1

15053 Tuesday March 5

bull Duality ndash The art of obtaining bounds ndash weak and strong duality

bull Handouts Lecture Notes

2

Bounds

bull One of the great contributions of optimization theory (and math programming) is the providing of upper bounds for maximization problems

bull We can prove that solutions are optimal

bull For other problems we can bound the distance from optimality

3

A 4-variable linear program

David has minerals that he will mix together and sellfor profit The minerals all contain some goldcontent and he wants to ensure that the mixture has3 gold and each bag will weigh 1 kilogram

Mineral 1 2 gold $3 profitkilo Mineral 2 3 gold $4 profitkiloMineral 3 4 gold $6 profitkiloMineral 4 5 gold $8 profitkilo

maximize z = 3x1 + 4x2 +6x3 + 8x4

subject to x1+ x2 + x3 + x4 = 1 2x1+ 3x2+4x3+ 5x4 = 3 x1 x2 x3 x4 ge 0

4

A 4-variable linear program

maximize z = 3x1 + 4x2 +6x3 + 8x4

subject to x1+ x2 + x3 + x4 = 1 2x1+ 3x2+4x3+ 5x4 = 3 x1 x2 x3 x4 ge 0

5

Obtaining a Bound

Subtract 8 times constraint 1 from the objective function

-z ndash 5 x1 ndash 4 x2 ndash 2 x3 = -8 z + 5 x1 + 4x2 + 2 x3 = 8

Does this show that z le8 YES

6

Obtaining a Second Bound Treat theoperation as pricing out

Subtract 3 constraint 1 and subtract constraint 2 from theobjective function

-z ndash 2x1 ndash 2 x2 ndash 1x3 = -6 z + 2 x1 + 2x2 + 1 x3 = 6Thus z le6 Which bound is better 6 or 8

Prices

7

Obtaining the Best Bound Formulatethe problem as an LP

Prices

A 3 - y1- 2y2 le0 y1 + 2y2 ge3

B 4 - y1- 3y2 le0 y1 + 3y2 ge4

C 6 - y1- 4y2 le0 y1 + 4y2 ge6

D 8 - y1- 5y2 le0 y1 + 5y2 ge8

minimizey1 + 3y2

8

The problem that we formed is calledthe dual problem

minimize y1+ 3y2

Subject to y1+ 2y2 ge

y1+ 3y2 ge

y1+ 5y2 ge

y1+ 4y2 ge

y1 a nd y2 are unconstrained in sign

9

Summary of previous slides

bull If ldquoreduced costsrdquo are non-positive then we have an upper bound on the objective value

bull The problem of finding the least upper bound is a linear program and is referred to as the dual of the original linear program

bull To do express duality in general notation

bull To do show that the shadow prices solve the dual and the bound is the optimal solution to the original problem

10

PRIMAL PROBLEM

maximize

subject toz = 3x1 + 4x2 +6x3 + 8x4

x1 + x2 + x3 + x4 = 1 2x1 + 3x2 +4x3 + 5x4 = 3 x1 x2 x3 x4 ge 0

maximize y1+ 3y2

subject to y1+ 2y2 ge

y1+ 3y2 ge

y1+ 4y2 ge

y1+ 5y2 ge

Observation 1

The constraint matrix in theprimal is the transpose of theconstraint matrix in the dual

Observation 2

The RHS coefficients in theprimal become the costcoefficients in the dual

DUSL PROBLEM

11

PRIMAL PROBLEM

maximize

subject toz = 3x1 + 4x2 +6x3 + 8x4

x1 + x2 + x3 + x4 = 1 2x1 + 3x2 +4x3 + 5x4 = 3 x1 x2 x3 x4 ge 0

DUSL PROBLEM

maximize y1+ 3y2

subject to y1+ 2y2 ge

y1+ 3y2 ge

y1+ 4y2 ge

y1+ 5y2 ge

Observation 3 The cost coefficientsin the primal become the RHScoefficients in the dual

Observation 4 The primal (in thiscase) is a max problem with equalityconstraints and non-negativevariables

The dual (in this case) is aminimization problem with geconstraints and variablesunconstrained in sign

12

PRIMAL PROBLEM (in standard form)

max z = c1x1 + c2x2 + c3x3 + hellip + cnxn

st a11x1 + a12x2 + a13x3 + hellip + a1nxn = b1

a21x1 + a22x2 + a23x3 + hellip + a2nxn = b2

hellip am1x1 + am2x2 + am3x3 + hellip + amnxn = bm xj ge0 for j = 1 to n

DUAL PROBLEM

max v = st

What is the dual problem in terms of the notation given above

13

PRIMAL PROBLEM (in standard form)

max z = c1x1 + c2x2 + c3x3 + hellip + cnxn

st a11x1 + a12x2 + a13x3 + hellip + a1nxn = b1

a21x1 + a22x2 + a23x3 + hellip + a2nxn = b2

hellip am1x1 + am2x2 + am3x3 + hellip + amnxn = bm

xj ge0 for j = 1 to n

DUAL PROBLEM

max v = b1y1 + b2y2 + b3y3 + hellip +bnyn

st a11y1 + a21y2 + a31y3 + hellip + am1ym ge c1

a12y1 + a22y2 + a32y3 + hellip + am2ym ge c2

hellip a1ny1 + a2ny2 + a3ny3 + hellip + amnym ge cn

14

Theorem Suppose that x is any feasiblesolution to the primal andy is any feasiblesolution to the dual Then

Σj=1n cjxj leΣi=1nyi bi (Max leMin)

Proof

Σj=1n cjxj leΣi=1nΣj=1m (yi aij )xi

leΣj=1m Σi=1n yi (aijxj)

leΣj=1m yi bi

Weak Duality Theorem

15

Unboundedness Property

Theorem Suppose that the primal (dual)problem has an unbounded solution Then thedual (primal) problem has no feasible solution

Proof

Suppose thaty was feasible for the dual Thenevery solution to the primal problem isunbounded above by

Σj=1m yi bi

16

Strong Duality Theorem

Theorem If the primal problem has a finiteoptimal solution value then so does the dualproblem and these two values are the same

17

Strong Duality Illustrated

Prices

Optimal dual solution y1 = ndash13 y2 = 53 v = 143

Optimal primal solution x1 = 23 x2 = 0 x3 = 0 x4 = 13 z = 143

18

Strong Duality Illustrated the final tableau

Optimal primal solution x1 = 23 x2 = 0 x3 = 0 x4 = 13 v = 143

Observation

The costcoeffients satisfycomplementaryslackness

19

Summary

bull If we take an LP in standard form (max) we can formulate a dual problem1048698bull Weak Duality Each solution of the dual gives an upper bound on the maximum objective value for the primal

bull Strong duality if there are feasible solutions to the primal and dual then the optimal objective value for both problems is the same

20

Shadow prices solve the dual

bull Theorem Suppose that the primal problem has a finite optimal solution value Then the shadow prices for the primal problem form an optimal solution to the dual problem (And the objective values are the same)

Davidrsquos Mineral Problem

21

FACTOptimalsimplexmultipliersare shadowprices

Shadowprices

22

Summary

bull The dual problem to a maximizing LP provides upper bounds on the optimal objective function

bull The maximum solution value for the primal is the same as the minimum solution value for the dual

bull The shadow prices are optimal for the dual LP

bull Next alternative optimality conditions

23

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij gecj for j = 1 to n

Optimality Condition 1A solution x is optimalfor the primal problem ifit is a basic feasiblesolution and the tableausatisfies the optimalityconditions

Optimality Condition 2

A solution x is optimal

for the primal problem if

it is feasible and if there is

a feasible solution y for

the dual with

Σj=1m cj xj = Σj=1m yi bi

24

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij ge cj for j = 1 to n

ComplementarySlackness Conditions

ndashSuppose that y is feasible

for the dual and let

1048698cj= cj - Σi=1m1048698yi aij

Suppose x is feasible for

the primal

Theorem (complementary

slackness) x and y are

optimal for the primal and

dual if and only if

cj xj = 0 for all j

25

Complementary Slackness Illustrated

prices

26

Next Lecture on Duality

bull Dual linear programs in general formbull Optimality Conditions in general formbull Illustrating Duality with 2-person 0-sum game theory

27

Coverage for first midterm(Closed book exam)

bull Formulations

bull 2D graphing and

finding opt solution

bull Setting up an LP in

standard form

bull The simplex algorithm

starting with a bfs

bull Phase 1 of the

simplex algorithm

bull Interpreting sensitivity analysis including shadow prices and ranges and reduced costs

bull Pricing out

bull Using tableaus to determine shadow prices reduced costs and ranges

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27

2

Bounds

bull One of the great contributions of optimization theory (and math programming) is the providing of upper bounds for maximization problems

bull We can prove that solutions are optimal

bull For other problems we can bound the distance from optimality

3

A 4-variable linear program

David has minerals that he will mix together and sellfor profit The minerals all contain some goldcontent and he wants to ensure that the mixture has3 gold and each bag will weigh 1 kilogram

Mineral 1 2 gold $3 profitkilo Mineral 2 3 gold $4 profitkiloMineral 3 4 gold $6 profitkiloMineral 4 5 gold $8 profitkilo

maximize z = 3x1 + 4x2 +6x3 + 8x4

subject to x1+ x2 + x3 + x4 = 1 2x1+ 3x2+4x3+ 5x4 = 3 x1 x2 x3 x4 ge 0

4

A 4-variable linear program

maximize z = 3x1 + 4x2 +6x3 + 8x4

subject to x1+ x2 + x3 + x4 = 1 2x1+ 3x2+4x3+ 5x4 = 3 x1 x2 x3 x4 ge 0

5

Obtaining a Bound

Subtract 8 times constraint 1 from the objective function

-z ndash 5 x1 ndash 4 x2 ndash 2 x3 = -8 z + 5 x1 + 4x2 + 2 x3 = 8

Does this show that z le8 YES

6

Obtaining a Second Bound Treat theoperation as pricing out

Subtract 3 constraint 1 and subtract constraint 2 from theobjective function

-z ndash 2x1 ndash 2 x2 ndash 1x3 = -6 z + 2 x1 + 2x2 + 1 x3 = 6Thus z le6 Which bound is better 6 or 8

Prices

7

Obtaining the Best Bound Formulatethe problem as an LP

Prices

A 3 - y1- 2y2 le0 y1 + 2y2 ge3

B 4 - y1- 3y2 le0 y1 + 3y2 ge4

C 6 - y1- 4y2 le0 y1 + 4y2 ge6

D 8 - y1- 5y2 le0 y1 + 5y2 ge8

minimizey1 + 3y2

8

The problem that we formed is calledthe dual problem

minimize y1+ 3y2

Subject to y1+ 2y2 ge

y1+ 3y2 ge

y1+ 5y2 ge

y1+ 4y2 ge

y1 a nd y2 are unconstrained in sign

9

Summary of previous slides

bull If ldquoreduced costsrdquo are non-positive then we have an upper bound on the objective value

bull The problem of finding the least upper bound is a linear program and is referred to as the dual of the original linear program

bull To do express duality in general notation

bull To do show that the shadow prices solve the dual and the bound is the optimal solution to the original problem

