OD Nonlinear Programming LARGE 2010

7/24/2019 OD Nonlinear Programming LARGE 2010

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NONLINEARPROGRAMMING


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Nonlinear Programming

Linear programming has a fundamental role in OR.

In linear programmingall it s funct ions(objective

function and constraint functions) are linear.

This assumption frequently does not hold, and

nonlinear programming problems are formulated:

Findx = (x1,x2,...,xn) to

Maximizef(x)

subject to

gi(x) bi , for i = 1, 2, ...,mandx0

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Nonlinear Programming

There are many types of nonlinear programming

problems, depending onf(x) andgi(x). Different algorithms are used for different types.

Some problems can be solved very efficiently, whilst

others, even small, can be very difficult.

Nonlinear programming is a particularly large subject

(all the animals that are not elephants).

Only some important types will be dealt with here.

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Application: product-mix problem

Inproduct-mixproblems (as Wyndor Glass Co.) the

goal is to determine optimal mix of production levels. Sometimesprice elast icit yis present: the amount of

sold product has an inverse relation to price charged:

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Price elasticity

p(x) is the price required to sellx units.

c is the unit cost for producing and distributingproduct.

Profit from producing and sellingx is:

P(x) =xp(x) cx

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Product-mix problem

If each product has a similar profit function, overall

objective function is

Other nonlinearity:marginal costvary with productionlevel.

It may decrease when production level is increased due

to thelearning-curve effect.

It may increase due to overtime or more expensiveproduction facilities when production increase.

335

1

( ) ( )n

j j

j

f P x

x


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Application: transportation problem

Determine optimal plan for shipping goods from

various sources to various destinations (see P&TCompany problem).

Cost per unit shippedmay not be fixed.Volume

discountsare sometimes available for large shipments.

Marginal costcan have a pattern like in the figure.

Cost of shippingx units is a piecewise linear function

C(x), with slope equal to the marginal cost.

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Volume discounts on shipping costs

Marginal cost Cost of shipping

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Transportation problem

If each combination of source and destination has a

similar shipping cost function, so that cost of shippingxij units from sourcei (i = 1, 2, ...,m)

to destinationj (j = 1, 2, ...,n) is given by a nonlinear

functionCij(xij).

The overall objective function is

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1 1

Minimize ( ) ( )m n

ij ij

i j

f C x

x


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Graphical illustration

Example: Wyndor Glass Co. problem with constraint

339

2 2

1 29 5 216x x


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Graphical illustration

Example: Wyndor Glass Co. problem with objective

function

340

2 21 1 2 2126 9 128 13Z x x x x


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Example: Wyndor Glass Co. (3)

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Global and local optimum

Example:f(x) with three local maxima (x = 0, 2,4), and

three local minima (x = 1, 3,5). Global?

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Guaranteed local maximum

Global maximum when:

Function always curving downward is a concavefunction (concave downward).

Function always curving upward is a convex function

(concave upward).

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2

2

( )0, for all

f x

x


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Guaranteed local optimum

Nonlinear programming with no constraints and

concaveobjective function, a local maximum is theglobal maximum.

Nonlinear programming with no constraints and

convexobjective function, a local minimum is the

global minimum.

With constraints, this guarantee still holds if the

feasible regionis a convex set.

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Ex: Wyndor Glass with one concavegi(x)

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Types of NP problems

Unconstrained Optimization: no constraints

necessarycondition for a solutionx* =x to be optimal:

whenf(x) is aconcavefunction this condition is

sufficient.

whenxj has a constraintxj0, sufficient condition

changes to:

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Maximize ( )f x

*( ) 0 at , for 1, 2, ,j

fj n

x

xx x

* *

* *

0 at , if 0( )

0 at , if 0

j

jj

xf

xx

x xx

x x


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Example: nonnegative constraint

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Linearly Constrained Optimization

All constraints are linear and objective function is nonlinear.

Special case: Quadratic Programming

Objective function isquadratic.

Many applications, e.g. portfolio selection, predictive control

Convex Programming assumptions for maximization

f(x) is aconcavefunction.

Eachgi(x) is aconvexfunction.

For a minimization problem,f(x) must be a convex function.

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Separable Programming convex programming where

f(x) andgi(x) are separable functions.

