ( Y ,

then S is the opposite vertex of S i.e.

f on a box X are distinct excepted in extreme cases for example i f g ( X ) = C L i

The real numbers H are classi ed in increasing order

0 < H H H < 1

Then an admissible path from S to S is found by modifying the coordinates of S in the order x , x , , x .

Unfortunately, the paths for a lower bound and an upper bound of a function

and g ( X ) = C for all i . When inequalities are strict in (17), the two paths are opposite which means

that S and S are respectively the initial vertices for lower and upper bounds s = x + x s for al l i , see gure

4 Optimization algorithm using these enclosure processes

The methods previously described are integrated, by the mean of Algo ri th m 1and Algo ri th m 2 in interval Branch and Bound algorithms [10] to solve the global minimization problem (1).

This is the topic of Algorithm 3-

Initial enclosure of the minimum on X , say [F , F ] Insert ( X , F ) in a list L

y ) - Divide Y into two parts (along it s larger edge): we get V and V

-

For i =

1 ,2

- Do

- Enclosure the global minimum on V = [F , F ] - i f F > f then

- Go to the next step

- i f F < f then . f F . Remove al l the elements of L which lower bound is greater than

the new f .- end i f - Insert (V , F ) in L following the increasing order of the F .

- end i f - end For

End While Return f and L .

Remark. The upper bound F may also be progressively computed in thealgorithm

We have three possibilities to enclose the global minimum:

i

i n i n 1 i 1 (17)

i 1 i 2 i n

1 U

2

k k k L

U k

(2).

L

(1 ) (2 ) .

( i ) L U L

- else U

U

( i ) L L

U


11/16

1. right simplex and, i f necessary, intersection with unbounded cylinder

which has for base the box X . 2. right simplex and, i f necessary, gradient relaxation.

3. research of an admissible simplex with two opposite vertices of X . When the size of a box X is su fficiently small, i t is no t necessary to computefor each sub-box of X the gradient according to inclusion monotonicity but this induces a slower convergence.

For every admissible simplex dened by the vertices S , S , , S the following inequalities hold:

min { f ( S ) } f

with f = in f

4.1 Complexity and convergence

Each underestimation of f requires an interval inclusion function of grad f n + 1 point evaluations of the function f

and nally the computation of { x , z } requires O (3 n ) elementary operationsfor a given box X .

This estimate is relative to a single box; but considering, the splitting of abox into adjacent sub-boxes, , which is done in global optimization problems, the number of evaluations of f may be perceptibly reduced. Indeed i f n verticesbelong to the interface of two sub-boxes, then one more vertex is necessary on each sub-box.

Moreover changing a single vertex does no t aff ect al l the coordinates of x and brings only a partial contribution in the computation of z .

Let z and z be respectively the computed lower andupper bounds.Le t w ( X ) and w (G ) be the width vectors, w ( X ) = x x and

w (G ) = (w (G ( X ) ) , w (G ( X ) ) , , w ( G ( X ) ) )

|g ( X ) | = (|g ( X ) | , | g ( X ) | , , | g ( X ) | )

where g ( X ) is dened by (2), then properties of convergence of affine estimatesru n from

Theo re m5. For a di ff erentiable function f : X IR IR; i f an admissible set { S } is used to get z and the opposite one { S } z fo r z on the box X , then, assuming g ( X )g ( X ) < 0, k, the following

|z f | min (w ( X ) , | g

z z (w ( X ) ,w (G )) n

And the excess width of a ffi ne estimation is at least of order 2 fo r a regular function.

U

U

=1 , 2 , ,n

k j =0 , 1 , ,n k j =0 , 1 , ,n

inequalities are valid:

L m in =0 , 1 , ,n

k 0 k 1 n of X ,

m n z (18)

m in x X

L

U U L

1 2 n T

and S S

1 S

2 S

n T

S

n L to get

U k L

k

S k ( X ) | ) n (19)

U L IR (20)


12/16

Proof. The proof is quite straightforward drawn from the arguments :

For al l j ,

|z f | |z f ( s ) | = ( x s ) g

and on equivalent inequality for z . For any x X ,

( x ) = g ( X ) + ( 1 ) g ( X ) , ]0,1[

Finally, E W excess width of affine estimate for a function f is given by

E W = (z z ) ( f f ) = z f

which induces a convergence of order 2 as soon as, for example, the rs tderivative of f satise s a Lipschitz condition.

5 Numerical experiments

First we search the global minimum of polynomial functions of 2 ,3, or 4 variableswith diff erent degrees and of functions involving trigonometric functions.

f ( x ) = 1 + ( x + 2 ) x + x x , with X = [1, 2] [ 10 , 10]

f ( x ) = 2 x 1 . 05 x + x x x + x , with X = [ 2 ,4]

f ( x ) = ( x 2 x 7 ) + ( 2 x + x 5 ) , with X = [ 2 .5 ,3 .5] [1 .5 ,4 .5] f ( x ) = [ 1 + ( x + x + 1 ) (1 9 14 x + 3 x 1 4 x + 6 x x + 3 x )]

[ 3 0 + ( 2 x 3 x ) (1 8 32 x + 1 2 x + 4 8 x 3 6 x x + 2 7 x )] with X = [2 ,2]

f ( x ) = ( x 1 ) ( x + 2 ) ( x + 1 ) ( x 2 ) x , with X = [ 2 , 2] f ( x ) = 4 x 2 x x + 4 x 2 x x + 4 x 2 x x + 4 x + 2 x x + 3 x