10

PRIMAL PROBLEM

maximize

subject toz = 3x1 + 4x2 +6x3 + 8x4

x1 + x2 + x3 + x4 = 1 2x1 + 3x2 +4x3 + 5x4 = 3 x1 x2 x3 x4 ge 0

maximize y1+ 3y2

subject to y1+ 2y2 ge

y1+ 3y2 ge

y1+ 4y2 ge

y1+ 5y2 ge

Observation 1

The constraint matrix in theprimal is the transpose of theconstraint matrix in the dual

Observation 2

The RHS coefficients in theprimal become the costcoefficients in the dual

DUSL PROBLEM

11

PRIMAL PROBLEM

maximize

subject toz = 3x1 + 4x2 +6x3 + 8x4

x1 + x2 + x3 + x4 = 1 2x1 + 3x2 +4x3 + 5x4 = 3 x1 x2 x3 x4 ge 0

DUSL PROBLEM

maximize y1+ 3y2

subject to y1+ 2y2 ge

y1+ 3y2 ge

y1+ 4y2 ge

y1+ 5y2 ge

Observation 3 The cost coefficientsin the primal become the RHScoefficients in the dual

Observation 4 The primal (in thiscase) is a max problem with equalityconstraints and non-negativevariables

The dual (in this case) is aminimization problem with geconstraints and variablesunconstrained in sign

12

PRIMAL PROBLEM (in standard form)

max z = c1x1 + c2x2 + c3x3 + hellip + cnxn

st a11x1 + a12x2 + a13x3 + hellip + a1nxn = b1

a21x1 + a22x2 + a23x3 + hellip + a2nxn = b2

hellip am1x1 + am2x2 + am3x3 + hellip + amnxn = bm xj ge0 for j = 1 to n

DUAL PROBLEM

max v = st

What is the dual problem in terms of the notation given above

13

PRIMAL PROBLEM (in standard form)

max z = c1x1 + c2x2 + c3x3 + hellip + cnxn

st a11x1 + a12x2 + a13x3 + hellip + a1nxn = b1

a21x1 + a22x2 + a23x3 + hellip + a2nxn = b2

hellip am1x1 + am2x2 + am3x3 + hellip + amnxn = bm

xj ge0 for j = 1 to n

DUAL PROBLEM

max v = b1y1 + b2y2 + b3y3 + hellip +bnyn

st a11y1 + a21y2 + a31y3 + hellip + am1ym ge c1

a12y1 + a22y2 + a32y3 + hellip + am2ym ge c2

hellip a1ny1 + a2ny2 + a3ny3 + hellip + amnym ge cn

14

Theorem Suppose that x is any feasiblesolution to the primal andy is any feasiblesolution to the dual Then

Σj=1n cjxj leΣi=1nyi bi (Max leMin)

Proof

Σj=1n cjxj leΣi=1nΣj=1m (yi aij )xi

leΣj=1m Σi=1n yi (aijxj)

leΣj=1m yi bi

Weak Duality Theorem

15

Unboundedness Property

Theorem Suppose that the primal (dual)problem has an unbounded solution Then thedual (primal) problem has no feasible solution

Proof

Suppose thaty was feasible for the dual Thenevery solution to the primal problem isunbounded above by

Σj=1m yi bi

16

Strong Duality Theorem

Theorem If the primal problem has a finiteoptimal solution value then so does the dualproblem and these two values are the same

17

Strong Duality Illustrated

Prices

Optimal dual solution y1 = ndash13 y2 = 53 v = 143

Optimal primal solution x1 = 23 x2 = 0 x3 = 0 x4 = 13 z = 143

18

Strong Duality Illustrated the final tableau

Optimal primal solution x1 = 23 x2 = 0 x3 = 0 x4 = 13 v = 143

Observation

The costcoeffients satisfycomplementaryslackness

19

Summary

bull If we take an LP in standard form (max) we can formulate a dual problem1048698bull Weak Duality Each solution of the dual gives an upper bound on the maximum objective value for the primal

bull Strong duality if there are feasible solutions to the primal and dual then the optimal objective value for both problems is the same

20

Shadow prices solve the dual

bull Theorem Suppose that the primal problem has a finite optimal solution value Then the shadow prices for the primal problem form an optimal solution to the dual problem (And the objective values are the same)

Davidrsquos Mineral Problem

21

FACTOptimalsimplexmultipliersare shadowprices

Shadowprices

22

Summary

bull The dual problem to a maximizing LP provides upper bounds on the optimal objective function

bull The maximum solution value for the primal is the same as the minimum solution value for the dual

bull The shadow prices are optimal for the dual LP

bull Next alternative optimality conditions

23

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij gecj for j = 1 to n

Optimality Condition 1A solution x is optimalfor the primal problem ifit is a basic feasiblesolution and the tableausatisfies the optimalityconditions

Optimality Condition 2

A solution x is optimal

for the primal problem if

it is feasible and if there is

a feasible solution y for

the dual with

Σj=1m cj xj = Σj=1m yi bi

24

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij ge cj for j = 1 to n

ComplementarySlackness Conditions

ndashSuppose that y is feasible

for the dual and let

1048698cj= cj - Σi=1m1048698yi aij

Suppose x is feasible for

the primal

Theorem (complementary

slackness) x and y are

optimal for the primal and

dual if and only if

cj xj = 0 for all j

25

Complementary Slackness Illustrated

prices

26

Next Lecture on Duality

bull Dual linear programs in general formbull Optimality Conditions in general formbull Illustrating Duality with 2-person 0-sum game theory

27

Coverage for first midterm(Closed book exam)

bull Formulations

bull 2D graphing and

finding opt solution

bull Setting up an LP in

standard form

bull The simplex algorithm

starting with a bfs

bull Phase 1 of the

simplex algorithm

bull Interpreting sensitivity analysis including shadow prices and ranges and reduced costs

bull Pricing out

bull Using tableaus to determine shadow prices reduced costs and ranges

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27

3

A 4-variable linear program

David has minerals that he will mix together and sellfor profit The minerals all contain some goldcontent and he wants to ensure that the mixture has3 gold and each bag will weigh 1 kilogram

Mineral 1 2 gold $3 profitkilo Mineral 2 3 gold $4 profitkiloMineral 3 4 gold $6 profitkiloMineral 4 5 gold $8 profitkilo

maximize z = 3x1 + 4x2 +6x3 + 8x4

subject to x1+ x2 + x3 + x4 = 1 2x1+ 3x2+4x3+ 5x4 = 3 x1 x2 x3 x4 ge 0

4

A 4-variable linear program

maximize z = 3x1 + 4x2 +6x3 + 8x4

subject to x1+ x2 + x3 + x4 = 1 2x1+ 3x2+4x3+ 5x4 = 3 x1 x2 x3 x4 ge 0

5

Obtaining a Bound

Subtract 8 times constraint 1 from the objective function

-z ndash 5 x1 ndash 4 x2 ndash 2 x3 = -8 z + 5 x1 + 4x2 + 2 x3 = 8

Does this show that z le8 YES

6

Obtaining a Second Bound Treat theoperation as pricing out

Subtract 3 constraint 1 and subtract constraint 2 from theobjective function

-z ndash 2x1 ndash 2 x2 ndash 1x3 = -6 z + 2 x1 + 2x2 + 1 x3 = 6Thus z le6 Which bound is better 6 or 8

Prices

7

Obtaining the Best Bound Formulatethe problem as an LP

Prices

A 3 - y1- 2y2 le0 y1 + 2y2 ge3

B 4 - y1- 3y2 le0 y1 + 3y2 ge4

C 6 - y1- 4y2 le0 y1 + 4y2 ge6

D 8 - y1- 5y2 le0 y1 + 5y2 ge8

minimizey1 + 3y2

8

The problem that we formed is calledthe dual problem

minimize y1+ 3y2

Subject to y1+ 2y2 ge

y1+ 3y2 ge

y1+ 5y2 ge

y1+ 4y2 ge

y1 a nd y2 are unconstrained in sign

9

Summary of previous slides

bull If ldquoreduced costsrdquo are non-positive then we have an upper bound on the objective value

bull The problem of finding the least upper bound is a linear program and is referred to as the dual of the original linear program

bull To do express duality in general notation

bull To do show that the shadow prices solve the dual and the bound is the optimal solution to the original problem

10

PRIMAL PROBLEM

maximize

subject toz = 3x1 + 4x2 +6x3 + 8x4

x1 + x2 + x3 + x4 = 1 2x1 + 3x2 +4x3 + 5x4 = 3 x1 x2 x3 x4 ge 0

maximize y1+ 3y2

subject to y1+ 2y2 ge

y1+ 3y2 ge

y1+ 4y2 ge

y1+ 5y2 ge

Observation 1

The constraint matrix in theprimal is the transpose of theconstraint matrix in the dual

Observation 2

The RHS coefficients in theprimal become the costcoefficients in the dual

DUSL PROBLEM

11

PRIMAL PROBLEM

maximize

subject toz = 3x1 + 4x2 +6x3 + 8x4

x1 + x2 + x3 + x4 = 1 2x1 + 3x2 +4x3 + 5x4 = 3 x1 x2 x3 x4 ge 0

DUSL PROBLEM

maximize y1+ 3y2

subject to y1+ 2y2 ge

y1+ 3y2 ge

y1+ 4y2 ge

y1+ 5y2 ge

Observation 3 The cost coefficientsin the primal become the RHScoefficients in the dual

Observation 4 The primal (in thiscase) is a max problem with equalityconstraints and non-negativevariables

The dual (in this case) is aminimization problem with geconstraints and variablesunconstrained in sign

12

PRIMAL PROBLEM (in standard form)

max z = c1x1 + c2x2 + c3x3 + hellip + cnxn

st a11x1 + a12x2 + a13x3 + hellip + a1nxn = b1

a21x1 + a22x2 + a23x3 + hellip + a2nxn = b2

hellip am1x1 + am2x2 + am3x3 + hellip + amnxn = bm xj ge0 for j = 1 to n

DUAL PROBLEM

max v = st

What is the dual problem in terms of the notation given above

13

PRIMAL PROBLEM (in standard form)

max z = c1x1 + c2x2 + c3x3 + hellip + cnxn

st a11x1 + a12x2 + a13x3 + hellip + a1nxn = b1

a21x1 + a22x2 + a23x3 + hellip + a2nxn = b2

hellip am1x1 + am2x2 + am3x3 + hellip + amnxn = bm

xj ge0 for j = 1 to n

DUAL PROBLEM

max v = b1y1 + b2y2 + b3y3 + hellip +bnyn

st a11y1 + a21y2 + a31y3 + hellip + am1ym ge c1

a12y1 + a22y2 + a32y3 + hellip + am2ym ge c2

hellip a1ny1 + a2ny2 + a3ny3 + hellip + amnym ge cn

14

Theorem Suppose that x is any feasiblesolution to the primal andy is any feasiblesolution to the dual Then

Σj=1n cjxj leΣi=1nyi bi (Max leMin)

Proof

Σj=1n cjxj leΣi=1nΣj=1m (yi aij )xi

leΣj=1m Σi=1n yi (aijxj)

leΣj=1m yi bi

Weak Duality Theorem

15

Unboundedness Property

Theorem Suppose that the primal (dual)problem has an unbounded solution Then thedual (primal) problem has no feasible solution

Proof

Suppose thaty was feasible for the dual Thenevery solution to the primal problem isunbounded above by

Σj=1m yi bi

16

Strong Duality Theorem

Theorem If the primal problem has a finiteoptimal solution value then so does the dualproblem and these two values are the same

