Math 407A: Linear Optimization
Lecture 4: LP Standard Form 2
2Author: James Burke, University of Washington
Lecture 4: LP Standard Form 3 Math 407A: Linear Optimization 1 / 27
1 LPs in Standard Form
2 Minimization → maximization
3 Linear equations to linear inequalities
4 Lower and upper bounded variables
5 Interval variable bounds
6 Free variable
7 Two Step Process to Standard Form
Lecture 4: LP Standard Form 4 Math 407A: Linear Optimization 2 / 27
LPs in Standard Form
We say that an LP is in standard form if its matrix representation has the form
max cT x It must be a maximization problem.
s.t. Ax ≤ b Only inequalities of the correct direction.
0 ≤ x All variables must be non-negative.
Lecture 4: LP Standard Form 5 Math 407A: Linear Optimization 3 / 27
LPs in Standard Form
We say that an LP is in standard form if its matrix representation has the form
max cT x
It must be a maximization problem.
s.t. Ax ≤ b
Only inequalities of the correct direction.
0 ≤ x
All variables must be non-negative.
5Author: James Burke, University of Washington
Lecture 4: LP Standard Form 5 Math 407A: Linear Optimization 3 / 27
LPs in Standard Form
We say that an LP is in standard form if its matrix representation has the form
max cT x It must be a maximization problem.
s.t. Ax ≤ b
Only inequalities of the correct direction.
0 ≤ x
All variables must be non-negative.
5Author: James Burke, University of Washington
Lecture 4: LP Standard Form 5 Math 407A: Linear Optimization 3 / 27
LPs in Standard Form
We say that an LP is in standard form if its matrix representation has the form
max cT x It must be a maximization problem.
s.t. Ax ≤ b Only inequalities of the correct direction.
0 ≤ x
All variables must be non-negative.
5Author: James Burke, University of Washington
Lecture 4: LP Standard Form 5 Math 407A: Linear Optimization 3 / 27
LPs in Standard Form
We say that an LP is in standard form if its matrix representation has the form
max cT x It must be a maximization problem.
s.t. Ax ≤ b Only inequalities of the correct direction.
0 ≤ x All variables must be non-negative.
5Author: James Burke, University of Washington
Lecture 4: LP Standard Form 6 Math 407A: Linear Optimization 3 / 27
Every LP can be Transformed to Standard Form
minimization → maximization
To transform a minimization problem to a maximization problem multiply theobjective function by −1.
linear inequalities
If an LP has an inequality constraint of the form
ai1x1 + ai2x2 + · · ·+ ainxn ≥ bi ,
it can be transformed to one in standard form by multiplying the inequalitythrough by −1 to get
−ai1x1 − ai2x2 − · · · − ainxn ≤ −bi .
Lecture 4: LP Standard Form 7 Math 407A: Linear Optimization 4 / 27
Every LP can be Transformed to Standard Form
minimization → maximization
To transform a minimization problem to a maximization problem multiply theobjective function by −1.
linear inequalities
If an LP has an inequality constraint of the form
ai1x1 + ai2x2 + · · ·+ ainxn ≥ bi ,
it can be transformed to one in standard form by multiplying the inequalitythrough by −1 to get
−ai1x1 − ai2x2 − · · · − ainxn ≤ −bi .
7Author: James Burke, University of Washington
Lecture 4: LP Standard Form 7 Math 407A: Linear Optimization 4 / 27
Every LP can be Transformed to Standard Form
minimization → maximization
To transform a minimization problem to a maximization problem multiply theobjective function by −1.
linear inequalities
If an LP has an inequality constraint of the form
ai1x1 + ai2x2 + · · ·+ ainxn ≥ bi ,
it can be transformed to one in standard form by multiplying the inequalitythrough by −1 to get
−ai1x1 − ai2x2 − · · · − ainxn ≤ −bi .