A separable function is a function where each term

involves only a single variable:

Nonconvex Programming: local optimum is not

assured to be a global optimum.

349

1( ) ( )

n

j j

jf f x

x


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Geometric Programming is applied to engineering

design as well as economics and statistics problems. Objective function and constraint functions are of the

form:

ci andaij are typically physical constraints.

When they are all strictly positive, functions are

generalized positive polynomials (posynomials), and a

convex programming algorithm can be applied.

350

1 2

1 21( ) ( ), where ( )

i i in

Na a a

i i i nig c P P x x x x x x


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Fractional Programming

maximizesf1(x) and minimizesf2(x).

whenf1(x) andf2(x) are linear:

can be transformed into a linear programming problem.

351

0

0

( ) c

fd

cxx

dx

1

2

( )Maximize ( )

( )

ff

f xx

x


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One-variable unconstrained optimization

Methods for solving unconstrained optimization with

only one variable (n = 1), where the differentiablefunctionf(x) isconcave.

Necessary and sufficientcondition for optimum:

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*( ) 0 at .f x

x xx

l i h i i i bl


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Solving the optimization problem

Iff(x) is not simple, problem cannot be solved

analytically. If not, search procedures can solve the problem

numerically.

We will describe two methods:

Bisection method

Newtons method

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i i h d


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Bisection method

We know that:

If derivative ofxisposit ive,x is alower boundofx*.

If derivative ofxisnegative,x is anupper boundofx*

.

354

*( ) 0 if ,df x

x xdx

*( ) 0 if ,df x

x xdx

*( ) 0 if .df x

x xdx

Bi i h d


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Bisection method

Having:

In the bisection method, new trial solution is the

midpont between the two current bounds.

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*

*

*

= current trial solution,

= current lower bound on ,

= current upper bound on ,

= error tolerance for .

x

x x

x x

x

Al i h f h Bi i M h d


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Algorithm of the Bisection Method

Initialization:Select. Find initial upper and lower

bounds. Select initial trial as: Iteration:

1. Evaluate

2. If

3. If

4. Select a new

Stopping rule:If stop. Otherwise, go to 1.

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2x xx

*( ) at ,df x

x xdx

( )

0, reset ,df x

x xdx

( ) 0, reset ,

df xx x

dx

2

x xx

2x x

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Example

Maximize

357

4 6 ( ) 12 3 2f x x x x

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Solution

First two derivatives:

358

3 5( ) 12(1 )df x

x x

dx

22 4

2

( ) 12(3 5 )

d f xx x

dx

Iteration x )/ x x x Newx x)

0 0 2 1 7.0000

1 12 0 1 0.5 5.7812

2 +10.12 0.5 1 0.75 7.6948

3 +4.09 0.75 1 0.875 7.8439

4 2.19 0.75 0.875 0.8125 7.8672

5 +1.31 0.8125 0.875 0.84375 7.8829

6 0.34 0.8125 0.84375 0.828125 7.8815

7 +0.51 0.828125 0.84375 0.8359375 7.8839

S l ti


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Solution

x* 0.836

0.828125


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Newtons method

This method approximatef(x) in neighborhood of

current trial solution by a quadratic function. This quadratic approximation uses Taylor series

truncated after second derivative term:

Maximized by settingf (xi+1) equal to zero (xi,f(xi) and

f (xi) are constants):

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2

1 1 1

( )( ) ( ) ( )( ) ( )

2

i

i i i i i i i

f xf x f x f x x x x x

1 1( ) ( ) ( )( ) 0i i i i if x f x f x x x

1

( )

( )i

i i

i

f xx x

f x

Al ith f N t M th d


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Algorithm of Newtons Method

Initialization:Select. Find initial trial solutionxi by

inspection. Seti = 1. Iterationi:

1.

2. Set

Stopping rule:If stop;xi+1 is optimal.

Otherwise,i =i + 1 (another iteration).

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Calculate ( ) and ( ).i if x f x

1

( ) .