17

Strong Duality Illustrated

Prices

Optimal dual solution y1 = ndash13 y2 = 53 v = 143

Optimal primal solution x1 = 23 x2 = 0 x3 = 0 x4 = 13 z = 143

18

Strong Duality Illustrated the final tableau

Optimal primal solution x1 = 23 x2 = 0 x3 = 0 x4 = 13 v = 143

Observation

The costcoeffients satisfycomplementaryslackness

19

Summary

bull If we take an LP in standard form (max) we can formulate a dual problem1048698bull Weak Duality Each solution of the dual gives an upper bound on the maximum objective value for the primal

bull Strong duality if there are feasible solutions to the primal and dual then the optimal objective value for both problems is the same

20

Shadow prices solve the dual

bull Theorem Suppose that the primal problem has a finite optimal solution value Then the shadow prices for the primal problem form an optimal solution to the dual problem (And the objective values are the same)

Davidrsquos Mineral Problem

21

FACTOptimalsimplexmultipliersare shadowprices

Shadowprices

22

Summary

bull The dual problem to a maximizing LP provides upper bounds on the optimal objective function

bull The maximum solution value for the primal is the same as the minimum solution value for the dual

bull The shadow prices are optimal for the dual LP

bull Next alternative optimality conditions

23

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij gecj for j = 1 to n

Optimality Condition 1A solution x is optimalfor the primal problem ifit is a basic feasiblesolution and the tableausatisfies the optimalityconditions

Optimality Condition 2

A solution x is optimal

for the primal problem if

it is feasible and if there is

a feasible solution y for

the dual with

Σj=1m cj xj = Σj=1m yi bi

24

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij ge cj for j = 1 to n

ComplementarySlackness Conditions

ndashSuppose that y is feasible

for the dual and let

1048698cj= cj - Σi=1m1048698yi aij

Suppose x is feasible for

the primal

Theorem (complementary

slackness) x and y are

optimal for the primal and

dual if and only if

cj xj = 0 for all j

25

Complementary Slackness Illustrated

prices

26

Next Lecture on Duality

bull Dual linear programs in general formbull Optimality Conditions in general formbull Illustrating Duality with 2-person 0-sum game theory

27

Coverage for first midterm(Closed book exam)

bull Formulations

bull 2D graphing and

finding opt solution

bull Setting up an LP in

standard form

bull The simplex algorithm

starting with a bfs

bull Phase 1 of the

simplex algorithm

bull Interpreting sensitivity analysis including shadow prices and ranges and reduced costs

bull Pricing out

bull Using tableaus to determine shadow prices reduced costs and ranges

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27

4

A 4-variable linear program

maximize z = 3x1 + 4x2 +6x3 + 8x4

subject to x1+ x2 + x3 + x4 = 1 2x1+ 3x2+4x3+ 5x4 = 3 x1 x2 x3 x4 ge 0

5

Obtaining a Bound

Subtract 8 times constraint 1 from the objective function

-z ndash 5 x1 ndash 4 x2 ndash 2 x3 = -8 z + 5 x1 + 4x2 + 2 x3 = 8

Does this show that z le8 YES

6

Obtaining a Second Bound Treat theoperation as pricing out

Subtract 3 constraint 1 and subtract constraint 2 from theobjective function

-z ndash 2x1 ndash 2 x2 ndash 1x3 = -6 z + 2 x1 + 2x2 + 1 x3 = 6Thus z le6 Which bound is better 6 or 8

Prices

7

Obtaining the Best Bound Formulatethe problem as an LP

Prices

A 3 - y1- 2y2 le0 y1 + 2y2 ge3

B 4 - y1- 3y2 le0 y1 + 3y2 ge4

C 6 - y1- 4y2 le0 y1 + 4y2 ge6

D 8 - y1- 5y2 le0 y1 + 5y2 ge8

minimizey1 + 3y2

8

The problem that we formed is calledthe dual problem

minimize y1+ 3y2

Subject to y1+ 2y2 ge

y1+ 3y2 ge

y1+ 5y2 ge

y1+ 4y2 ge

y1 a nd y2 are unconstrained in sign

9

Summary of previous slides

bull If ldquoreduced costsrdquo are non-positive then we have an upper bound on the objective value

bull The problem of finding the least upper bound is a linear program and is referred to as the dual of the original linear program

bull To do express duality in general notation

bull To do show that the shadow prices solve the dual and the bound is the optimal solution to the original problem

10

PRIMAL PROBLEM

maximize

subject toz = 3x1 + 4x2 +6x3 + 8x4

x1 + x2 + x3 + x4 = 1 2x1 + 3x2 +4x3 + 5x4 = 3 x1 x2 x3 x4 ge 0

maximize y1+ 3y2

subject to y1+ 2y2 ge

y1+ 3y2 ge

y1+ 4y2 ge

y1+ 5y2 ge

Observation 1

The constraint matrix in theprimal is the transpose of theconstraint matrix in the dual

Observation 2

The RHS coefficients in theprimal become the costcoefficients in the dual

DUSL PROBLEM

11

PRIMAL PROBLEM

maximize

subject toz = 3x1 + 4x2 +6x3 + 8x4

x1 + x2 + x3 + x4 = 1 2x1 + 3x2 +4x3 + 5x4 = 3 x1 x2 x3 x4 ge 0

DUSL PROBLEM

maximize y1+ 3y2

subject to y1+ 2y2 ge

y1+ 3y2 ge

y1+ 4y2 ge

y1+ 5y2 ge

Observation 3 The cost coefficientsin the primal become the RHScoefficients in the dual

Observation 4 The primal (in thiscase) is a max problem with equalityconstraints and non-negativevariables

The dual (in this case) is aminimization problem with geconstraints and variablesunconstrained in sign

12

PRIMAL PROBLEM (in standard form)

max z = c1x1 + c2x2 + c3x3 + hellip + cnxn

st a11x1 + a12x2 + a13x3 + hellip + a1nxn = b1

a21x1 + a22x2 + a23x3 + hellip + a2nxn = b2

hellip am1x1 + am2x2 + am3x3 + hellip + amnxn = bm xj ge0 for j = 1 to n

DUAL PROBLEM

max v = st

What is the dual problem in terms of the notation given above

13

PRIMAL PROBLEM (in standard form)

max z = c1x1 + c2x2 + c3x3 + hellip + cnxn

st a11x1 + a12x2 + a13x3 + hellip + a1nxn = b1

a21x1 + a22x2 + a23x3 + hellip + a2nxn = b2

hellip am1x1 + am2x2 + am3x3 + hellip + amnxn = bm

xj ge0 for j = 1 to n

DUAL PROBLEM

max v = b1y1 + b2y2 + b3y3 + hellip +bnyn

st a11y1 + a21y2 + a31y3 + hellip + am1ym ge c1

a12y1 + a22y2 + a32y3 + hellip + am2ym ge c2

hellip a1ny1 + a2ny2 + a3ny3 + hellip + amnym ge cn

14

Theorem Suppose that x is any feasiblesolution to the primal andy is any feasiblesolution to the dual Then

Σj=1n cjxj leΣi=1nyi bi (Max leMin)

Proof

Σj=1n cjxj leΣi=1nΣj=1m (yi aij )xi

leΣj=1m Σi=1n yi (aijxj)

leΣj=1m yi bi

Weak Duality Theorem

15

Unboundedness Property

Theorem Suppose that the primal (dual)problem has an unbounded solution Then thedual (primal) problem has no feasible solution

Proof

Suppose thaty was feasible for the dual Thenevery solution to the primal problem isunbounded above by

Σj=1m yi bi

16

Strong Duality Theorem

Theorem If the primal problem has a finiteoptimal solution value then so does the dualproblem and these two values are the same

17

Strong Duality Illustrated

Prices

Optimal dual solution y1 = ndash13 y2 = 53 v = 143

Optimal primal solution x1 = 23 x2 = 0 x3 = 0 x4 = 13 z = 143

18

Strong Duality Illustrated the final tableau

Optimal primal solution x1 = 23 x2 = 0 x3 = 0 x4 = 13 v = 143

Observation

The costcoeffients satisfycomplementaryslackness

19

Summary

bull If we take an LP in standard form (max) we can formulate a dual problem1048698bull Weak Duality Each solution of the dual gives an upper bound on the maximum objective value for the primal

bull Strong duality if there are feasible solutions to the primal and dual then the optimal objective value for both problems is the same

20

Shadow prices solve the dual

bull Theorem Suppose that the primal problem has a finite optimal solution value Then the shadow prices for the primal problem form an optimal solution to the dual problem (And the objective values are the same)

Davidrsquos Mineral Problem

21

FACTOptimalsimplexmultipliersare shadowprices

Shadowprices

22

Summary

bull The dual problem to a maximizing LP provides upper bounds on the optimal objective function

bull The maximum solution value for the primal is the same as the minimum solution value for the dual

bull The shadow prices are optimal for the dual LP

bull Next alternative optimality conditions

23

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij gecj for j = 1 to n

Optimality Condition 1A solution x is optimalfor the primal problem ifit is a basic feasiblesolution and the tableausatisfies the optimalityconditions

Optimality Condition 2

A solution x is optimal

for the primal problem if

it is feasible and if there is

a feasible solution y for

the dual with

Σj=1m cj xj = Σj=1m yi bi

24

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij ge cj for j = 1 to n

ComplementarySlackness Conditions

ndashSuppose that y is feasible

for the dual and let

1048698cj= cj - Σi=1m1048698yi aij

Suppose x is feasible for

the primal

Theorem (complementary

slackness) x and y are

optimal for the primal and

dual if and only if

cj xj = 0 for all j

25

Complementary Slackness Illustrated

prices

26

Next Lecture on Duality

bull Dual linear programs in general formbull Optimality Conditions in general formbull Illustrating Duality with 2-person 0-sum game theory

27

Coverage for first midterm(Closed book exam)

bull Formulations

bull 2D graphing and

finding opt solution

bull Setting up an LP in

standard form

bull The simplex algorithm

starting with a bfs

bull Phase 1 of the

simplex algorithm

bull Interpreting sensitivity analysis including shadow prices and ranges and reduced costs

bull Pricing out

bull Using tableaus to determine shadow prices reduced costs and ranges

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27

5

Obtaining a Bound

Subtract 8 times constraint 1 from the objective function

-z ndash 5 x1 ndash 4 x2 ndash 2 x3 = -8 z + 5 x1 + 4x2 + 2 x3 = 8

Does this show that z le8 YES

6

Obtaining a Second Bound Treat theoperation as pricing out

Subtract 3 constraint 1 and subtract constraint 2 from theobjective function

-z ndash 2x1 ndash 2 x2 ndash 1x3 = -6 z + 2 x1 + 2x2 + 1 x3 = 6Thus z le6 Which bound is better 6 or 8

Prices

7

Obtaining the Best Bound Formulatethe problem as an LP

Prices

A 3 - y1- 2y2 le0 y1 + 2y2 ge3

B 4 - y1- 3y2 le0 y1 + 3y2 ge4

C 6 - y1- 4y2 le0 y1 + 4y2 ge6

D 8 - y1- 5y2 le0 y1 + 5y2 ge8

minimizey1 + 3y2

8

The problem that we formed is calledthe dual problem

minimize y1+ 3y2

Subject to y1+ 2y2 ge

y1+ 3y2 ge

y1+ 5y2 ge

y1+ 4y2 ge

y1 a nd y2 are unconstrained in sign

9

Summary of previous slides

bull If ldquoreduced costsrdquo are non-positive then we have an upper bound on the objective value

bull The problem of finding the least upper bound is a linear program and is referred to as the dual of the original linear program

bull To do express duality in general notation

bull To do show that the shadow prices solve the dual and the bound is the optimal solution to the original problem