7Author: James Burke, University of Washington
Lecture 4: LP Standard Form 7 Math 407A: Linear Optimization 4 / 27
Every LP can be Transformed to Standard Form
minimization → maximization
To transform a minimization problem to a maximization problem multiply theobjective function by −1.
linear inequalities
If an LP has an inequality constraint of the form
ai1x1 + ai2x2 + · · ·+ ainxn ≥ bi ,
it can be transformed to one in standard form by multiplying the inequalitythrough by −1 to get
−ai1x1 − ai2x2 − · · · − ainxn ≤ −bi .
7Author: James Burke, University of Washington
Lecture 4: LP Standard Form 7 Math 407A: Linear Optimization 4 / 27
Every LP can be Transformed to Standard Form
minimization → maximization
To transform a minimization problem to a maximization problem multiply theobjective function by −1.
linear inequalities
If an LP has an inequality constraint of the form
ai1x1 + ai2x2 + · · ·+ ainxn ≥ bi ,
it can be transformed to one in standard form by multiplying the inequalitythrough by −1 to get
−ai1x1 − ai2x2 − · · · − ainxn ≤ −bi .
7Author: James Burke, University of Washington
Lecture 4: LP Standard Form 8 Math 407A: Linear Optimization 4 / 27
Every LP can be Transformed to Standard Form
linear equations
The linear equationai1xi + · · ·+ ainxn = bi
can be written as two linear inequalities
ai1x1 + · · ·+ ainxn ≤ bi
andai1x1 + · · ·+ ainxn ≥ bi .
or equivalentlyai1x1 + · · ·+ ainxn ≤ bi
−ai1x1 − · · · − ainxn ≤ −bi
Lecture 4: LP Standard Form 9 Math 407A: Linear Optimization 5 / 27
Every LP can be Transformed to Standard Form
linear equations
The linear equationai1xi + · · ·+ ainxn = bi
can be written as two linear inequalities
ai1x1 + · · ·+ ainxn ≤ bi
andai1x1 + · · ·+ ainxn ≥ bi .
or equivalentlyai1x1 + · · ·+ ainxn ≤ bi
−ai1x1 − · · · − ainxn ≤ −bi
9Author: James Burke, University of Washington
Lecture 4: LP Standard Form 10 Math 407A: Linear Optimization 5 / 27
Every LP can be Transformed to Standard Form
variables with lower bounds
If a variable xi has lower bound li which is not zero (li ≤ xi ) or equivalently,0 ≤ xi − li , one obtains a non-negative variable wi := xi − li yielding thesubstitution
xi = wi + li .
In this case, the bound li ≤ xi is equivalent to the bound 0 ≤ wi .
variables with upper bounds
If a variable xi has an upper bound ui (xi ≤ ui ), or equivalently, 0 ≤ ui − xi ,one obtains a non-negative variable wi := ui − xi yielding the substitution
xi = ui − wi .
In this case, the bound xi ≤ ui is equivalent to the bound 0 ≤ wi .
Lecture 4: LP Standard Form 11 Math 407A: Linear Optimization 6 / 27
Every LP can be Transformed to Standard Form
variables with lower bounds
If a variable xi has lower bound li which is not zero (li ≤ xi ) or equivalently,0 ≤ xi − li , one obtains a non-negative variable wi := xi − li yielding thesubstitution
xi = wi + li .
In this case, the bound li ≤ xi is equivalent to the bound 0 ≤ wi .
variables with upper bounds
If a variable xi has an upper bound ui (xi ≤ ui ), or equivalently, 0 ≤ ui − xi ,one obtains a non-negative variable wi := ui − xi yielding the substitution
xi = ui − wi .
In this case, the bound xi ≤ ui is equivalent to the bound 0 ≤ wi .
11Author: James Burke, University of Washington
Lecture 4: LP Standard Form 11 Math 407A: Linear Optimization 6 / 27
Every LP can be Transformed to Standard Form
variables with lower bounds
If a variable xi has lower bound li which is not zero (li ≤ xi ) or equivalently,0 ≤ xi − li , one obtains a non-negative variable wi := xi − li yielding thesubstitution
xi = wi + li .