+ 5 x , with X = [ 1 , 3 ] [ 10 , 1 0 ] [1 , 4] [ 1 , 5 ] f ( x ) = x + x cos18 x + x sin18 x + x cos x + x x x

with X = [1,500]

. N: Number of iterations,

. T(s): CPU time in seconds,

U max m in L

L m in

z ) (w ( X ) ,w (G )) n

L k L k T S k ( X )

U

f x i

Li U

i

U L

max m in ( f

IR

1 21 2 1 2

2 21 41 22 1 2 1 6

62 2

3 1 2 2

1 2 4 1 2 1 21 2 1 2 22

1 2 2 1 21 2 1 2 22 2

5 1 1 2 2 23 3

6 21 1 2 22 2 3 23 3 4 24 1 2 3 4

21 22

3

Pb f 1 N T(s) C

f 2 N T(s) C

f 3 N T(s) C

f 4 N T(s) C

f 2 N T(s) C

NE

T1

T1 + M

1550 16,80 286

114

2,29

9

27 0,32 5

649 7,73 150

158 3,76

21

18 0,27 6

218 1,08 36

174 1,64

5

23 0,23 3

- - - 456 4,02 83

331 8,16

26

25 0,53 8 5373

356,73

42

4769 129,4 38

RS RS+M

64 0,65 8 23 0,29 5

87 1,33 25 18 0,29 6

55 0,51 3 19 0,16 3

3379 84,83 24 3318 83,5 24

131 1,76 32 22 0,50 9


13/16


14/16

Methods used to compute upper and lower bounds are:

. f ( X ): range of f over X , i = 1 , , 6,

. NE : natural inclusion extension function,

. T1 : Taylor inclusion function,

. RS1: right simplex and intersection of the polyhedral cone with the un-bounded cylinder,

. RS2: right simplex and gradient relaxation,

. AS: research of an admissible simplex.

First, we can see that our enclosure methods are no t automatically better thanclassical natural inclusion extension of the function. However, when the width of the boxes decreases, our algorithms are more efficient and in some cases, theyare very accurate and they give the true bounds (method RS2 over box X ).

6 Conclusion

The methods developed in this paper, give a lower bound and/or an upper boundover a box X for a diff erentiable multivariate function, and therefore may be used as an evaluation function, as well as enclosure methods of a global optimum.For a function dened over a box X of IR , one estimation of a minimum, for example, requires point evaluations in n + 1 vertices of the box X and an intervalevaluation of the partial derivatives.

Efficiency of algorithms is obtained by suitable choices of these vertices.Numerical results are very satisfactory on unconstrained problems; the rs t

results on optimization problems with constraints are promising.

References 1. G. ALEFEL D Bounding the Slope of Polynomial Operators and some Applica-

tions , Computing : 26 (227-237) - (1980). 2. K. DU, R. B. KEARFOTT The Cluster Problem in Multivariate Global Optimiza-

tion , Journal of Global Optimization : 10 (27-32)

-

(1996).

3. E. HANSEN Global optimization Using Interval Analysis , MARCEL DEKKER,Inc. 270 Madison Avenue, New York, New York 10016 - (1992).

4. K. ICHIDA, Y. FUJII An Interval Arithmetic Method for Global Optimization ,Computing : 23 (85-97) - (1979).

5. P. HANSEN, B. JAUMARD, J. XIONG Decomposition and Interval Ari thmetic

Applied to Global Minimization of Polynomial and Rational Functions , Journal of Global Optimization : 3 (421-437)

- (1992). 6. R. B. KEARFOTT Rigorous Global Search: Continuous Problems , Kluwer Aca-

demic Publishers, Dordrecht, Boston, London - (1996).

Boxes Methods

X 1 X 2 X 3 X 4 X 5 X 6

f ( X i )

NE

T1 RS1 RS2

AS

[3 ,29]

[17 ,33]

[55 ,55] [44 ,82]

[45 .3,77 .4] [32 .2,55 .7]

[1.6,20]

[12 ,28]

[24 ,32] [7.4,28 .6] [7.4,28 .6] [7.8,28 .1]

[0.1,20]

[7 ,26]

[13 .5,27 .5] [2 ,23] [2 ,23]

[2.2,22 .8]

[1.6,16]

[11 ,24]

[21 ,27] [7.2,24 .8] [7.2,24 .8] [7.6,24 .4]

[1.6,16]

[9 ,21]

[19 .1,24 .9] [6 ,22]

[1.6,19] [6 ,22]

[4,19]

[1.5,24 .5]

[1.75 ,22 .3] [3.8,19.3]

[4,19] [3.8,19.3]

i i

6

n


15/16


16/16

merical Analysis, T. 326, S global dune fonction di ff

7. J.L. LAGOUANELLE, F. MESSINE Algorithme dencadremen t de loptimu m erentiable , Compte Rendu `a lAcademi e des Sciences, Nu-

erie I , (1998).

8. R.H. MLADINEO An Algor ithm for Finding the Global Maximum of a Multi-modal, Multivariate Function , Mathematical Programming : 34 (188-200) - (1986). 9. R. E. MOORE Interval Analysis , Prentice Hall, Inc. Englewood Cliff s, N.J. -

(1966). 10. H. RATSCHEK, J. ROKNE New Computer Methods for Global optimization , EL-

LIS HORWOOD LIMITED Market Cross House, Cooper Street, Chichester, West Sussex, PO19 1EB, England - (1988).

11. V. VISWESWARAN, C.A. FLOUDAS Uncontrained and Constrained Global Op-timization of Polynomial Function i n one Variable ,JournalofGlobalOptimization: 2 (73-99) - (1992).

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