10

PRIMAL PROBLEM

maximize

subject toz = 3x1 + 4x2 +6x3 + 8x4

x1 + x2 + x3 + x4 = 1 2x1 + 3x2 +4x3 + 5x4 = 3 x1 x2 x3 x4 ge 0

maximize y1+ 3y2

subject to y1+ 2y2 ge

y1+ 3y2 ge

y1+ 4y2 ge

y1+ 5y2 ge

Observation 1

The constraint matrix in theprimal is the transpose of theconstraint matrix in the dual

Observation 2

The RHS coefficients in theprimal become the costcoefficients in the dual

DUSL PROBLEM

11

PRIMAL PROBLEM

maximize

subject toz = 3x1 + 4x2 +6x3 + 8x4

x1 + x2 + x3 + x4 = 1 2x1 + 3x2 +4x3 + 5x4 = 3 x1 x2 x3 x4 ge 0

DUSL PROBLEM

maximize y1+ 3y2

subject to y1+ 2y2 ge

y1+ 3y2 ge

y1+ 4y2 ge

y1+ 5y2 ge

Observation 3 The cost coefficientsin the primal become the RHScoefficients in the dual

Observation 4 The primal (in thiscase) is a max problem with equalityconstraints and non-negativevariables

The dual (in this case) is aminimization problem with geconstraints and variablesunconstrained in sign

12

PRIMAL PROBLEM (in standard form)

max z = c1x1 + c2x2 + c3x3 + hellip + cnxn

st a11x1 + a12x2 + a13x3 + hellip + a1nxn = b1

a21x1 + a22x2 + a23x3 + hellip + a2nxn = b2

hellip am1x1 + am2x2 + am3x3 + hellip + amnxn = bm xj ge0 for j = 1 to n

DUAL PROBLEM

max v = st

What is the dual problem in terms of the notation given above

13

PRIMAL PROBLEM (in standard form)

max z = c1x1 + c2x2 + c3x3 + hellip + cnxn

st a11x1 + a12x2 + a13x3 + hellip + a1nxn = b1

a21x1 + a22x2 + a23x3 + hellip + a2nxn = b2

hellip am1x1 + am2x2 + am3x3 + hellip + amnxn = bm

xj ge0 for j = 1 to n

DUAL PROBLEM

max v = b1y1 + b2y2 + b3y3 + hellip +bnyn

st a11y1 + a21y2 + a31y3 + hellip + am1ym ge c1

a12y1 + a22y2 + a32y3 + hellip + am2ym ge c2

hellip a1ny1 + a2ny2 + a3ny3 + hellip + amnym ge cn

14

Theorem Suppose that x is any feasiblesolution to the primal andy is any feasiblesolution to the dual Then

Σj=1n cjxj leΣi=1nyi bi (Max leMin)

Proof

Σj=1n cjxj leΣi=1nΣj=1m (yi aij )xi

leΣj=1m Σi=1n yi (aijxj)

leΣj=1m yi bi

Weak Duality Theorem

15

Unboundedness Property

Theorem Suppose that the primal (dual)problem has an unbounded solution Then thedual (primal) problem has no feasible solution

Proof

Suppose thaty was feasible for the dual Thenevery solution to the primal problem isunbounded above by

Σj=1m yi bi

16

Strong Duality Theorem

Theorem If the primal problem has a finiteoptimal solution value then so does the dualproblem and these two values are the same

17

Strong Duality Illustrated

Prices

Optimal dual solution y1 = ndash13 y2 = 53 v = 143

Optimal primal solution x1 = 23 x2 = 0 x3 = 0 x4 = 13 z = 143

18

Strong Duality Illustrated the final tableau

Optimal primal solution x1 = 23 x2 = 0 x3 = 0 x4 = 13 v = 143

Observation

The costcoeffients satisfycomplementaryslackness

19

Summary

bull If we take an LP in standard form (max) we can formulate a dual problem1048698bull Weak Duality Each solution of the dual gives an upper bound on the maximum objective value for the primal

bull Strong duality if there are feasible solutions to the primal and dual then the optimal objective value for both problems is the same

20

Shadow prices solve the dual

bull Theorem Suppose that the primal problem has a finite optimal solution value Then the shadow prices for the primal problem form an optimal solution to the dual problem (And the objective values are the same)

Davidrsquos Mineral Problem

21

FACTOptimalsimplexmultipliersare shadowprices

Shadowprices

22

Summary

bull The dual problem to a maximizing LP provides upper bounds on the optimal objective function

bull The maximum solution value for the primal is the same as the minimum solution value for the dual

bull The shadow prices are optimal for the dual LP

bull Next alternative optimality conditions

23

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij gecj for j = 1 to n

Optimality Condition 1A solution x is optimalfor the primal problem ifit is a basic feasiblesolution and the tableausatisfies the optimalityconditions

Optimality Condition 2

A solution x is optimal

for the primal problem if

it is feasible and if there is

a feasible solution y for

the dual with

Σj=1m cj xj = Σj=1m yi bi

24

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij ge cj for j = 1 to n

ComplementarySlackness Conditions

ndashSuppose that y is feasible

for the dual and let

1048698cj= cj - Σi=1m1048698yi aij

Suppose x is feasible for

the primal

Theorem (complementary

slackness) x and y are

optimal for the primal and

dual if and only if

cj xj = 0 for all j

25

Complementary Slackness Illustrated

prices

26

Next Lecture on Duality

bull Dual linear programs in general formbull Optimality Conditions in general formbull Illustrating Duality with 2-person 0-sum game theory

27

Coverage for first midterm(Closed book exam)

bull Formulations

bull 2D graphing and

finding opt solution

bull Setting up an LP in

standard form

bull The simplex algorithm

starting with a bfs

bull Phase 1 of the

simplex algorithm

bull Interpreting sensitivity analysis including shadow prices and ranges and reduced costs

bull Pricing out

bull Using tableaus to determine shadow prices reduced costs and ranges

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27

6

Obtaining a Second Bound Treat theoperation as pricing out

Subtract 3 constraint 1 and subtract constraint 2 from theobjective function

-z ndash 2x1 ndash 2 x2 ndash 1x3 = -6 z + 2 x1 + 2x2 + 1 x3 = 6Thus z le6 Which bound is better 6 or 8

Prices

7

Obtaining the Best Bound Formulatethe problem as an LP

Prices

A 3 - y1- 2y2 le0 y1 + 2y2 ge3

B 4 - y1- 3y2 le0 y1 + 3y2 ge4

C 6 - y1- 4y2 le0 y1 + 4y2 ge6

D 8 - y1- 5y2 le0 y1 + 5y2 ge8

minimizey1 + 3y2

8

The problem that we formed is calledthe dual problem

minimize y1+ 3y2

Subject to y1+ 2y2 ge

y1+ 3y2 ge

y1+ 5y2 ge

y1+ 4y2 ge

y1 a nd y2 are unconstrained in sign

9

Summary of previous slides

bull If ldquoreduced costsrdquo are non-positive then we have an upper bound on the objective value

bull The problem of finding the least upper bound is a linear program and is referred to as the dual of the original linear program

bull To do express duality in general notation

bull To do show that the shadow prices solve the dual and the bound is the optimal solution to the original problem

10

PRIMAL PROBLEM

maximize

subject toz = 3x1 + 4x2 +6x3 + 8x4

x1 + x2 + x3 + x4 = 1 2x1 + 3x2 +4x3 + 5x4 = 3 x1 x2 x3 x4 ge 0

maximize y1+ 3y2

subject to y1+ 2y2 ge

y1+ 3y2 ge

y1+ 4y2 ge

y1+ 5y2 ge

Observation 1

The constraint matrix in theprimal is the transpose of theconstraint matrix in the dual

Observation 2

The RHS coefficients in theprimal become the costcoefficients in the dual

DUSL PROBLEM

11

PRIMAL PROBLEM

maximize

subject toz = 3x1 + 4x2 +6x3 + 8x4

x1 + x2 + x3 + x4 = 1 2x1 + 3x2 +4x3 + 5x4 = 3 x1 x2 x3 x4 ge 0

DUSL PROBLEM

maximize y1+ 3y2

subject to y1+ 2y2 ge

y1+ 3y2 ge

y1+ 4y2 ge

y1+ 5y2 ge

Observation 3 The cost coefficientsin the primal become the RHScoefficients in the dual

Observation 4 The primal (in thiscase) is a max problem with equalityconstraints and non-negativevariables

The dual (in this case) is aminimization problem with geconstraints and variablesunconstrained in sign

12

PRIMAL PROBLEM (in standard form)

max z = c1x1 + c2x2 + c3x3 + hellip + cnxn

st a11x1 + a12x2 + a13x3 + hellip + a1nxn = b1

a21x1 + a22x2 + a23x3 + hellip + a2nxn = b2

hellip am1x1 + am2x2 + am3x3 + hellip + amnxn = bm xj ge0 for j = 1 to n

DUAL PROBLEM

max v = st

What is the dual problem in terms of the notation given above

13

PRIMAL PROBLEM (in standard form)

max z = c1x1 + c2x2 + c3x3 + hellip + cnxn

st a11x1 + a12x2 + a13x3 + hellip + a1nxn = b1

a21x1 + a22x2 + a23x3 + hellip + a2nxn = b2

hellip am1x1 + am2x2 + am3x3 + hellip + amnxn = bm

xj ge0 for j = 1 to n

DUAL PROBLEM

max v = b1y1 + b2y2 + b3y3 + hellip +bnyn

st a11y1 + a21y2 + a31y3 + hellip + am1ym ge c1

a12y1 + a22y2 + a32y3 + hellip + am2ym ge c2

hellip a1ny1 + a2ny2 + a3ny3 + hellip + amnym ge cn

14

Theorem Suppose that x is any feasiblesolution to the primal andy is any feasiblesolution to the dual Then

Σj=1n cjxj leΣi=1nyi bi (Max leMin)

Proof

Σj=1n cjxj leΣi=1nΣj=1m (yi aij )xi

leΣj=1m Σi=1n yi (aijxj)

leΣj=1m yi bi

Weak Duality Theorem

15

Unboundedness Property

Theorem Suppose that the primal (dual)problem has an unbounded solution Then thedual (primal) problem has no feasible solution

Proof

Suppose thaty was feasible for the dual Thenevery solution to the primal problem isunbounded above by

Σj=1m yi bi

16

Strong Duality Theorem

Theorem If the primal problem has a finiteoptimal solution value then so does the dualproblem and these two values are the same

17

Strong Duality Illustrated

Prices

Optimal dual solution y1 = ndash13 y2 = 53 v = 143

Optimal primal solution x1 = 23 x2 = 0 x3 = 0 x4 = 13 z = 143

18

Strong Duality Illustrated the final tableau

Optimal primal solution x1 = 23 x2 = 0 x3 = 0 x4 = 13 v = 143

Observation

The costcoeffients satisfycomplementaryslackness

19

Summary

bull If we take an LP in standard form (max) we can formulate a dual problem1048698bull Weak Duality Each solution of the dual gives an upper bound on the maximum objective value for the primal

bull Strong duality if there are feasible solutions to the primal and dual then the optimal objective value for both problems is the same