In this case, the bound li ≤ xi is equivalent to the bound 0 ≤ wi .
variables with upper bounds
If a variable xi has an upper bound ui (xi ≤ ui ), or equivalently, 0 ≤ ui − xi ,one obtains a non-negative variable wi := ui − xi yielding the substitution
xi = ui − wi .
In this case, the bound xi ≤ ui is equivalent to the bound 0 ≤ wi .
11Author: James Burke, University of Washington
Lecture 4: LP Standard Form 11 Math 407A: Linear Optimization 6 / 27
Every LP can be Transformed to Standard Form
variables with lower bounds
If a variable xi has lower bound li which is not zero (li ≤ xi ) or equivalently,0 ≤ xi − li , one obtains a non-negative variable wi := xi − li yielding thesubstitution
xi = wi + li .
In this case, the bound li ≤ xi is equivalent to the bound 0 ≤ wi .
variables with upper bounds
If a variable xi has an upper bound ui (xi ≤ ui ), or equivalently, 0 ≤ ui − xi ,one obtains a non-negative variable wi := ui − xi yielding the substitution
xi = ui − wi .
In this case, the bound xi ≤ ui is equivalent to the bound 0 ≤ wi .
11Author: James Burke, University of Washington
Lecture 4: LP Standard Form 12 Math 407A: Linear Optimization 6 / 27
Every LP can be Transformed to Standard Form
variables with interval bounds
An interval bound of the form li ≤ xi ≤ ui can be transformed into onenon-negativity constraint and one linear inequality constraint in standardform by making the substitution
xi = wi + li .
In this case, the bounds li ≤ xi ≤ ui are equivalent to the constraints
0 ≤ wi and wi ≤ ui − li .
Lecture 4: LP Standard Form 13 Math 407A: Linear Optimization 7 / 27
Every LP can be Transformed to Standard Form
variables with interval bounds
An interval bound of the form li ≤ xi ≤ ui can be transformed into onenon-negativity constraint and one linear inequality constraint in standardform by making the substitution
xi = wi + li .
In this case, the bounds li ≤ xi ≤ ui are equivalent to the constraints
0 ≤ wi and wi ≤ ui − li .
13Author: James Burke, University of Washington
Lecture 4: LP Standard Form 13 Math 407A: Linear Optimization 7 / 27
Every LP can be Transformed to Standard Form
variables with interval bounds
An interval bound of the form li ≤ xi ≤ ui can be transformed into onenon-negativity constraint and one linear inequality constraint in standardform by making the substitution
xi = wi + li .
In this case, the bounds li ≤ xi ≤ ui are equivalent to the constraints
0 ≤ wi and wi ≤ ui − li .
13Author: James Burke, University of Washington
Lecture 4: LP Standard Form 14 Math 407A: Linear Optimization 7 / 27
Every LP can be Transformed to Standard Form
free variables
Sometimes a variable is given without any bounds. Such variables are calledfree variables. To obtain standard form every free variable must be replacedby the difference of two non-negative variables. That is, if xi is free, then weget
xi = ui − vi
with 0 ≤ ui and 0 ≤ vi .
Lecture 4: LP Standard Form 15 Math 407A: Linear Optimization 8 / 27
Every LP can be Transformed to Standard Form
free variables
Sometimes a variable is given without any bounds. Such variables are calledfree variables. To obtain standard form every free variable must be replacedby the difference of two non-negative variables. That is, if xi is free, then weget
xi = ui − vi
with 0 ≤ ui and 0 ≤ vi .
15Author: James Burke, University of Washington
Lecture 4: LP Standard Form 16 Math 407A: Linear Optimization 8 / 27
Transformation to Standard Form
Put the following LP into standard form.
minimize 3x1 − x2subject to −x1 + 6x2 − x3 + x4 ≥ −3
7x2 + x4 = 5x3 + x4 ≤ 2
−1 ≤ x2, x3 ≤ 5, −2 ≤ x4 ≤ 2.
