Math Models of OR: Standard Formeaton.math.rpi.edu/.../notesMATP4700/lecture02/02A_standardformb… · Mitchell Math Models of OR: Standard Form 4 / 23. Introduction Two important

Math Models of OR: Standard Form

John E. Mitchell

Department of Mathematical SciencesRPI, Troy, NY 12180 USA

August 2018

Mitchell Math Models of OR: Standard Form 1 / 23

Introduction

Outline

1 Introduction

2 Notation

3 Reformulating any LOP into standard formMaximization problemsInequality constraintsNonpositive variablesFree variables

4 An example


Introduction

Standard formA general linear optimization problem has the form:

maximizeor a linear function of several variables in IRn

minimize

subject to linear inequality constraintsand / or

linear equality constraints

In order to make it easier to present algorithms and theoretical results,we work with a standard form:

a minimization problemwith equality constraints only,with all variables nonnegative.


Introduction



minimize






Introduction



minimize






Introduction

Two important points

1 Any linear program is equivalent to one in standard form, as wewill see shortly.

2 In practice: if you are using a computational package to solve alinear program, do not reformulate it into standard form. Theremay be some structure in the original formulation that can beexploited by algorithms, and which can be lost if the problem isreformulated.


Introduction

Two important points

1 Any linear program is equivalent to one in standard form, as wewill see shortly.

2 In practice: if you are using a computational package to solve alinear program, do not reformulate it into standard form. Theremay be some structure in the original formulation that can beexploited by algorithms, and which can be lost if the problem isreformulated.


Notation

Outline

1 Introduction

2 Notation


4 An example


Notation

Notation

We take our variables x ∈ IRn. We typically denote the objectivefunction coefficients as cj for j = 1, . . . ,n, so we have vector c ∈ IRn ofobjective function coefficients.

We have m linear equality constraints. We use bi to denote the righthand side of the i th constraint, so we have a vector b ∈ IRm of righthand side values.

Each constraint has a coefficient for each variable, and we use aij todenote the coefficient of xj in the i th constraint. The values aij form anm × n matrix A.

We also allow the objective function to contain a constant term d , ascalar.


Notation

A general problem in standard formThe standard form linear optimization problem has the form

minx∈IRn∑n

j=1 cjxj + dsubject to

∑nj=1 aijxj = bi , i = 1, . . . ,m

xj ≥ 0, j = 1, . . . ,n

For example, the standard form problem

minx∈IR3 3x1 − 2x2 + 4x3 − 6subject to 2x1 − x2 + 4x3 = 7

x1 + x2 + 2x3 = 5xj ≥ 0, j = 1, . . . ,3

has

c =

3−2

4

, b =

[75

], A =

[2 −1 41 1 2

], d = −6, with x =

x1x2x3

.


Notation

Matrix notationThe objective function is equal to the dot product or vector productbetween the vectors c and x .

We think of all vectors as column vectors, so we could also think of cand x as n × 1 matrices.

Then the dot product can be written using matrix notation:n∑

j=1

cjxj = cT x ,

where the superscript T denotes the transcript of a matrix.

The equality constraints can be written in matrix terms as Ax = b.Thus, we can represent our standard form in matrix terms as

minx∈IRn cT x + dsubject to Ax = b

x ≥ 0


Notation

Terminology

minx∈IRn cT x + dsubject to Ax = b

x ≥ 0

This is a linear optimization problem or LOP.

The older terminology is to call it a linear program or LP, where weuse the word “program” in the sense of developing a “plan”, not as acomputer program.


Reformulating any LOP into standard form

Outline

1 Introduction

2 Notation


4 An example


Reformulating any LOP into standard form Maximization problems

Outline

1 Introduction

2 Notation


4 An example


Reformulating any LOP into standard form Maximization problems

Maximization problems

Note that

max{cT x : Ax = b, x ≥ 0} = − min{−cT x : Ax = b, x ≥ 0}.

So can change the sign of the coefficients in the objective function.