20

Shadow prices solve the dual

bull Theorem Suppose that the primal problem has a finite optimal solution value Then the shadow prices for the primal problem form an optimal solution to the dual problem (And the objective values are the same)

Davidrsquos Mineral Problem

21

FACTOptimalsimplexmultipliersare shadowprices

Shadowprices

22

Summary

bull The dual problem to a maximizing LP provides upper bounds on the optimal objective function

bull The maximum solution value for the primal is the same as the minimum solution value for the dual

bull The shadow prices are optimal for the dual LP

bull Next alternative optimality conditions

23

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij gecj for j = 1 to n

Optimality Condition 1A solution x is optimalfor the primal problem ifit is a basic feasiblesolution and the tableausatisfies the optimalityconditions

Optimality Condition 2

A solution x is optimal

for the primal problem if

it is feasible and if there is

a feasible solution y for

the dual with

Σj=1m cj xj = Σj=1m yi bi

24

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij ge cj for j = 1 to n

ComplementarySlackness Conditions

ndashSuppose that y is feasible

for the dual and let

1048698cj= cj - Σi=1m1048698yi aij

Suppose x is feasible for

the primal

Theorem (complementary

slackness) x and y are

optimal for the primal and

dual if and only if

cj xj = 0 for all j

25

Complementary Slackness Illustrated

prices

26

Next Lecture on Duality

bull Dual linear programs in general formbull Optimality Conditions in general formbull Illustrating Duality with 2-person 0-sum game theory

27

Coverage for first midterm(Closed book exam)

bull Formulations

bull 2D graphing and

finding opt solution

bull Setting up an LP in

standard form

bull The simplex algorithm

starting with a bfs

bull Phase 1 of the

simplex algorithm

bull Interpreting sensitivity analysis including shadow prices and ranges and reduced costs

bull Pricing out

bull Using tableaus to determine shadow prices reduced costs and ranges

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27

7

Obtaining the Best Bound Formulatethe problem as an LP

Prices

A 3 - y1- 2y2 le0 y1 + 2y2 ge3

B 4 - y1- 3y2 le0 y1 + 3y2 ge4

C 6 - y1- 4y2 le0 y1 + 4y2 ge6

D 8 - y1- 5y2 le0 y1 + 5y2 ge8

minimizey1 + 3y2

8

The problem that we formed is calledthe dual problem

minimize y1+ 3y2

Subject to y1+ 2y2 ge

y1+ 3y2 ge

y1+ 5y2 ge

y1+ 4y2 ge

y1 a nd y2 are unconstrained in sign

9

Summary of previous slides

bull If ldquoreduced costsrdquo are non-positive then we have an upper bound on the objective value

bull The problem of finding the least upper bound is a linear program and is referred to as the dual of the original linear program

bull To do express duality in general notation

bull To do show that the shadow prices solve the dual and the bound is the optimal solution to the original problem

10

PRIMAL PROBLEM

maximize

subject toz = 3x1 + 4x2 +6x3 + 8x4

x1 + x2 + x3 + x4 = 1 2x1 + 3x2 +4x3 + 5x4 = 3 x1 x2 x3 x4 ge 0

maximize y1+ 3y2

subject to y1+ 2y2 ge

y1+ 3y2 ge

y1+ 4y2 ge

y1+ 5y2 ge

Observation 1

The constraint matrix in theprimal is the transpose of theconstraint matrix in the dual

Observation 2

The RHS coefficients in theprimal become the costcoefficients in the dual

DUSL PROBLEM

11

PRIMAL PROBLEM

maximize

subject toz = 3x1 + 4x2 +6x3 + 8x4

x1 + x2 + x3 + x4 = 1 2x1 + 3x2 +4x3 + 5x4 = 3 x1 x2 x3 x4 ge 0

DUSL PROBLEM

maximize y1+ 3y2

subject to y1+ 2y2 ge

y1+ 3y2 ge

y1+ 4y2 ge

y1+ 5y2 ge

Observation 3 The cost coefficientsin the primal become the RHScoefficients in the dual

Observation 4 The primal (in thiscase) is a max problem with equalityconstraints and non-negativevariables

The dual (in this case) is aminimization problem with geconstraints and variablesunconstrained in sign

12

PRIMAL PROBLEM (in standard form)

max z = c1x1 + c2x2 + c3x3 + hellip + cnxn

st a11x1 + a12x2 + a13x3 + hellip + a1nxn = b1

a21x1 + a22x2 + a23x3 + hellip + a2nxn = b2

hellip am1x1 + am2x2 + am3x3 + hellip + amnxn = bm xj ge0 for j = 1 to n

DUAL PROBLEM

max v = st

What is the dual problem in terms of the notation given above

13

PRIMAL PROBLEM (in standard form)

max z = c1x1 + c2x2 + c3x3 + hellip + cnxn

st a11x1 + a12x2 + a13x3 + hellip + a1nxn = b1

a21x1 + a22x2 + a23x3 + hellip + a2nxn = b2

hellip am1x1 + am2x2 + am3x3 + hellip + amnxn = bm

xj ge0 for j = 1 to n

DUAL PROBLEM

max v = b1y1 + b2y2 + b3y3 + hellip +bnyn

st a11y1 + a21y2 + a31y3 + hellip + am1ym ge c1

a12y1 + a22y2 + a32y3 + hellip + am2ym ge c2

hellip a1ny1 + a2ny2 + a3ny3 + hellip + amnym ge cn

14

Theorem Suppose that x is any feasiblesolution to the primal andy is any feasiblesolution to the dual Then

Σj=1n cjxj leΣi=1nyi bi (Max leMin)

Proof

Σj=1n cjxj leΣi=1nΣj=1m (yi aij )xi

leΣj=1m Σi=1n yi (aijxj)

leΣj=1m yi bi

Weak Duality Theorem

15

Unboundedness Property

Theorem Suppose that the primal (dual)problem has an unbounded solution Then thedual (primal) problem has no feasible solution

Proof

Suppose thaty was feasible for the dual Thenevery solution to the primal problem isunbounded above by

Σj=1m yi bi

16

Strong Duality Theorem

Theorem If the primal problem has a finiteoptimal solution value then so does the dualproblem and these two values are the same

17

Strong Duality Illustrated

Prices

Optimal dual solution y1 = ndash13 y2 = 53 v = 143

Optimal primal solution x1 = 23 x2 = 0 x3 = 0 x4 = 13 z = 143

18

Strong Duality Illustrated the final tableau

Optimal primal solution x1 = 23 x2 = 0 x3 = 0 x4 = 13 v = 143

Observation

The costcoeffients satisfycomplementaryslackness

19

Summary

bull If we take an LP in standard form (max) we can formulate a dual problem1048698bull Weak Duality Each solution of the dual gives an upper bound on the maximum objective value for the primal

bull Strong duality if there are feasible solutions to the primal and dual then the optimal objective value for both problems is the same

20

Shadow prices solve the dual

bull Theorem Suppose that the primal problem has a finite optimal solution value Then the shadow prices for the primal problem form an optimal solution to the dual problem (And the objective values are the same)

Davidrsquos Mineral Problem

21

FACTOptimalsimplexmultipliersare shadowprices

Shadowprices

22

Summary

bull The dual problem to a maximizing LP provides upper bounds on the optimal objective function

bull The maximum solution value for the primal is the same as the minimum solution value for the dual

bull The shadow prices are optimal for the dual LP

bull Next alternative optimality conditions

23

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij gecj for j = 1 to n

Optimality Condition 1A solution x is optimalfor the primal problem ifit is a basic feasiblesolution and the tableausatisfies the optimalityconditions

Optimality Condition 2

A solution x is optimal

for the primal problem if

it is feasible and if there is

a feasible solution y for

the dual with

Σj=1m cj xj = Σj=1m yi bi

24

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij ge cj for j = 1 to n

ComplementarySlackness Conditions

ndashSuppose that y is feasible

for the dual and let

1048698cj= cj - Σi=1m1048698yi aij

Suppose x is feasible for

the primal

Theorem (complementary

slackness) x and y are

optimal for the primal and

dual if and only if

cj xj = 0 for all j

25

Complementary Slackness Illustrated

prices

26

Next Lecture on Duality

bull Dual linear programs in general formbull Optimality Conditions in general formbull Illustrating Duality with 2-person 0-sum game theory

27

Coverage for first midterm(Closed book exam)

bull Formulations

bull 2D graphing and

finding opt solution

bull Setting up an LP in

standard form

bull The simplex algorithm

starting with a bfs

bull Phase 1 of the

simplex algorithm

bull Interpreting sensitivity analysis including shadow prices and ranges and reduced costs

bull Pricing out

bull Using tableaus to determine shadow prices reduced costs and ranges

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27

8

The problem that we formed is calledthe dual problem

minimize y1+ 3y2

Subject to y1+ 2y2 ge

y1+ 3y2 ge

y1+ 5y2 ge

y1+ 4y2 ge

y1 a nd y2 are unconstrained in sign

9

Summary of previous slides

bull If ldquoreduced costsrdquo are non-positive then we have an upper bound on the objective value

bull The problem of finding the least upper bound is a linear program and is referred to as the dual of the original linear program

bull To do express duality in general notation

bull To do show that the shadow prices solve the dual and the bound is the optimal solution to the original problem

10

PRIMAL PROBLEM

maximize

subject toz = 3x1 + 4x2 +6x3 + 8x4

x1 + x2 + x3 + x4 = 1 2x1 + 3x2 +4x3 + 5x4 = 3 x1 x2 x3 x4 ge 0

maximize y1+ 3y2

subject to y1+ 2y2 ge

y1+ 3y2 ge

y1+ 4y2 ge

y1+ 5y2 ge

Observation 1

The constraint matrix in theprimal is the transpose of theconstraint matrix in the dual

Observation 2

The RHS coefficients in theprimal become the costcoefficients in the dual

DUSL PROBLEM

11

PRIMAL PROBLEM

maximize

subject toz = 3x1 + 4x2 +6x3 + 8x4

x1 + x2 + x3 + x4 = 1 2x1 + 3x2 +4x3 + 5x4 = 3 x1 x2 x3 x4 ge 0

DUSL PROBLEM

maximize y1+ 3y2

subject to y1+ 2y2 ge

y1+ 3y2 ge

y1+ 4y2 ge

y1+ 5y2 ge

Observation 3 The cost coefficientsin the primal become the RHScoefficients in the dual

Observation 4 The primal (in thiscase) is a max problem with equalityconstraints and non-negativevariables

The dual (in this case) is aminimization problem with geconstraints and variablesunconstrained in sign

12

PRIMAL PROBLEM (in standard form)

max z = c1x1 + c2x2 + c3x3 + hellip + cnxn

st a11x1 + a12x2 + a13x3 + hellip + a1nxn = b1

a21x1 + a22x2 + a23x3 + hellip + a2nxn = b2

hellip am1x1 + am2x2 + am3x3 + hellip + amnxn = bm xj ge0 for j = 1 to n

DUAL PROBLEM

max v = st

What is the dual problem in terms of the notation given above

13

PRIMAL PROBLEM (in standard form)

max z = c1x1 + c2x2 + c3x3 + hellip + cnxn

st a11x1 + a12x2 + a13x3 + hellip + a1nxn = b1

a21x1 + a22x2 + a23x3 + hellip + a2nxn = b2

hellip am1x1 + am2x2 + am3x3 + hellip + amnxn = bm

xj ge0 for j = 1 to n

DUAL PROBLEM

max v = b1y1 + b2y2 + b3y3 + hellip +bnyn

st a11y1 + a21y2 + a31y3 + hellip + am1ym ge c1

a12y1 + a22y2 + a32y3 + hellip + am2ym ge c2

hellip a1ny1 + a2ny2 + a3ny3 + hellip + amnym ge cn

14

Theorem Suppose that x is any feasiblesolution to the primal andy is any feasiblesolution to the dual Then