Lecture 4: LP Standard Form 17 Math 407A: Linear Optimization 9 / 27
Transformation to Standard Form
The hardest part of the translation to standard form, or at least the part mostsusceptible to error, is the replacement of existing variables with non-negativevariables.
Reduce errors by doing the transformation in two steps.
Step 1: Make all of the changes that do not involve a variable substitution.
Step 2: Make all of the variable substitutions.
Lecture 4: LP Standard Form 18 Math 407A: Linear Optimization 10 / 27
Transformation to Standard Form
The hardest part of the translation to standard form, or at least the part mostsusceptible to error, is the replacement of existing variables with non-negativevariables.
Reduce errors by doing the transformation in two steps.
Step 1: Make all of the changes that do not involve a variable substitution.
Step 2: Make all of the variable substitutions.
18Author: James Burke, University of Washington
Lecture 4: LP Standard Form 18 Math 407A: Linear Optimization 10 / 27
Transformation to Standard Form
The hardest part of the translation to standard form, or at least the part mostsusceptible to error, is the replacement of existing variables with non-negativevariables.
Reduce errors by doing the transformation in two steps.
Step 1: Make all of the changes that do not involve a variable substitution.
Step 2: Make all of the variable substitutions.
18Author: James Burke, University of Washington
Lecture 4: LP Standard Form 18 Math 407A: Linear Optimization 10 / 27
Transformation to Standard Form
The hardest part of the translation to standard form, or at least the part mostsusceptible to error, is the replacement of existing variables with non-negativevariables.
Reduce errors by doing the transformation in two steps.
Step 1: Make all of the changes that do not involve a variable substitution.
Step 2: Make all of the variable substitutions.
18Author: James Burke, University of Washington
Lecture 4: LP Standard Form 19 Math 407A: Linear Optimization 10 / 27
Must be a maximization problem
minimize 3x1 − x2subject to −x1 + 6x2 − x3 + x4 ≥ −3
7x2 + x4 = 5x3 + x4 ≤ 2
−1 ≤ x2, x3 ≤ 5, −2 ≤ x4 ≤ 2.
Minimization =⇒ maximization
maximize − 3x1 + x2.
Lecture 4: LP Standard Form 20 Math 407A: Linear Optimization 11 / 27
Must be a maximization problem
minimize 3x1 − x2subject to −x1 + 6x2 − x3 + x4 ≥ −3
7x2 + x4 = 5x3 + x4 ≤ 2
−1 ≤ x2, x3 ≤ 5, −2 ≤ x4 ≤ 2.
Minimization =⇒ maximization
maximize − 3x1 + x2.
20Author: James Burke, University of Washington
Lecture 4: LP Standard Form 20 Math 407A: Linear Optimization 11 / 27
Must be a maximization problem
minimize 3x1 − x2subject to −x1 + 6x2 − x3 + x4 ≥ −3
7x2 + x4 = 5x3 + x4 ≤ 2
−1 ≤ x2, x3 ≤ 5, −2 ≤ x4 ≤ 2.
Minimization =⇒ maximization
maximize − 3x1 + x2.
20Author: James Burke, University of Washington
Lecture 4: LP Standard Form 21 Math 407A: Linear Optimization 11 / 27
Inequalities must go the right way.
− x1 + 6x2 − x3 + x4 ≥ −3
becomes
x1 − 6x2 + x3 − x4 ≤ 3.
Lecture 4: LP Standard Form 22 Math 407A: Linear Optimization 12 / 27
Inequalities must go the right way.
− x1 + 6x2 − x3 + x4 ≥ −3
becomes
x1 − 6x2 + x3 − x4 ≤ 3.
22Author: James Burke, University of Washington
Lecture 4: LP Standard Form 22 Math 407A: Linear Optimization 12 / 27
Inequalities must go the right way.