( )i

i i

i

f xx x

f x

1 ,i ix x

E l


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Example

Maximize again

New solution is given by:

Selectingx1 = 1, and= 0.00001,

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4 6 ( ) 12 3 2f x x x x

3 5 3 5

1 2 4 2 4

( ) 12(1 ) 1

( ) 12(3 5 ) 3 5i

i i i i

i

f x x x x xx x x x

f x x x x x

Iteration i x

i

x

i

) x

i

) x

i

) x

i

+1

1 1 7 12 96 0.875

2 0.875 7.8439 2.1940 62.733 0.84003

3 0.84003 7.8838 0.1325 55.279 0.83763

4 0.83763 7.8839 0.0006 54.790 0.83762

M lti i bl t i d ti i ti


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Multivariable unconstrained optimization

Problem: maximizing a concavefunctionf(x) of

multiple variables x = (x1,x2,...,xn) with no constraints. Necessary and sufficient condition for optimality:

partial derivatives equal to zero.

No analytical solution numerical searchprocedure

must be used.

One of these is thegradient search procedure:

It identifies the direction that maximizes the rate at

whichf(x) is increased.

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Gradient search procedure


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Use values of partial derivatives to select the specific

direction to move, using the gradient. Gradient of a pointx =x is thevectorwithpartial

derivativesevaluated atx =x:

Moves in the direction of this gradient untilf(x) stops

increasing. Each iteration changes the trial solutionx:

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1 2( ) , , , at

n

f f f

f x x x

x x x

*Next ( )t f x x x



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wheret* is the value oftthatmaximizesf(x+tf(x)):

The functionf(x +tf(x)) is simplyf(x) where:

Iterations continue untilf(x) = 0 withtolerance:

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, for 1,2, , .j

f j nx

, for 1, 2, ,j jj

fx x t j nx

x x

*

0 ( ( )) max ( ( ))

tf t f f t f

x x x x

Summary of gradient search procedure


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Init ializat ion:Selectand any initial trial solutionx.

Go to stopping rule.Iterat ion:

1. Expressf(x +tf(x)) as a function oftby setting

and substitute these expressions intof(x).

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, for 1,2, ,j j

j

fx x t j n

x

x x



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Iterat ion (concl.):

2. Use search procedure to findt=t* that maximizesf(x +tf(x)) overt0.

3. Resetx =x +t* f(x). Go to stopping rule.

Stopping rule:Evaluatef(x) atx =x. Check if:

If so, stop with currentx as the approximation ofx*.

Otherwise, perform another iteration.

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, for 1,2, , .j

fj n

x

Example


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Example

Maximize

Thus,

Verify thatf(x) is concave (see Appendix 2 of Hilliers

book).

Suppose thatx = (0, 0) is initial trial solution. Thus,

368

2 2

1 2 2 1 2( ) 2 2 2 .f x x x x x x

2 1

1

2 2f

x xx

1 2

2

2 2 4f

x xx

(0, 0) (0, 2)f

Example (2)


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Example (2)

Iterat ion 1:Step 1 sets

by substituting these expressions intof(x):

Because

369

1

2

0 (0) 00 (2) 2

x t

x t t

2 2

2

( ( )) (0,2 )2(0)(2 ) 2(2 ) 0 2(2 )

4 8

f t f f tt t t

t t

x x

* 2

0 0 (0, 2 ) max (0, 2 ) max {4 8 }

t tf t f t f t t

Example (3)


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Example (3)

and

it follows that

so

This completes first iteration. For new trial gradient is:

370

24 8 4 16 0d

t t tdt

* 1

4t

1 1

Reset (0,0) (0, 2) 0,4 2

x

10, (1,0)

2

f

Example (4)


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Example (4)

As < 1,Iterat ion 2:

so

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1 10, (1,0) ,2 2

t t

x

2

2

2

1 1 ( ( )) 0 , 0 ,

2 21 1 1

(2 ) 2 22 2 2

1

2

f t f f t t f t

t t

t t

x x

* 2

0 0 (0,2 ) max (0,2 ) max {4 8 }

t tf t f t f t t

Example (5)


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Example (5)

Because

and

then

so

This completes second iteration. See figure.

