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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
Mitchell Math Models of OR: Standard Form 2 / 23
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.
Mitchell Math Models of OR: Standard Form 3 / 23
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.
Mitchell Math Models of OR: Standard Form 3 / 23
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.
Mitchell Math Models of OR: Standard Form 3 / 23
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.
Mitchell Math Models of OR: Standard Form 4 / 23
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.
Mitchell Math Models of OR: Standard Form 4 / 23
Notation
Outline
1 Introduction
2 Notation
3 Reformulating any LOP into standard formMaximization problemsInequality constraintsNonpositive variablesFree variables
4 An example
Mitchell Math Models of OR: Standard Form 5 / 23
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.
Mitchell Math Models of OR: Standard Form 6 / 23
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
.
Mitchell Math Models of OR: Standard Form 7 / 23
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
Mitchell Math Models of OR: Standard Form 8 / 23
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.
Mitchell Math Models of OR: Standard Form 9 / 23
Reformulating any LOP into standard form
Outline
1 Introduction
2 Notation
3 Reformulating any LOP into standard formMaximization problemsInequality constraintsNonpositive variablesFree variables
4 An example
Mitchell Math Models of OR: Standard Form 10 / 23
Reformulating any LOP into standard form Maximization problems
Outline
1 Introduction
2 Notation
3 Reformulating any LOP into standard formMaximization problemsInequality constraintsNonpositive variablesFree variables
4 An example
Mitchell Math Models of OR: Standard Form 11 / 23
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.
Mitchell Math Models of OR: Standard Form 12 / 23
Reformulating any LOP into standard form Inequality constraints
Outline
1 Introduction
2 Notation
3 Reformulating any LOP into standard formMaximization problemsInequality constraintsNonpositive variablesFree variables
4 An example
Mitchell Math Models of OR: Standard Form 13 / 23
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
Mitchell Math Models of OR: Standard Form 14 / 23
Reformulating any LOP into standard form Nonpositive variables
Outline
1 Introduction
2 Notation
3 Reformulating any LOP into standard formMaximization problemsInequality constraintsNonpositive variablesFree variables
4 An example
Mitchell Math Models of OR: Standard Form 15 / 23
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.
Mitchell Math Models of OR: Standard Form 16 / 23
Reformulating any LOP into standard form Free variables
Outline
1 Introduction
2 Notation
3 Reformulating any LOP into standard formMaximization problemsInequality constraintsNonpositive variablesFree variables
4 An example
Mitchell Math Models of OR: Standard Form 17 / 23
Reformulating any LOP into standard form Free variables
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.
Mitchell Math Models of OR: Standard Form 18 / 23
Reformulating any LOP into standard form Free 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).
Mitchell Math Models of OR: Standard Form 19 / 23
Reformulating any LOP into standard form Free variables
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
Mitchell Math Models of OR: Standard Form 20 / 23
An example
Outline
1 Introduction
2 Notation
3 Reformulating any LOP into standard formMaximization problemsInequality constraintsNonpositive variablesFree variables
4 An example
Mitchell Math Models of OR: Standard Form 21 / 23
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.
Mitchell Math Models of OR: Standard Form 22 / 23
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.
Mitchell Math Models of OR: Standard Form 22 / 23
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.
Mitchell Math Models of OR: Standard Form 22 / 23
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.
Mitchell Math Models of OR: Standard Form 22 / 23
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:
maxx∈IR3 3x1 + 5x2 − 2x3subject to 2x1 + x2 + x3 ≥ 2
3x1 − x2 + 2x3 = 11x1 − 4x2 + 3x3 ≤ 20
x1 free, x2 ≤ 0, x3 ≥ 0
Mitchell Math Models of OR: Standard Form 23 / 23
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