− x1 + 6x2 − x3 + x4 ≥ −3
becomes
x1 − 6x2 + x3 − x4 ≤ 3.
22Author: James Burke, University of Washington
Lecture 4: LP Standard Form 22 Math 407A: Linear Optimization 12 / 27
Inequalities must go the right way.
− x1 + 6x2 − x3 + x4 ≥ −3
becomes
x1 − 6x2 + x3 − x4 ≤ 3.
22Author: James Burke, University of Washington
Lecture 4: LP Standard Form 23 Math 407A: Linear Optimization 12 / 27
Equalities are replaced by 2 inequalities.
7x2 + x4 = 5
becomes
7x2 + x4 ≤ 5
and
− 7x2 − x4 ≤ −5
Lecture 4: LP Standard Form 24 Math 407A: Linear Optimization 13 / 27
Equalities are replaced by 2 inequalities.
7x2 + x4 = 5
becomes
7x2 + x4 ≤ 5
and
− 7x2 − x4 ≤ −5
24Author: James Burke, University of Washington
Lecture 4: LP Standard Form 24 Math 407A: Linear Optimization 13 / 27
Equalities are replaced by 2 inequalities.
7x2 + x4 = 5
becomes
7x2 + x4 ≤ 5
and
− 7x2 − x4 ≤ −5
24Author: James Burke, University of Washington
Lecture 4: LP Standard Form 24 Math 407A: Linear Optimization 13 / 27
Equalities are replaced by 2 inequalities.
7x2 + x4 = 5
becomes
7x2 + x4 ≤ 5
and
− 7x2 − x4 ≤ −5
24Author: James Burke, University of Washington
Lecture 4: LP Standard Form 24 Math 407A: Linear Optimization 13 / 27
Equalities are replaced by 2 inequalities.
7x2 + x4 = 5
becomes
7x2 + x4 ≤ 5
and
− 7x2 − x4 ≤ −5
24Author: James Burke, University of Washington
Lecture 4: LP Standard Form 25 Math 407A: Linear Optimization 13 / 27
Grouping upper bounds with the linear inequalities.
The double bound −2 ≤ x4 ≤ 2 indicates that we should group the upper boundwith the linear inequalities.
x4 ≤ 2
Lecture 4: LP Standard Form 26 Math 407A: Linear Optimization 14 / 27
Grouping upper bounds with the linear inequalities.
The double bound −2 ≤ x4 ≤ 2 indicates that we should group the upper boundwith the linear inequalities.
x4 ≤ 2
26Author: James Burke, University of Washington
Lecture 4: LP Standard Form 27 Math 407A: Linear Optimization 14 / 27
Step 1: Transformation to Standard Form
Combining all of these changes gives the LP
maximize −3x1 + x2subject to x1 − 6x2 + x3 − x4 ≤ 3
7x2 + x4 ≤ 5− 7x2 − x4 ≤ −5
x3 + x4 ≤ 2x4 ≤ 2
−1 ≤ x2, x3 ≤ 5, −2 ≤ x4.
Lecture 4: LP Standard Form 28 Math 407A: Linear Optimization 15 / 27
Step 2: Variable Replacement
The variable x1 is free, so we replace it by
x1 = z+1 − z−1 with 0 ≤ z+1 , 0 ≤ z−1 .
x2 has a non-zero lower bound (−1 ≤ x2) so we replace it by
z2 = x2 + 1 or x2 = z2 − 1 with 0 ≤ z2.
x3 is bounded above (x3 ≤ 5), so we replace it by
z3 = 5− x3 or x3 = 5− z3 with 0 ≤ z3.
x4 is bounded below (−2 ≤ x4), so we replace it by
z4 = x4 + 2 or x4 = z4 − 2 with 0 ≤ z4.
