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ENGINEERING OPTIMIZATIONMethods and Applications

A. Ravindran, K. M. Ragsdell, G. V. Reklaitis

Book Review

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Chapter 9: Direction Generation Methods

Based on Linearization

Part 1: Ferhat Dikbiyik

Part 2:Mohammad F. Habib

Review Session

July 30, 2010

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The Linearization-based algorithms in Ch. 8

LP solution techniques to specify the sequence ofintermediate solution points.

)(tx

The linearized

subproblem atthis point is

updated

The exact location

of next iterate isdetermined by LP

)1( tx

The linearized subproblem cannot be expected to

give a very good estimate of either boundaries ofthe feasible solution region or the contours of the

objective function

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GoodDirection Search

Rather than relying on the admittedly inaccuratelinearization to define the precise location of a

point, it is more realistic to utilize the linear

approximations only to determine a locally good

direction for search.

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Outline

9.1 Method of Feasible Directions

9.2 Simplex Extensions for LinearlyConstrained Problems

9.3 Generalized Reduced Gradient Method

9.4 Design Application

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G. Zoutendijk

Mehtods of Feasible

Directions,

Elsevier, Amsterdam,

1960

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Preliminaries

Source: Dr. Muhammad Al-Slamah, Industrial Engineering, KFUPM

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Preliminaries


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Suppose that is a starting point that satisfies all

constraints.

and suppose that a certain subset of these constraints

are binding at .

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Suppose is a feasible point

Definex as

The first order Taylor approximation off(x) is given by

In order for , we have to have

A direction satisfying this relationship is called adescent

direction

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This relationship dictates the angle between dand

to be greater than 90 and less than 270 .

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The first order Taylor approximation forconstraints

And with assumption

(because its binding)

In order forx to be a feasible, hence

Any direction dsatisfying this relationship

called afeasible direction

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This relationship dictates the angle between dand

has to be between 0 and than 90 .

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In order forx to solve the inequality constrained problem,

the direction dhas to be both a descent and feasible solution.

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Zoutendijks basic idea is at each stage ofiteration to determine a vector dthat will be both

a feasible direction and a descent direction. This

is accomplished numerically by finding a

normalized direction vector dand a scalarparameter > 0 such that

and is as large as possible.

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9.1.1 Basic Algorithm

The active constraint set is defined as

for some small

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Step 1. Solve the linear programming problem

Label the solution and

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Step 2. If the iteration terminates,since no further improvement is possible.Otherwise, determine

If no exists, set

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Step 3. Find such that

Set and continue.

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Example 9.1

, since g_1 is the only bindingconstraint.

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Example 9.1

We must search along the ray

to find the point at which boundary of feasible region is intersected

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Example 9.1

Since

is positive for all 0 and is not violated as is increased. Todetermine the point at which will be intersected, we solve

Finally, we search on over range todetermine the optimum of

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Example 9.1

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9.1.2 Active Constraint Sets and Jamming

Example 9.2

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9.1.2 Active Constraint Sets and Jamming

The active constraint set used in the basic formof feasible direction algorithm, namely,

cannot only slow down the process of iterations

but also lead to convergence to points that arenot Kuhn-Tucker points.

This type of false convergence is known as

jamming

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9.1.2.1 -Perturbation Method

1. At iteration point and with given ,define and carry out step 1 of the basic

algorithm.

2. Modify step 2 with the following: If ,

set and continue. However, if, set and proceed with line search of

the basic method. If , then a Kuhn-

Tucker point has been found.

With this modification, it is efficient to set rather looselyinitially so as to include the constraints in a larger neighborhood

of the point . Then, as the iterations proceed, the size of the

neighborhood will be reduced only when it is found to be

necessary.

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9.1.2.2 Topkis-Veinott Variant

This approach simply dispense with the activeconstraint concept altogether and redefine the

direction-finding subproblem as follows:

If the constraint loose at , then the selection ofdis less affected by

constraintj, because the positive constraint value will counterbalance the

effect of the gradient term. This ensures that no sudden changes are

introduced in the search direction.

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9.2 Simplex Extensions for Linearly Constrained

Problems

At a given point, the number of directions thatare both descent and feasible directions isgenerally infinite.

In the case of linear programs, the generation ofsearch directions was simplified by changing

one variable at a time; feasibility was ensured bychecking sign restrictions, and descent wasensured by selecting a variable with negativerelative-cost coefficient.

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9.2.1 Convex Simplex Method

M rows N components

Given a feasible point thex variable is partitioned intotwo sets:

the basic variables , which are all positive

the nonbasic variables , which are all zero

anMvector

an N-Mvector

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The relative-cost coefficients

The nonbasic variable to enter is selected by finding

such that

The basic variable to leave the basis is selected using the

minimum-ratio rule. That is, we find rsuch that

elements of matrix

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The new feasible solution

and all other variables zero. At this point, the

variables and are relabeled. Since an

exchange will have occurred,will be redefined. The matrix is recomputed

and another cycle of iterations is begun.

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The application of same algorithm to linearizedform of a non-linear objective function:

The relative-cost factor:

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Example 9.4

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Example 9.4

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Example 9.4

The relative-cost factor:

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Example 9.4

The nonbasic variable to enter will be , since

The basic variable to leave will be , since

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Example 9.4

The new point is thus

A line search between and is now required

to locate minimum of . Note that

remains at 0, while changes as given by

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Example 9.4

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Convex Simplex Algorithm

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Convex Simplex Algorithm

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9.2.2 Reduced Gradient Method

The nonbasic variable direction vector:

This definition ensures that when for all

i, the Kuhn-Tucker conditions are satisfied.

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9.2.2 Reduced Gradient Method

In the first case, the limiting value will be given by

If all , then set

In the second case,

If all , then set

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Reduced Gradient Algorithm

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