Σj=1n cjxj leΣi=1nyi bi (Max leMin)

Proof

Σj=1n cjxj leΣi=1nΣj=1m (yi aij )xi

leΣj=1m Σi=1n yi (aijxj)

leΣj=1m yi bi

Weak Duality Theorem

15

Unboundedness Property

Theorem Suppose that the primal (dual)problem has an unbounded solution Then thedual (primal) problem has no feasible solution

Proof

Suppose thaty was feasible for the dual Thenevery solution to the primal problem isunbounded above by

Σj=1m yi bi

16

Strong Duality Theorem

Theorem If the primal problem has a finiteoptimal solution value then so does the dualproblem and these two values are the same

17

Strong Duality Illustrated

Prices

Optimal dual solution y1 = ndash13 y2 = 53 v = 143

Optimal primal solution x1 = 23 x2 = 0 x3 = 0 x4 = 13 z = 143

18

Strong Duality Illustrated the final tableau

Optimal primal solution x1 = 23 x2 = 0 x3 = 0 x4 = 13 v = 143

Observation

The costcoeffients satisfycomplementaryslackness

19

Summary

bull If we take an LP in standard form (max) we can formulate a dual problem1048698bull Weak Duality Each solution of the dual gives an upper bound on the maximum objective value for the primal

bull Strong duality if there are feasible solutions to the primal and dual then the optimal objective value for both problems is the same

20

Shadow prices solve the dual

bull Theorem Suppose that the primal problem has a finite optimal solution value Then the shadow prices for the primal problem form an optimal solution to the dual problem (And the objective values are the same)

Davidrsquos Mineral Problem

21

FACTOptimalsimplexmultipliersare shadowprices

Shadowprices

22

Summary

bull The dual problem to a maximizing LP provides upper bounds on the optimal objective function

bull The maximum solution value for the primal is the same as the minimum solution value for the dual

bull The shadow prices are optimal for the dual LP

bull Next alternative optimality conditions

23

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij gecj for j = 1 to n

Optimality Condition 1A solution x is optimalfor the primal problem ifit is a basic feasiblesolution and the tableausatisfies the optimalityconditions

Optimality Condition 2

A solution x is optimal

for the primal problem if

it is feasible and if there is

a feasible solution y for

the dual with

Σj=1m cj xj = Σj=1m yi bi

24

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij ge cj for j = 1 to n

ComplementarySlackness Conditions

ndashSuppose that y is feasible

for the dual and let

1048698cj= cj - Σi=1m1048698yi aij

Suppose x is feasible for

the primal

Theorem (complementary

slackness) x and y are

optimal for the primal and

dual if and only if

cj xj = 0 for all j

25

Complementary Slackness Illustrated

prices

26

Next Lecture on Duality

bull Dual linear programs in general formbull Optimality Conditions in general formbull Illustrating Duality with 2-person 0-sum game theory

27

Coverage for first midterm(Closed book exam)

bull Formulations

bull 2D graphing and

finding opt solution

bull Setting up an LP in

standard form

bull The simplex algorithm

starting with a bfs

bull Phase 1 of the

simplex algorithm

bull Interpreting sensitivity analysis including shadow prices and ranges and reduced costs

bull Pricing out

bull Using tableaus to determine shadow prices reduced costs and ranges

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27

9

Summary of previous slides

bull If ldquoreduced costsrdquo are non-positive then we have an upper bound on the objective value

bull The problem of finding the least upper bound is a linear program and is referred to as the dual of the original linear program

bull To do express duality in general notation

bull To do show that the shadow prices solve the dual and the bound is the optimal solution to the original problem

10

PRIMAL PROBLEM

maximize

subject toz = 3x1 + 4x2 +6x3 + 8x4

x1 + x2 + x3 + x4 = 1 2x1 + 3x2 +4x3 + 5x4 = 3 x1 x2 x3 x4 ge 0

maximize y1+ 3y2

subject to y1+ 2y2 ge

y1+ 3y2 ge

y1+ 4y2 ge

y1+ 5y2 ge

Observation 1

The constraint matrix in theprimal is the transpose of theconstraint matrix in the dual

Observation 2

The RHS coefficients in theprimal become the costcoefficients in the dual

DUSL PROBLEM

11

PRIMAL PROBLEM

maximize

subject toz = 3x1 + 4x2 +6x3 + 8x4

x1 + x2 + x3 + x4 = 1 2x1 + 3x2 +4x3 + 5x4 = 3 x1 x2 x3 x4 ge 0

DUSL PROBLEM

maximize y1+ 3y2

subject to y1+ 2y2 ge

y1+ 3y2 ge

y1+ 4y2 ge

y1+ 5y2 ge

Observation 3 The cost coefficientsin the primal become the RHScoefficients in the dual

Observation 4 The primal (in thiscase) is a max problem with equalityconstraints and non-negativevariables

The dual (in this case) is aminimization problem with geconstraints and variablesunconstrained in sign

12

PRIMAL PROBLEM (in standard form)

max z = c1x1 + c2x2 + c3x3 + hellip + cnxn

st a11x1 + a12x2 + a13x3 + hellip + a1nxn = b1

a21x1 + a22x2 + a23x3 + hellip + a2nxn = b2

hellip am1x1 + am2x2 + am3x3 + hellip + amnxn = bm xj ge0 for j = 1 to n

DUAL PROBLEM

max v = st

What is the dual problem in terms of the notation given above

13

PRIMAL PROBLEM (in standard form)

max z = c1x1 + c2x2 + c3x3 + hellip + cnxn

st a11x1 + a12x2 + a13x3 + hellip + a1nxn = b1

a21x1 + a22x2 + a23x3 + hellip + a2nxn = b2

hellip am1x1 + am2x2 + am3x3 + hellip + amnxn = bm

xj ge0 for j = 1 to n

DUAL PROBLEM

max v = b1y1 + b2y2 + b3y3 + hellip +bnyn

st a11y1 + a21y2 + a31y3 + hellip + am1ym ge c1

a12y1 + a22y2 + a32y3 + hellip + am2ym ge c2

hellip a1ny1 + a2ny2 + a3ny3 + hellip + amnym ge cn

14

Theorem Suppose that x is any feasiblesolution to the primal andy is any feasiblesolution to the dual Then

Σj=1n cjxj leΣi=1nyi bi (Max leMin)

Proof

Σj=1n cjxj leΣi=1nΣj=1m (yi aij )xi

leΣj=1m Σi=1n yi (aijxj)

leΣj=1m yi bi

Weak Duality Theorem

15

Unboundedness Property

Theorem Suppose that the primal (dual)problem has an unbounded solution Then thedual (primal) problem has no feasible solution

Proof

Suppose thaty was feasible for the dual Thenevery solution to the primal problem isunbounded above by

Σj=1m yi bi

16

Strong Duality Theorem

Theorem If the primal problem has a finiteoptimal solution value then so does the dualproblem and these two values are the same

17

Strong Duality Illustrated

Prices

Optimal dual solution y1 = ndash13 y2 = 53 v = 143

Optimal primal solution x1 = 23 x2 = 0 x3 = 0 x4 = 13 z = 143

18

Strong Duality Illustrated the final tableau

Optimal primal solution x1 = 23 x2 = 0 x3 = 0 x4 = 13 v = 143

Observation

The costcoeffients satisfycomplementaryslackness

19

Summary

bull If we take an LP in standard form (max) we can formulate a dual problem1048698bull Weak Duality Each solution of the dual gives an upper bound on the maximum objective value for the primal

bull Strong duality if there are feasible solutions to the primal and dual then the optimal objective value for both problems is the same

20

Shadow prices solve the dual

bull Theorem Suppose that the primal problem has a finite optimal solution value Then the shadow prices for the primal problem form an optimal solution to the dual problem (And the objective values are the same)

Davidrsquos Mineral Problem

21

FACTOptimalsimplexmultipliersare shadowprices

Shadowprices

22

Summary

bull The dual problem to a maximizing LP provides upper bounds on the optimal objective function

bull The maximum solution value for the primal is the same as the minimum solution value for the dual

bull The shadow prices are optimal for the dual LP

bull Next alternative optimality conditions

23

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij gecj for j = 1 to n

Optimality Condition 1A solution x is optimalfor the primal problem ifit is a basic feasiblesolution and the tableausatisfies the optimalityconditions

Optimality Condition 2

A solution x is optimal

for the primal problem if

it is feasible and if there is

a feasible solution y for

the dual with

Σj=1m cj xj = Σj=1m yi bi

24

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij ge cj for j = 1 to n

ComplementarySlackness Conditions

ndashSuppose that y is feasible

for the dual and let

1048698cj= cj - Σi=1m1048698yi aij

Suppose x is feasible for

the primal

Theorem (complementary

slackness) x and y are

optimal for the primal and

dual if and only if

cj xj = 0 for all j

25

Complementary Slackness Illustrated

prices

26

Next Lecture on Duality

bull Dual linear programs in general formbull Optimality Conditions in general formbull Illustrating Duality with 2-person 0-sum game theory

27

Coverage for first midterm(Closed book exam)

bull Formulations

bull 2D graphing and

finding opt solution

bull Setting up an LP in

standard form

bull The simplex algorithm

starting with a bfs

bull Phase 1 of the

simplex algorithm

bull Interpreting sensitivity analysis including shadow prices and ranges and reduced costs

bull Pricing out

bull Using tableaus to determine shadow prices reduced costs and ranges

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27

10

PRIMAL PROBLEM

maximize

subject toz = 3x1 + 4x2 +6x3 + 8x4

x1 + x2 + x3 + x4 = 1 2x1 + 3x2 +4x3 + 5x4 = 3 x1 x2 x3 x4 ge 0

maximize y1+ 3y2

subject to y1+ 2y2 ge

y1+ 3y2 ge

y1+ 4y2 ge

y1+ 5y2 ge

Observation 1

The constraint matrix in theprimal is the transpose of theconstraint matrix in the dual

Observation 2

The RHS coefficients in theprimal become the costcoefficients in the dual

DUSL PROBLEM

11

PRIMAL PROBLEM

maximize

subject toz = 3x1 + 4x2 +6x3 + 8x4

x1 + x2 + x3 + x4 = 1 2x1 + 3x2 +4x3 + 5x4 = 3 x1 x2 x3 x4 ge 0

DUSL PROBLEM

maximize y1+ 3y2

subject to y1+ 2y2 ge

y1+ 3y2 ge

y1+ 4y2 ge

y1+ 5y2 ge

Observation 3 The cost coefficientsin the primal become the RHScoefficients in the dual

Observation 4 The primal (in thiscase) is a max problem with equalityconstraints and non-negativevariables

The dual (in this case) is aminimization problem with geconstraints and variablesunconstrained in sign

