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Introduction to Linear and Nonlinear Programming

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OutlineOutline IntroductionIntroduction - types of problems- types of problems - size of problems- size of problems - iterative algorithms and - iterative algorithms and

convergenceconvergence Basic properties of Linear Basic properties of Linear

ProgramsPrograms - introduction- introduction - examples of linear programming- examples of linear programming problemsproblems

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Types of Problems Types of Problems

Three partsThree parts

- linear programming- linear programming

- unconstrained problems- unconstrained problems

- constrained problems- constrained problems The last two parts comprise the The last two parts comprise the

subjectsubject

of nonlinear programmingof nonlinear programming

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Linear ProgrammingLinear Programming It is characterized by linearIt is characterized by linear functions of functions of the unknowns; the the unknowns; the objectiveobjective is linear in is linear in the unknowns, and the the unknowns, and the constraintsconstraints are are linear equalities or linear inequalities.linear equalities or linear inequalities. Why are linear forms for objectives andWhy are linear forms for objectives and constraints so popular in problem constraints so popular in problem

formulation?formulation? - a great number of constraints and objectives that arise - a great number of constraints and objectives that arise

inin practice are indisputably linear (ex: budget constraint)practice are indisputably linear (ex: budget constraint) - they are often the least difficult to define - they are often the least difficult to define

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Unconstrained Unconstrained ProblemsProblems

Are unconstrained problems devoid of structuralAre unconstrained problems devoid of structural

properties as to preclude their applicability asproperties as to preclude their applicability as

useful models of meaningful problems?useful models of meaningful problems? - if the scope of a problem is broadened to the consideration of- if the scope of a problem is broadened to the consideration of

all relevant decision variables, there may then be no all relevant decision variables, there may then be no

constraintsconstraints

- many constrained problems are sometimes easily converted- many constrained problems are sometimes easily converted

to unconstrained problems to unconstrained problems

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Constrained ProblemsConstrained Problems

Many complex problems cannot be directlyMany complex problems cannot be directly

treated in its entirety accounting for all treated in its entirety accounting for all

possible choices, but instead must be possible choices, but instead must be decomposeddecomposed

into separate subproblemsinto separate subproblems ““Continuous variable programming”Continuous variable programming”

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Size of ProblemsSize of Problems

Three classes of problemsThree classes of problems - small scale: five or fewer unknowns and constraints- small scale: five or fewer unknowns and constraints

- intermediate scale: from five to a hundred - intermediate scale: from five to a hundred variablesvariables

- large scale: from a hundred to thousands variables- large scale: from a hundred to thousands variables

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Iterative Algorithms Iterative Algorithms and Convergenceand Convergence

Most algorithms designed to solve largeMost algorithms designed to solve large

optimization problems are iterative.optimization problems are iterative. For LP problems, the generated sequenceFor LP problems, the generated sequence

is of finite length, reaching the solutionis of finite length, reaching the solution

point exactly after a finite number of steps.point exactly after a finite number of steps. For non-LP problems, the sequence generally For non-LP problems, the sequence generally

does not ever exactly reach the solution does not ever exactly reach the solution point, but converges toward it. point, but converges toward it.

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Iterative Algorithms Iterative Algorithms and Convergence and Convergence

((cont’d)cont’d) Iterative algorithmsIterative algorithms

1. the creation of the algorithms themselves1. the creation of the algorithms themselves

2. the verification that a given algorithm will in fact2. the verification that a given algorithm will in fact

generate a sequence that converges to a solutiongenerate a sequence that converges to a solution

3. the rate at which the generated sequence of points3. the rate at which the generated sequence of points

converges to the solutionconverges to the solution Convergence rate theoryConvergence rate theory

1. the theory is, for the most part, extremely simple in 1. the theory is, for the most part, extremely simple in naturenature

2. a large class of seemingly distinct algorithms turns out 2. a large class of seemingly distinct algorithms turns out toto

have a common convergence rate have a common convergence rate

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LP Problems’ Standard LP Problems’ Standard FormForm

1 1 2 2 n

11 1 12 2 1 1

21 1 22 2 2 2

minimize c c ..........c

subject to ...........

...........

.

.