Lecture 4: LP Standard Form 29 Math 407A: Linear Optimization 16 / 27
Step 2: Variable Replacement
The variable x1 is free, so we replace it by
x1 = z+1 − z−1 with 0 ≤ z+1 , 0 ≤ z−1 .
x2 has a non-zero lower bound (−1 ≤ x2) so we replace it by
z2 = x2 + 1 or x2 = z2 − 1 with 0 ≤ z2.
x3 is bounded above (x3 ≤ 5), so we replace it by
z3 = 5− x3 or x3 = 5− z3 with 0 ≤ z3.
x4 is bounded below (−2 ≤ x4), so we replace it by
z4 = x4 + 2 or x4 = z4 − 2 with 0 ≤ z4.
29Author: James Burke, University of Washington
Lecture 4: LP Standard Form 29 Math 407A: Linear Optimization 16 / 27
Step 2: Variable Replacement
The variable x1 is free, so we replace it by
x1 = z+1 − z−1 with 0 ≤ z+1 , 0 ≤ z−1 .
x2 has a non-zero lower bound (−1 ≤ x2) so we replace it by
z2 = x2 + 1 or x2 = z2 − 1 with 0 ≤ z2.
x3 is bounded above (x3 ≤ 5), so we replace it by
z3 = 5− x3 or x3 = 5− z3 with 0 ≤ z3.
x4 is bounded below (−2 ≤ x4), so we replace it by
z4 = x4 + 2 or x4 = z4 − 2 with 0 ≤ z4.
29Author: James Burke, University of Washington
Lecture 4: LP Standard Form 29 Math 407A: Linear Optimization 16 / 27
Step 2: Variable Replacement
The variable x1 is free, so we replace it by
x1 = z+1 − z−1 with 0 ≤ z+1 , 0 ≤ z−1 .
x2 has a non-zero lower bound (−1 ≤ x2) so we replace it by
z2 = x2 + 1 or x2 = z2 − 1 with 0 ≤ z2.
x3 is bounded above (x3 ≤ 5), so we replace it by
z3 = 5− x3 or x3 = 5− z3 with 0 ≤ z3.
x4 is bounded below (−2 ≤ x4), so we replace it by
z4 = x4 + 2 or x4 = z4 − 2 with 0 ≤ z4.
29Author: James Burke, University of Washington
Lecture 4: LP Standard Form 29 Math 407A: Linear Optimization 16 / 27
Step 2: Variable Replacement
The variable x1 is free, so we replace it by
x1 = z+1 − z−1 with 0 ≤ z+1 , 0 ≤ z−1 .
x2 has a non-zero lower bound (−1 ≤ x2) so we replace it by
z2 = x2 + 1 or x2 = z2 − 1 with 0 ≤ z2.
x3 is bounded above (x3 ≤ 5), so we replace it by
z3 = 5− x3 or x3 = 5− z3 with 0 ≤ z3.
x4 is bounded below (−2 ≤ x4), so we replace it by
z4 = x4 + 2 or x4 = z4 − 2 with 0 ≤ z4.
29Author: James Burke, University of Washington
Lecture 4: LP Standard Form 29 Math 407A: Linear Optimization 16 / 27
Step 2: Variable Replacement
The variable x1 is free, so we replace it by
x1 = z+1 − z−1 with 0 ≤ z+1 , 0 ≤ z−1 .
x2 has a non-zero lower bound (−1 ≤ x2) so we replace it by
z2 = x2 + 1 or x2 = z2 − 1 with 0 ≤ z2.
x3 is bounded above (x3 ≤ 5), so we replace it by
z3 = 5− x3 or x3 = 5− z3 with 0 ≤ z3.
x4 is bounded below (−2 ≤ x4), so we replace it by
z4 = x4 + 2 or x4 = z4 − 2 with 0 ≤ z4.
29Author: James Burke, University of Washington
Lecture 4: LP Standard Form 29 Math 407A: Linear Optimization 16 / 27
Step 2: Variable Replacement
The variable x1 is free, so we replace it by
x1 = z+1 − z−1 with 0 ≤ z+1 , 0 ≤ z−1 .
x2 has a non-zero lower bound (−1 ≤ x2) so we replace it by
z2 = x2 + 1 or x2 = z2 − 1 with 0 ≤ z2.
x3 is bounded above (x3 ≤ 5), so we replace it by
z3 = 5− x3 or x3 = 5− z3 with 0 ≤ z3.
x4 is bounded below (−2 ≤ x4), so we replace it by
z4 = x4 + 2 or x4 = z4 − 2 with 0 ≤ z4.