12

PRIMAL PROBLEM (in standard form)

max z = c1x1 + c2x2 + c3x3 + hellip + cnxn

st a11x1 + a12x2 + a13x3 + hellip + a1nxn = b1

a21x1 + a22x2 + a23x3 + hellip + a2nxn = b2

hellip am1x1 + am2x2 + am3x3 + hellip + amnxn = bm xj ge0 for j = 1 to n

DUAL PROBLEM

max v = st

What is the dual problem in terms of the notation given above

13

PRIMAL PROBLEM (in standard form)

max z = c1x1 + c2x2 + c3x3 + hellip + cnxn

st a11x1 + a12x2 + a13x3 + hellip + a1nxn = b1

a21x1 + a22x2 + a23x3 + hellip + a2nxn = b2

hellip am1x1 + am2x2 + am3x3 + hellip + amnxn = bm

xj ge0 for j = 1 to n

DUAL PROBLEM

max v = b1y1 + b2y2 + b3y3 + hellip +bnyn

st a11y1 + a21y2 + a31y3 + hellip + am1ym ge c1

a12y1 + a22y2 + a32y3 + hellip + am2ym ge c2

hellip a1ny1 + a2ny2 + a3ny3 + hellip + amnym ge cn

14

Theorem Suppose that x is any feasiblesolution to the primal andy is any feasiblesolution to the dual Then

Σj=1n cjxj leΣi=1nyi bi (Max leMin)

Proof

Σj=1n cjxj leΣi=1nΣj=1m (yi aij )xi

leΣj=1m Σi=1n yi (aijxj)

leΣj=1m yi bi

Weak Duality Theorem

15

Unboundedness Property

Theorem Suppose that the primal (dual)problem has an unbounded solution Then thedual (primal) problem has no feasible solution

Proof

Suppose thaty was feasible for the dual Thenevery solution to the primal problem isunbounded above by

Σj=1m yi bi

16

Strong Duality Theorem

Theorem If the primal problem has a finiteoptimal solution value then so does the dualproblem and these two values are the same

17

Strong Duality Illustrated

Prices

Optimal dual solution y1 = ndash13 y2 = 53 v = 143

Optimal primal solution x1 = 23 x2 = 0 x3 = 0 x4 = 13 z = 143

18

Strong Duality Illustrated the final tableau

Optimal primal solution x1 = 23 x2 = 0 x3 = 0 x4 = 13 v = 143

Observation

The costcoeffients satisfycomplementaryslackness

19

Summary

bull If we take an LP in standard form (max) we can formulate a dual problem1048698bull Weak Duality Each solution of the dual gives an upper bound on the maximum objective value for the primal

bull Strong duality if there are feasible solutions to the primal and dual then the optimal objective value for both problems is the same

20

Shadow prices solve the dual

bull Theorem Suppose that the primal problem has a finite optimal solution value Then the shadow prices for the primal problem form an optimal solution to the dual problem (And the objective values are the same)

Davidrsquos Mineral Problem

21

FACTOptimalsimplexmultipliersare shadowprices

Shadowprices

22

Summary

bull The dual problem to a maximizing LP provides upper bounds on the optimal objective function

bull The maximum solution value for the primal is the same as the minimum solution value for the dual

bull The shadow prices are optimal for the dual LP

bull Next alternative optimality conditions

23

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij gecj for j = 1 to n

Optimality Condition 1A solution x is optimalfor the primal problem ifit is a basic feasiblesolution and the tableausatisfies the optimalityconditions

Optimality Condition 2

A solution x is optimal

for the primal problem if

it is feasible and if there is

a feasible solution y for

the dual with

Σj=1m cj xj = Σj=1m yi bi

24

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij ge cj for j = 1 to n

ComplementarySlackness Conditions

ndashSuppose that y is feasible

for the dual and let

1048698cj= cj - Σi=1m1048698yi aij

Suppose x is feasible for

the primal

Theorem (complementary

slackness) x and y are

optimal for the primal and

dual if and only if

cj xj = 0 for all j

25

Complementary Slackness Illustrated

prices

26

Next Lecture on Duality

bull Dual linear programs in general formbull Optimality Conditions in general formbull Illustrating Duality with 2-person 0-sum game theory

27

Coverage for first midterm(Closed book exam)

bull Formulations

bull 2D graphing and

finding opt solution

bull Setting up an LP in

standard form

bull The simplex algorithm

starting with a bfs

bull Phase 1 of the

simplex algorithm

bull Interpreting sensitivity analysis including shadow prices and ranges and reduced costs

bull Pricing out

bull Using tableaus to determine shadow prices reduced costs and ranges

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27

11

PRIMAL PROBLEM

maximize

subject toz = 3x1 + 4x2 +6x3 + 8x4

x1 + x2 + x3 + x4 = 1 2x1 + 3x2 +4x3 + 5x4 = 3 x1 x2 x3 x4 ge 0

DUSL PROBLEM

maximize y1+ 3y2

subject to y1+ 2y2 ge

y1+ 3y2 ge

y1+ 4y2 ge

y1+ 5y2 ge

Observation 3 The cost coefficientsin the primal become the RHScoefficients in the dual

Observation 4 The primal (in thiscase) is a max problem with equalityconstraints and non-negativevariables

The dual (in this case) is aminimization problem with geconstraints and variablesunconstrained in sign

12

PRIMAL PROBLEM (in standard form)

max z = c1x1 + c2x2 + c3x3 + hellip + cnxn

st a11x1 + a12x2 + a13x3 + hellip + a1nxn = b1

a21x1 + a22x2 + a23x3 + hellip + a2nxn = b2

hellip am1x1 + am2x2 + am3x3 + hellip + amnxn = bm xj ge0 for j = 1 to n

DUAL PROBLEM

max v = st

What is the dual problem in terms of the notation given above

13

PRIMAL PROBLEM (in standard form)

max z = c1x1 + c2x2 + c3x3 + hellip + cnxn

st a11x1 + a12x2 + a13x3 + hellip + a1nxn = b1

a21x1 + a22x2 + a23x3 + hellip + a2nxn = b2

hellip am1x1 + am2x2 + am3x3 + hellip + amnxn = bm

xj ge0 for j = 1 to n

DUAL PROBLEM

max v = b1y1 + b2y2 + b3y3 + hellip +bnyn

st a11y1 + a21y2 + a31y3 + hellip + am1ym ge c1

a12y1 + a22y2 + a32y3 + hellip + am2ym ge c2

hellip a1ny1 + a2ny2 + a3ny3 + hellip + amnym ge cn

14

Theorem Suppose that x is any feasiblesolution to the primal andy is any feasiblesolution to the dual Then

Σj=1n cjxj leΣi=1nyi bi (Max leMin)

Proof

Σj=1n cjxj leΣi=1nΣj=1m (yi aij )xi

leΣj=1m Σi=1n yi (aijxj)

leΣj=1m yi bi

Weak Duality Theorem

15

Unboundedness Property

Theorem Suppose that the primal (dual)problem has an unbounded solution Then thedual (primal) problem has no feasible solution

Proof

Suppose thaty was feasible for the dual Thenevery solution to the primal problem isunbounded above by

Σj=1m yi bi

16

Strong Duality Theorem

Theorem If the primal problem has a finiteoptimal solution value then so does the dualproblem and these two values are the same

17

Strong Duality Illustrated

Prices

Optimal dual solution y1 = ndash13 y2 = 53 v = 143

Optimal primal solution x1 = 23 x2 = 0 x3 = 0 x4 = 13 z = 143

18

Strong Duality Illustrated the final tableau

Optimal primal solution x1 = 23 x2 = 0 x3 = 0 x4 = 13 v = 143

Observation

The costcoeffients satisfycomplementaryslackness

19

Summary

bull If we take an LP in standard form (max) we can formulate a dual problem1048698bull Weak Duality Each solution of the dual gives an upper bound on the maximum objective value for the primal

bull Strong duality if there are feasible solutions to the primal and dual then the optimal objective value for both problems is the same

20

Shadow prices solve the dual

bull Theorem Suppose that the primal problem has a finite optimal solution value Then the shadow prices for the primal problem form an optimal solution to the dual problem (And the objective values are the same)

Davidrsquos Mineral Problem

21

FACTOptimalsimplexmultipliersare shadowprices

Shadowprices

22

Summary

bull The dual problem to a maximizing LP provides upper bounds on the optimal objective function

bull The maximum solution value for the primal is the same as the minimum solution value for the dual

bull The shadow prices are optimal for the dual LP

bull Next alternative optimality conditions

23

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij gecj for j = 1 to n

Optimality Condition 1A solution x is optimalfor the primal problem ifit is a basic feasiblesolution and the tableausatisfies the optimalityconditions

Optimality Condition 2

A solution x is optimal

for the primal problem if

it is feasible and if there is

a feasible solution y for

the dual with

Σj=1m cj xj = Σj=1m yi bi

24

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij ge cj for j = 1 to n

ComplementarySlackness Conditions

ndashSuppose that y is feasible

for the dual and let

1048698cj= cj - Σi=1m1048698yi aij

Suppose x is feasible for

the primal

Theorem (complementary

slackness) x and y are

optimal for the primal and

dual if and only if

cj xj = 0 for all j

25

Complementary Slackness Illustrated

prices

26

Next Lecture on Duality

bull Dual linear programs in general formbull Optimality Conditions in general formbull Illustrating Duality with 2-person 0-sum game theory

27

Coverage for first midterm(Closed book exam)

bull Formulations

bull 2D graphing and

finding opt solution

bull Setting up an LP in

standard form

bull The simplex algorithm

starting with a bfs

bull Phase 1 of the

simplex algorithm

bull Interpreting sensitivity analysis including shadow prices and ranges and reduced costs

bull Pricing out

bull Using tableaus to determine shadow prices reduced costs and ranges

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27

12

PRIMAL PROBLEM (in standard form)

max z = c1x1 + c2x2 + c3x3 + hellip + cnxn

st a11x1 + a12x2 + a13x3 + hellip + a1nxn = b1

a21x1 + a22x2 + a23x3 + hellip + a2nxn = b2

hellip am1x1 + am2x2 + am3x3 + hellip + amnxn = bm xj ge0 for j = 1 to n

DUAL PROBLEM

max v = st

What is the dual problem in terms of the notation given above

13

PRIMAL PROBLEM (in standard form)

max z = c1x1 + c2x2 + c3x3 + hellip + cnxn

st a11x1 + a12x2 + a13x3 + hellip + a1nxn = b1

a21x1 + a22x2 + a23x3 + hellip + a2nxn = b2

hellip am1x1 + am2x2 + am3x3 + hellip + amnxn = bm

xj ge0 for j = 1 to n

DUAL PROBLEM

max v = b1y1 + b2y2 + b3y3 + hellip +bnyn

st a11y1 + a21y2 + a31y3 + hellip + am1ym ge c1

a12y1 + a22y2 + a32y3 + hellip + am2ym ge c2

hellip a1ny1 + a2ny2 + a3ny3 + hellip + amnym ge cn

14

Theorem Suppose that x is any feasiblesolution to the primal andy is any feasiblesolution to the dual Then

Σj=1n cjxj leΣi=1nyi bi (Max leMin)

Proof

Σj=1n cjxj leΣi=1nΣj=1m (yi aij )xi

leΣj=1m Σi=1n yi (aijxj)

leΣj=1m yi bi

Weak Duality Theorem

15

Unboundedness Property

Theorem Suppose that the primal (dual)problem has an unbounded solution Then thedual (primal) problem has no feasible solution