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* 2

0 0

1 1 1 , max , max

2 2 2t t

f t f t t t

2 1 1 2 0

2

dt t t

dt

* 1

2t

1 1 1 1Reset 0, (1,0) ,

2 2 2 2

x

Illustration of example


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Illustration of example

Optimal solution is (1, 1), as f(1, 1) = (0, 0)

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Newtons method


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Newton s method

It is aquadratic approximat ionof objective function

f(x). When objective function isconcaveand its gradient

f(x) are written ascolumn vectors,

The solutionx that maximizes the approximating

quadratic function is:

where2f(x) is thennHessian matrix.

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2 1[ ( )] ( ),f f x x x x

Newtons method


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Newton s method

Theinverseof the Hessian matrix is commonly

approximated in various ways. These approximations are referred to as quasi-Newton

methods (orvariable metr ic methods).

Recall that this topic was mentioned in the discipline

Intelligent Systems, e.g. in neural network learning.

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Conditions for optimality


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Conditions for optimality

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Problem

Necessary conditions

for optimality

Also sufficient if:

One-variableunconstrained

f(x) concave

Multivariableunconstrained f(x) concave

Constrained, nonnegativeconstraints only

f(x) concave

General constrainedproblem Karush-Kuhn-Tuckerconditions

f(x) concave and gi(x)

convex (j = 1, 2,...,n)

Karush-Kuhn-Tucker conditions


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Theorem: Assume thatf(x),g1(x),g2(x),...,gm(x) are

differentiablefunctions satisfying regularityconditions. Then

x = (x1*,x1

*,...,xn*)

can be anopt imal solutionfor the NP problem if

there arem numbersu1,u2,...,um such thatalltheKKT condit ionsare satisfied:

1.

2.

377

1*

*

1

0

at , for 1, 2 , .

0

mi

i

ij j

mi

j i

ij j

gfu

x xj n

gfx ux x

x x



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3.

4.

5.

6.

Conditions 2. and 4. require that one of the twoquantities must be zero.

Thus, conditions 3. and 4. can be combined:

(3,4)

378

*

*

( ) 0for 1,2, , .

[ ( ) ] 0

i i

i i i

g bj m

u g b

x

x

* 0, for 1, 2, , .j

x j n

0, for 1,2, , .iu j m

*( ) 0

(or 0, if 0), for 1, 2, , .

i i

i

g b

u j m

x



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Similarly, conditions 1. and 2. can be combined:

(1,2)

Corollary: assume thatf(x) isconcaveandg1(x),g2(x),...,gm(x) areconvexfunctions, where all

functions satisfy the regularity conditions. Then,

x = (x1*,x1

*,...,xn*) is anopt imal solutionif and only if

all the conditions of the theorem are satisfied.

379

1*

0

(or 0 if 0), for 1,2 , .

m ii

ij j

j

gf ux x

x j n

Example


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Example

Thus,m = 1, andg1(x) = 2x1 +x2 is convex.

Further,f(x) is concave (check it!).

Thus, the KKT conditions gives conditions to find anoptimal solution.

380

1 2Maximize ( ) ln( 1)f x x x

1 20, 0x x

subject to

1 22 3x x and

Example: KKT conditions


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Example: KKT conditions

1. (j= 1)

2. (j= 1)

1. (j= 2)

2. (j= 2)3.

4.

5.

6.

381

1

1

12 0

1u

x

1 1

1

12 0

1x u

x

11 0u

2 11 0x u 1 22 3 0x x

1 1 2(2 3) 0u x x

1 2

0, 0x x

1 0u

Example: solving KKT conditions


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Example: solving KKT conditions

From condition 1 (j= 2)u11.x10 from condition 5.

Therefore,

Therefore,x1 = 0, from condition 2 (j= 1).

u10 implies that 2x1 +x2 3 = 0 from condition 4.

Two previous steps implies thatx2 = 3.

x20 implies thatu1 = 1 from condition 2 (j= 2).

No conditions are violated forx1 = 0,x2 = 3,u1 = 1.

Consequentlyx*

= (0,3).