.

n

n n

n n

x x x

a x a x a x b

a x a x a x b

1 1 2 2

1 2

..........

and 0, 0,............ 0m m mn n m

n

a x a x a x b

x x x

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LP Problems’ Standard LP Problems’ Standard Form Form ((cont’d)cont’d)

Here x is an n-dimensional column vector,Here x is an n-dimensional column vector, is an n-dimensional row vector, A is anis an n-dimensional row vector, A is an m×n matrix, and b is an m-dimensionalm×n matrix, and b is an m-dimensional column vector.column vector.

Tc

Tmin imize c x

subject to Ax=b and x 0

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Example 1 (Slack Example 1 (Slack Variables)Variables)

1 1 2 2 n

11 1 12 2 1 1

21 1 22 2 2 2

minimize c c ..........c

subject to ...........

...........

.

.

.

n

n n

n n

x x x

a x a x a x b

a x a x a x b

1 1 2 2

1 2

..........

and 0, 0,............ 0m m mn n m

n

a x a x a x b

x x x

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Example 1 (Slack Example 1 (Slack Variables) Variables) ((cont’d)cont’d)

1 1 2 2 n

11 1 12 2 1 1 1

21 1 22 2 2 2 2

minimize c c ..........c

subject to ...........

...........

.

.

.

n

n n

n n

x x x

a x a x a x y b

a x a x a x y b

1 1 2 2

1 2

1 2

..........

and y 0, 0,............ 0

and 0, 0,............ 0

m m mn n m m

m

n

a x a x a x y b

y y

x x x

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Example 2 (Free Example 2 (Free Variables)Variables)

XX11 is free to take on either positive or is free to take on either positive or negative values.negative values. We then write XWe then write X11=U=U11-V-V11, where U, where U11≧0≧0 and Vand V11≧0.≧0. Substitute USubstitute U11-V-V11 for X for X11, then the , then the

linearitylinearity of the constraints is preserved and allof the constraints is preserved and all variables are now required to be variables are now required to be

nonnegative.nonnegative.

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The Diet ProblemThe Diet Problem

We assume that there are available at the marketWe assume that there are available at the market

nn different foods and that the different foods and that the iith food sells at a th food sells at a

price per unit. In addition there are price per unit. In addition there are mm basic basic

nutritional ingredients and, to achieve a balancednutritional ingredients and, to achieve a balanced

diet, each individual must receive at least unitsdiet, each individual must receive at least units

of the of the jjth nutrient per day. Finally, we assume that th nutrient per day. Finally, we assume that each unit of foodeach unit of food ii contains units of the contains units of the jjth th nutrient.nutrient.

ic

jb

jia

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The Diet Problem The Diet Problem ((cont’d)cont’d)

1 1 2 2 n

11 1 12 2 1 1

21 1 22 2 2 2

minimize c c ..........c

subject to ...........

...........

.

.

.

n

n n

n n

x x x

a x a x a x b

a x a x a x b

1 1 2 2

1 2

..........

and 0, 0,............ 0m m mn n m

n

a x a x a x b

x x x

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The Transportation The Transportation ProblemProblem

Quantities a1, a2,…, aQuantities a1, a2,…, amm, respectively, of a certain, respectively, of a certainproduct are to be shipped from each of m locationsproduct are to be shipped from each of m locationsand received in amounts b1, b2,…, band received in amounts b1, b2,…, bnn, respectively,, respectively,at each of n destinations. Associated with the at each of n destinations. Associated with the Shipping of a unit of product from origin i to Shipping of a unit of product from origin i to destination j is a unit shipping cost cdestination j is a unit shipping cost cijij. It is desired . It is desired

totodetermine the amounts xdetermine the amounts xij ij to be shipped between to be shipped between each origin-destination pair (i=1,2,…,m; j=1,2,…,n)each origin-destination pair (i=1,2,…,m; j=1,2,…,n)

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The Transportation The Transportation Problem Problem ((cont’d)cont’d)

ij

n

j=1

m

i=1

minimize

subject to for i=1,2,......,m

for j=1,2,......,n

0 for i=1,2,......m

ij ij

ij i

ij j

ij

c x

x a

x b

x

j=1,2,......n