29Author: James Burke, University of Washington
Lecture 4: LP Standard Form 29 Math 407A: Linear Optimization 16 / 27
Step 2: Variable Replacement
The variable x1 is free, so we replace it by
x1 = z+1 − z−1 with 0 ≤ z+1 , 0 ≤ z−1 .
x2 has a non-zero lower bound (−1 ≤ x2) so we replace it by
z2 = x2 + 1 or x2 = z2 − 1 with 0 ≤ z2.
x3 is bounded above (x3 ≤ 5), so we replace it by
z3 = 5− x3 or x3 = 5− z3 with 0 ≤ z3.
x4 is bounded below (−2 ≤ x4), so we replace it by
z4 = x4 + 2 or x4 = z4 − 2 with 0 ≤ z4.
29Author: James Burke, University of Washington
Lecture 4: LP Standard Form 30 Math 407A: Linear Optimization 16 / 27
Step 2: Transformation to Standard Form
Substituting x1 = z+1 − z−1 into
maximize −3x1 + x2subject to x1 − 6x2 + x3 − x4 ≤ 3
7x2 + x4 ≤ 5− 7x2 − x4 ≤ −5
x3 + x4 ≤ 2x4 ≤ 2
−1 ≤ x2, x3 ≤ 5, −2 ≤ x4.
gives
Lecture 4: LP Standard Form 31 Math 407A: Linear Optimization 17 / 27
Step 2: Transformation to Standard Form
maximize −3z+1 + 3z−1 + x2subject to z+1 − z−1 − 6x2 + x3 − x4 ≤ 3
7x2 + x4 ≤ 5− 7x2 − x4 ≤ −5
x3 + x4 ≤ 2x4 ≤ 2
0 ≤ z+1 , 0 ≤ z−1 , −1 ≤ x2, x3 ≤ 5, −2 ≤ x4.
Lecture 4: LP Standard Form 32 Math 407A: Linear Optimization 18 / 27
Step 2: Transformation to Standard Form
Substituting x2 = z2 − 1 into
maximize −3z+1 + 3z−1 + x2subject to z+1 − z−1 − 6x2 + x3 − x4 ≤ 3
7x2 + x4 ≤ 5− 7x2 − x4 ≤ −5
x3 + x4 ≤ 2x4 ≤ 2
0 ≤ z+1 , 0 ≤ z−1 , −1 ≤ x2, x3 ≤ 5, −2 ≤ x4.
gives
Lecture 4: LP Standard Form 33 Math 407A: Linear Optimization 19 / 27
Step 2: Transformation to Standard Form
maximize −3z+1 + 3z−1 + z2 − 1subject to z+1 − z−1 − 6z2 + x3 − x4 ≤ 3− 6 = −3
7z2 + x4 ≤ 5 + 7 = 12− 7z2 − x4 ≤ −5− 7 = −12
x3 + x4 ≤ 2x4 ≤ 2
0 ≤ z+1 , 0 ≤ z−1 , −0 ≤ z2, x3 ≤ 5, −2 ≤ x4.
Lecture 4: LP Standard Form 34 Math 407A: Linear Optimization 20 / 27
Step 2: Transformation to Standard Form
maximize −3z+1 + 3z−1 + z2subject to z+1 − z−1 − 6z2 + x3 − x4 ≤ −3
7z2 + x4 ≤ 12− 7z2 − x4 ≤ −12
x3 + x4 ≤ 2x4 ≤ 2
0 ≤ z+1 , 0 ≤ z−1 , −0 ≤ z2, x3 ≤ 5, −2 ≤ x4.