Reformulating any LOP into standard form Inequality constraints

Outline

1 Introduction

2 Notation


4 An example


Reformulating any LOP into standard form Inequality constraints

Inequality constraintsWe can introduce slack variables. For example, the linearoptimization problem

minx∈IR3 3x1 + 5x2 − 2x3subject to 2x1 − x2 + x3 ≥ 2

3x1 + x2 + 2x3 = 11x1 + 4x2 + 3x3 ≤ 20

xj ≥ 0, j = 1, . . . ,3

is equivalent to the standard form problem

minx∈IR3 3x1 + 5x2 − 2x3subject to 2x1 − x2 + x3 − x4 = 2

3x1 + x2 + 2x3 = 11x1 + 4x2 + 3x3 + x5 = 20

xj ≥ 0, j = 1, . . . ,5


Reformulating any LOP into standard form Nonpositive variables

Outline

1 Introduction

2 Notation


4 An example


Reformulating any LOP into standard form Nonpositive variables

Nonpositive variables

If the variable xj is required to be no larger than zero, we can define anew variable equal to −xj , which must then be nonnegative.


Reformulating any LOP into standard form Free variables

Outline

1 Introduction

2 Notation


4 An example



Free variables

A free variable is one that has no restriction on its sign, so it can bepositive or negative, or zero.

Now, any number can be written as the difference of two nonnegativenumbers. For example,

5 = 5− 0, −7 = 2− 9.

We replace each free variable by the difference of two newnonnegative variables.



Example

minx∈IR3 3x1 + 5x2 − 2x3subject to 2x1 − x2 + x3 = 3

3x1 + x2 + 2x3 = 11x1 free, x2, x3 ≥ 0

is equivalent to the problem

minx∈IR4 3x4 − 3x5 + 5x2 − 2x3subject to 2x4 − 2x5 − x2 + x3 = 3

3x4 − 3x5 + x2 + 2x3 = 11xj ≥ 0, j = 2, . . . ,5

with the understanding that x1 = x4 − x5.

The feasible point x1 = 1, x2 = 2, x3 = 3 in the original formulationcorresponds to the point x4 = 1, x5 = 0, x2 = 2, x3 = 3(and also to, for example, x4 = 7, x5 = 6, x2 = 2, x3 = 3).



In matrix notation

Say we have a linear optimization problem in variables x ∈ IRn andy ∈ IRp:

minx∈IRn,y∈IRp cT x + gT ysubject to Ax + Hy = b

x free, y ≥ 0

We introduce nonnegative variables u, v ∈ IRn and get the equivalentproblem

minu∈IRn,v∈IRn,y∈IRp cT u − cT v + gT ysubject to Ax − Av + Hy = b

u, v , y ≥ 0


An example

Outline

1 Introduction

2 Notation


4 An example


An example

An exampleConsider the linear optimization problem

maxx∈IR3 3x1 + 5x2 − 2x3subject to 2x1 + x2 + x3 ≥ 2

3x1 − x2 + 2x3 = 11x1 − 4x2 + 3x3 ≤ 20

x1 free, x2 ≤ 0, x3 ≥ 0

Wechange the sign of the objective to min−3x1 − 5x2 + 2x3,split x1 = x4 − x5 for two new nonnegative variables x4 and x5,replace the nonpositive variable x2 by the nonnegative variablex6 = −x2,introduce slack variables x7 and x8 in the two inequalityconstraints.


An example



3x1 − x2 + 2x3 = 11x1 − 4x2 + 3x3 ≤ 20

x1 free, x2 ≤ 0, x3 ≥ 0



An example



3x1 − x2 + 2x3 = 11x1 − 4x2 + 3x3 ≤ 20

x1 free, x2 ≤ 0, x3 ≥ 0



An example



3x1 − x2 + 2x3 = 11x1 − 4x2 + 3x3 ≤ 20

x1 free, x2 ≤ 0, x3 ≥ 0



An example

Equivalent problemThis gives the equivalent problem

minx∈IR6 −3x4 + 3x5 + 5x6 + 2x3subject to 2x4 − 2x5 − x6 + x3 − x7 = 2

3x4 − 3x5 + x6 + 2x3 = 11x4 − x5 + 4x6 + 3x3 + x8 = 20

xj ≥ 0, j = 3, . . . ,8

Original was:


3x1 − x2 + 2x3 = 11x1 − 4x2 + 3x3 ≤ 20

x1 free, x2 ≤ 0, x3 ≥ 0


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Math Models of OR: Standard Formeaton.math.rpi.edu/.../notesMATP4700/lecture02/02A_standardformb… · Mitchell Math Models of OR: Standard Form 4 / 23. Introduction Two important