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1

1

1 2 0.1

ux

Quadratic Programming


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Quadratic Programming

Objective function can be expressed as:

383

1Maximize ( ) 2

Tf x cx x Qx

subject to

, and Ax b x 0

1 1 1

1 1( )

2 2

n n nT

j j ij i j

j i j

f c x q x x

x cx x Qx

Example


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Example

In this case,

384

2 2

1 2 1 2 1 2Maximize ( ) 15 30 4 2 4f x x x x x x xsubject to

1 2 1 22 30, and 0, 0x x x x

[15 30]c

[1 2]A

1

2

x

x

x

[30]b

4 4

4 8

Q

Solving QP problems


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Solving QP problems

Some KKT conditions for quadratic programming

problems can be transformed in equality constraintsby introducing slack variables.

KKT conditions can be condensed due to the

complementary variables, introducing

complementary constraints. As this, except for the complementary constraints, all

KKT conditions are linear programming constraints.

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Solving QP problems


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Solving QP problems

Using the previous properties, QP problems can be

solved using a modified simplex method. See example of a QP problem in Hilliers book (pages

580-581).

Excel, LINGO, LINDO, Matlab and MPL/CPLEX all can

solve quadratic programming problems.

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Separable Programming


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Separable Programming

Assumed thatf(x) is concave andgi(x) are convex.

f(x) is a (concave)piecewise linear function (see

example).

Ifgi(x) are linear, this problem can be reformulated as

an LP problem by using a separate variable for each

line segment.

The same technique can be used for nonlineargi(x).

387

1

( ) ( )n

j j

j

f f x

x

Example


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Example

388

Example


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Example

389

Convex Programming


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Convex Programming

Many algorithms can be used, falling into 3 categories:

1. Gradient algorithms, where the gradient searchprocedure is modified to avoid violating a constraint.

Example: generalized reduced gradient(GRG).

2. Sequential unconstrained algorithms, includepenalty

function and barrier function methods. Example: sequential unconstrained minimization

technique (SUMT).

3. Sequential approximation algorithms, include linearand quadratic approximation methods.

Example: Frank-Wolfe algorithm.

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Frank-Wolfe algorithm


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Frank Wolfe algorithm

It is a sequential linear approximation algorithm.

It replaces the objective functionf(x) by the first-orderTaylor expansion off(x) aroundx =x, namely:

Asf(x) and f(x)x have fixed values, they can be

dropped to give a linear objective function:

391

1

( )( ) ( ) ( ) ( ) ( )( )

n

j j

j j

ff f x x f f

x

x

x x x x x x

1

( )

( ) ( ) , where at .

n

j j jj j

f

g f c x c x

x

x x x x x

Frank-Wolfe algorithm


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g

Simplex method is applied to find a solutionxLP.

Then, chose the point that maximizes the nonlinearobjective function along the line segment.

This can be done using an one-variable unconstrained

optimization algorithm.

The algorithm continues the iterations until the stop

condition is satisfied.

392

Summary of Frank-Wolfe algorithm


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y g

Init ializat ion:Find feasible initial trial solutionx(0), e.g.

using LP to find initial BF solution, Setk= 1.Iterat ion k:

1. Forj = 1, 2, ...,n, evaluate

and setcj equal to this value.

2. Find optimal solution by solving LP problem:

393

( )LP

kx

( 1)( ) at .k

j

f

x

xx x

1

Maximize ( ) ,n

j j

j

g c x

x

subject toand Ax b x 0

Summary of Frank-Wolfe algorithm


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y g

3. For the variablet [0,1], set

so thath(t) gives value off(x) on line segment

between (wheret= 0) and (wheret= 1).

Use one-variable unconstrained optimization to

maximizeh(t) to findx(k).

Stopping rule:Ifx(k1) andx(k) are sufficiently close

stop.x(k) is the estimate of optimal solution.

Otherwise, resetk=k+ 1.

394

( 1) ( ) ( 1)

LP LP( ) ( ) for ( ),k k k

h t f t

x x x x x

( 1)kx

( )

LP

kx

Example


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p

As

the unconstrained maximumx = (2.5, 2) violates the

functional constraint.

395

2 2

1 1 2 2Maximize ( ) 5 8 2f x x x x x

subject to

1 2 1 23 2 6, and 0, 0x x x x

1 2

1 2

5 2 , 8 4f f

x xx x

Example (2)


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p ( )

Iterat ion 1:x = (0, 0) is feasible (initial trialx(0)).

Step 1 givesc1 = 5 andc2 = 8, sog(x) = 5x1 + 8x2. Step 2: solving graphically yields = (0, 3).