Lecture 4: LP Standard Form 35 Math 407A: Linear Optimization 21 / 27
Step 2: Transformation to Standard Form
Substituting x3 = 5− z3 into
maximize −3z+1 + 3z−1 + z2subject to z+1 − z−1 − 6z2 + x3 − x4 ≤ −3
7z2 + x4 ≤ 12− 7z2 − x4 ≤ −12
x3 + x4 ≤ 2x4 ≤ 2
0 ≤ z+1 , 0 ≤ z−1 , −0 ≤ z2, x3 ≤ 5, −2 ≤ x4.
gives
Lecture 4: LP Standard Form 36 Math 407A: Linear Optimization 22 / 27
Step 2: Transformation to Standard Form
maximize −3z+1 + 3z−1 + z2subject to z+1 − z−1 − 6z2 − z3 − x4 ≤ −3− 5 = −8
7z2 + x4 ≤ 12− 7z2 − x4 ≤ −12
− z3 + x4 ≤ 2− 5 = −3x4 ≤ 2
0 ≤ z+1 , 0 ≤ z−1 , −0 ≤ z2, 0 ≤ z3, −2 ≤ x4.
Lecture 4: LP Standard Form 37 Math 407A: Linear Optimization 23 / 27
Step 2: Transformation to Standard Form
Substituting x4 = z4 − 2 into
maximize −3z+1 + 3z−1 + z2subject to z+1 − z−1 − 6z2 − z3 − x4 ≤ −8
7z2 + x4 ≤ 12− 7z2 − x4 ≤ −12
− z3 + x4 ≤ −3x4 ≤ 2
0 ≤ z+1 , 0 ≤ z−1 , −0 ≤ z2, 0 ≤ z3, −2 ≤ x4.
gives
Lecture 4: LP Standard Form 38 Math 407A: Linear Optimization 24 / 27
Step 2: Transformation to Standard Form
maximize −3z+1 + 3z−1 + z2subject to z+1 − z−1 − 6z2 − z3 − z4 ≤ −8− 2 = −10
7z2 + z4 ≤ 12 + 2 = 14− 7z2 − z4 ≤ −12− 2 = −14
− z3 + z4 ≤ −3 + 2 = −1z4 ≤ 2 + 2 = 4
0 ≤ z+1 , 0 ≤ z−1 , −0 ≤ z2, 0 ≤ z3, 0 ≤ z4.
which is in standard form.
Lecture 4: LP Standard Form 39 Math 407A: Linear Optimization 25 / 27
Step 2: Transformation to Standard Form
After making these substitutions, we get the following LP in standard form:
maximize −3z+1 + 3z−1 + z2subject to z+1 − z−1 − 6z2 − z3 − z4 ≤ −10
7z2 + z4 ≤ 14− 7z2 − z4 ≤ −14
− z3 + z4 ≤ −1z4 ≤ 4
0 ≤ z+1 , z−1 , z2, z3, z4.
Lecture 4: LP Standard Form 40 Math 407A: Linear Optimization 26 / 27
Step 2: Transformation to Standard Form
After making these substitutions, we get the following LP in standard form:
maximize −3z+1 + 3z−1 + z2subject to z+1 − z−1 − 6z2 − z3 − z4 ≤ −10
7z2 + z4 ≤ 14− 7z2 − z4 ≤ −14
− z3 + z4 ≤ −1z4 ≤ 4
0 ≤ z+1 , z−1 , z2, z3, z4.
40Author: James Burke, University of Washington
Lecture 4: LP Standard Form 41 Math 407A: Linear Optimization 26 / 27
Transformation to Standard Form: Practice
Transform the following LP to an LP in standard form.
minimize x1 − 12x2 + 2x3
subject to −5x1 − x2 + 3x3 = −152x1 + x2 − 20x3 ≥ −300 ≤ x2 , 1 ≤ x3 ≤ 4
Lecture 4: LP Standard Form 42 Math 407A: Linear Optimization 27 / 27