Step 3: points between (0, 0) and (0, 3) are:

This expression gives

396

(1)

LPx

1 2( , ) (0,0) [(0,3) (0,0)] for [0,1](0,3 )

x x t tt

2

2

( ) (0,3 ) 8(3 ) 2(3 )

24 18

h t f t t t

t t

Example (3)


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p ( )

the valuet=t* that maximizesh(t) is given by

sot* = 2/3. This results leads to the next trial solution,

(see figure):

Iterat ion 2:following the same procedure leads to the

next trial solutionx(2)

=(5/6, 7/6).

397

( ) 24 36 0dh t tdt

(1) 2(0,0) [(0,3) (0,0)]3

(0,2)

x

Example (4)


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p ( )

398

Example (5)


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p ( )

Figure shows next iterations.

Note that trial solutions alternate between twotrajectories that intersect at pointx = (1, 1.5).

This is the optimal solution (satisfy KKT conditions).

Usingquadraticinstead oflinearapproximations lead

to a much faster convergence.

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Sequential unconstrained minimization


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Main versions ofSUMT:

exterior-pointalgorithm: deals withinfeasiblesolutionsand apenaltyfunction,

interior-pointalgorithm: deals withfeasiblesolutions

and abarrierfunction.

Uses the advantage of solving unconstrainedproblems, which are much easier to solve.

Each unconstrained problem in the sequence chooses

a smaller and smaller value ofr, and solves forx to

400

Maximize ( ; ) ( ) ( )P r f rB x x x

SUMT


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B(x) is a barrier function with following properties:

1.B(x) issmallwhenx isfarfrom boundary of feasibleregion.

2.B(x) islargewhenx isclosefrom boundary of feasible

region.

3.B(x) as distance from boundary of feasible region

0.

Most common choice ofB(x):

401

1 1

1 1( )( )

m n

i ji i j

Bb g x

xx

Summary of SUMT


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Init ializat ion:Find feasible initial trial solutionx(0), not

on the boundary of feasible region. Setk= 1. Choosevalue forrand< 1 (e.g.r= 1 and= 0.01).

Iterat ion k:starting fromx(k1), apply a multivariable

unconstrained optimization procedure (e.g. gradient

search procedure) to find local maximumx(k) of

402

1 1

1 1( ; ) ( )

( )

m n

i ji i j

P r f r b g x

x x

x

Summary of SUMT


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Stopping rule:If change fromx(k1) tox(k) is very small

stop and usesx(k)

aslocal maximum. Otherwise, setk=k+ 1 andr= rfor other iteration.

SUMT can be extended for equality constraints.

Note that SUMT is quite sensitive tonumerical

instability, so it should be applied cautiously.

403

Example


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is convex, but is not concave. Initialization: (x1,x2) =x

(0) = (1, 1),r= 1 and= 0.01.

For each iteration:

404

1 2

Maximize ( )f x xx

subject to2

1 2 1 23, and 0, 0x x x x

2

1 1 2( )g x x x 1 2( )f x xx

1 2 2

1 2 1 2

1 1 1( ; ) 3

P r x x r x x x x

x

Example (2)


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Forr= 1, maximization leads tox(1) = (0.90, 1.36).

Table below shows convergence to (1, 2).

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k r x

1

k

)

x

2

k

)

0 1 1

1 1 0.90 1.36

2 102 0.987 1.925

3 104 0.998 1.993

1 2

Nonconvex Programming


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Assumptions of convex programming often fail.

Nonconvex programming problems can be much moredifficult to solve.

Dealing with non differentiable and non continuous

objective functions is usually very complicated.

LINDO, LINGO and MPL have efficient algorithms to

deal with these problems.

Simple problems can be solved using hill-climbing to

find alocal maximumseveral times.

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Nonconvex Programming


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An example is given in Hilliers book using Excel Solver

to solve simple problems. More difficult problems can use Evolutionary Solver.

It uses metaheuristics based on genetics, evolution

and survival of the fittest: a genetic algorithm.

We presented several well known metaheuristics to

solve this type of problems.

OD Nonlinear Programming LARGE 2010

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Transcript of OD Nonlinear Programming LARGE 2010