Proof

Suppose thaty was feasible for the dual Thenevery solution to the primal problem isunbounded above by

Σj=1m yi bi

16

Strong Duality Theorem

Theorem If the primal problem has a finiteoptimal solution value then so does the dualproblem and these two values are the same

17

Strong Duality Illustrated

Prices

Optimal dual solution y1 = ndash13 y2 = 53 v = 143

Optimal primal solution x1 = 23 x2 = 0 x3 = 0 x4 = 13 z = 143

18

Strong Duality Illustrated the final tableau

Optimal primal solution x1 = 23 x2 = 0 x3 = 0 x4 = 13 v = 143

Observation

The costcoeffients satisfycomplementaryslackness

19

Summary

bull If we take an LP in standard form (max) we can formulate a dual problem1048698bull Weak Duality Each solution of the dual gives an upper bound on the maximum objective value for the primal

bull Strong duality if there are feasible solutions to the primal and dual then the optimal objective value for both problems is the same

20

Shadow prices solve the dual

bull Theorem Suppose that the primal problem has a finite optimal solution value Then the shadow prices for the primal problem form an optimal solution to the dual problem (And the objective values are the same)

Davidrsquos Mineral Problem

21

FACTOptimalsimplexmultipliersare shadowprices

Shadowprices

22

Summary

bull The dual problem to a maximizing LP provides upper bounds on the optimal objective function

bull The maximum solution value for the primal is the same as the minimum solution value for the dual

bull The shadow prices are optimal for the dual LP

bull Next alternative optimality conditions

23

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij gecj for j = 1 to n

Optimality Condition 1A solution x is optimalfor the primal problem ifit is a basic feasiblesolution and the tableausatisfies the optimalityconditions

Optimality Condition 2

A solution x is optimal

for the primal problem if

it is feasible and if there is

a feasible solution y for

the dual with

Σj=1m cj xj = Σj=1m yi bi

24

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij ge cj for j = 1 to n

ComplementarySlackness Conditions

ndashSuppose that y is feasible

for the dual and let

1048698cj= cj - Σi=1m1048698yi aij

Suppose x is feasible for

the primal

Theorem (complementary

slackness) x and y are

optimal for the primal and

dual if and only if

cj xj = 0 for all j

25

Complementary Slackness Illustrated

prices

26

Next Lecture on Duality

bull Dual linear programs in general formbull Optimality Conditions in general formbull Illustrating Duality with 2-person 0-sum game theory

27

Coverage for first midterm(Closed book exam)

bull Formulations

bull 2D graphing and

finding opt solution

bull Setting up an LP in

standard form

bull The simplex algorithm

starting with a bfs

bull Phase 1 of the

simplex algorithm

bull Interpreting sensitivity analysis including shadow prices and ranges and reduced costs

bull Pricing out

bull Using tableaus to determine shadow prices reduced costs and ranges

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27

13

PRIMAL PROBLEM (in standard form)

max z = c1x1 + c2x2 + c3x3 + hellip + cnxn

st a11x1 + a12x2 + a13x3 + hellip + a1nxn = b1

a21x1 + a22x2 + a23x3 + hellip + a2nxn = b2

hellip am1x1 + am2x2 + am3x3 + hellip + amnxn = bm

xj ge0 for j = 1 to n

DUAL PROBLEM

max v = b1y1 + b2y2 + b3y3 + hellip +bnyn

st a11y1 + a21y2 + a31y3 + hellip + am1ym ge c1

a12y1 + a22y2 + a32y3 + hellip + am2ym ge c2

hellip a1ny1 + a2ny2 + a3ny3 + hellip + amnym ge cn

14

Theorem Suppose that x is any feasiblesolution to the primal andy is any feasiblesolution to the dual Then

Σj=1n cjxj leΣi=1nyi bi (Max leMin)

Proof

Σj=1n cjxj leΣi=1nΣj=1m (yi aij )xi

leΣj=1m Σi=1n yi (aijxj)

leΣj=1m yi bi

Weak Duality Theorem

15

Unboundedness Property

Theorem Suppose that the primal (dual)problem has an unbounded solution Then thedual (primal) problem has no feasible solution

Proof

Suppose thaty was feasible for the dual Thenevery solution to the primal problem isunbounded above by

Σj=1m yi bi

16

Strong Duality Theorem

Theorem If the primal problem has a finiteoptimal solution value then so does the dualproblem and these two values are the same

17

Strong Duality Illustrated

Prices

Optimal dual solution y1 = ndash13 y2 = 53 v = 143

Optimal primal solution x1 = 23 x2 = 0 x3 = 0 x4 = 13 z = 143

18

Strong Duality Illustrated the final tableau

Optimal primal solution x1 = 23 x2 = 0 x3 = 0 x4 = 13 v = 143

Observation

The costcoeffients satisfycomplementaryslackness

19

Summary

bull If we take an LP in standard form (max) we can formulate a dual problem1048698bull Weak Duality Each solution of the dual gives an upper bound on the maximum objective value for the primal

bull Strong duality if there are feasible solutions to the primal and dual then the optimal objective value for both problems is the same

20

Shadow prices solve the dual

bull Theorem Suppose that the primal problem has a finite optimal solution value Then the shadow prices for the primal problem form an optimal solution to the dual problem (And the objective values are the same)

Davidrsquos Mineral Problem

21

FACTOptimalsimplexmultipliersare shadowprices

Shadowprices

22

Summary

bull The dual problem to a maximizing LP provides upper bounds on the optimal objective function

bull The maximum solution value for the primal is the same as the minimum solution value for the dual

bull The shadow prices are optimal for the dual LP

bull Next alternative optimality conditions

23

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij gecj for j = 1 to n

Optimality Condition 1A solution x is optimalfor the primal problem ifit is a basic feasiblesolution and the tableausatisfies the optimalityconditions

Optimality Condition 2

A solution x is optimal

for the primal problem if

it is feasible and if there is

a feasible solution y for

the dual with

Σj=1m cj xj = Σj=1m yi bi

24

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij ge cj for j = 1 to n

ComplementarySlackness Conditions

ndashSuppose that y is feasible

for the dual and let

1048698cj= cj - Σi=1m1048698yi aij

Suppose x is feasible for

the primal

Theorem (complementary

slackness) x and y are

optimal for the primal and

dual if and only if

cj xj = 0 for all j

25

Complementary Slackness Illustrated

prices

26

Next Lecture on Duality

bull Dual linear programs in general formbull Optimality Conditions in general formbull Illustrating Duality with 2-person 0-sum game theory

27

Coverage for first midterm(Closed book exam)

bull Formulations

bull 2D graphing and

finding opt solution

bull Setting up an LP in

standard form

bull The simplex algorithm

starting with a bfs

bull Phase 1 of the

simplex algorithm

bull Interpreting sensitivity analysis including shadow prices and ranges and reduced costs

bull Pricing out

bull Using tableaus to determine shadow prices reduced costs and ranges

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27

14

Theorem Suppose that x is any feasiblesolution to the primal andy is any feasiblesolution to the dual Then

Σj=1n cjxj leΣi=1nyi bi (Max leMin)

Proof

Σj=1n cjxj leΣi=1nΣj=1m (yi aij )xi

leΣj=1m Σi=1n yi (aijxj)

leΣj=1m yi bi

Weak Duality Theorem

15

Unboundedness Property

Theorem Suppose that the primal (dual)problem has an unbounded solution Then thedual (primal) problem has no feasible solution

Proof

Suppose thaty was feasible for the dual Thenevery solution to the primal problem isunbounded above by

Σj=1m yi bi

16

Strong Duality Theorem

Theorem If the primal problem has a finiteoptimal solution value then so does the dualproblem and these two values are the same

17

Strong Duality Illustrated

Prices

Optimal dual solution y1 = ndash13 y2 = 53 v = 143

Optimal primal solution x1 = 23 x2 = 0 x3 = 0 x4 = 13 z = 143

18

Strong Duality Illustrated the final tableau

Optimal primal solution x1 = 23 x2 = 0 x3 = 0 x4 = 13 v = 143

Observation

The costcoeffients satisfycomplementaryslackness

19

Summary

bull If we take an LP in standard form (max) we can formulate a dual problem1048698bull Weak Duality Each solution of the dual gives an upper bound on the maximum objective value for the primal

bull Strong duality if there are feasible solutions to the primal and dual then the optimal objective value for both problems is the same

20

Shadow prices solve the dual

bull Theorem Suppose that the primal problem has a finite optimal solution value Then the shadow prices for the primal problem form an optimal solution to the dual problem (And the objective values are the same)

Davidrsquos Mineral Problem

21

FACTOptimalsimplexmultipliersare shadowprices

Shadowprices

22

Summary

bull The dual problem to a maximizing LP provides upper bounds on the optimal objective function

bull The maximum solution value for the primal is the same as the minimum solution value for the dual

bull The shadow prices are optimal for the dual LP

bull Next alternative optimality conditions

23

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij gecj for j = 1 to n

Optimality Condition 1A solution x is optimalfor the primal problem ifit is a basic feasiblesolution and the tableausatisfies the optimalityconditions

Optimality Condition 2

A solution x is optimal

for the primal problem if

it is feasible and if there is

a feasible solution y for

the dual with

Σj=1m cj xj = Σj=1m yi bi

24

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij ge cj for j = 1 to n

ComplementarySlackness Conditions

ndashSuppose that y is feasible

for the dual and let

1048698cj= cj - Σi=1m1048698yi aij

Suppose x is feasible for

the primal

Theorem (complementary

slackness) x and y are

optimal for the primal and

dual if and only if

cj xj = 0 for all j

25

Complementary Slackness Illustrated

prices

26

Next Lecture on Duality

bull Dual linear programs in general formbull Optimality Conditions in general formbull Illustrating Duality with 2-person 0-sum game theory

27

Coverage for first midterm(Closed book exam)

bull Formulations

bull 2D graphing and

finding opt solution

bull Setting up an LP in

standard form

bull The simplex algorithm

starting with a bfs

bull Phase 1 of the

simplex algorithm

bull Interpreting sensitivity analysis including shadow prices and ranges and reduced costs

bull Pricing out

bull Using tableaus to determine shadow prices reduced costs and ranges

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27

15

Unboundedness Property

Theorem Suppose that the primal (dual)problem has an unbounded solution Then thedual (primal) problem has no feasible solution

Proof

Suppose thaty was feasible for the dual Thenevery solution to the primal problem isunbounded above by

Σj=1m yi bi

16

Strong Duality Theorem

Theorem If the primal problem has a finiteoptimal solution value then so does the dualproblem and these two values are the same

17

Strong Duality Illustrated

Prices

Optimal dual solution y1 = ndash13 y2 = 53 v = 143

Optimal primal solution x1 = 23 x2 = 0 x3 = 0 x4 = 13 z = 143

18

Strong Duality Illustrated the final tableau

Optimal primal solution x1 = 23 x2 = 0 x3 = 0 x4 = 13 v = 143

Observation

The costcoeffients satisfycomplementaryslackness

19

Summary

bull If we take an LP in standard form (max) we can formulate a dual problem1048698bull Weak Duality Each solution of the dual gives an upper bound on the maximum objective value for the primal

bull Strong duality if there are feasible solutions to the primal and dual then the optimal objective value for both problems is the same

20

Shadow prices solve the dual

bull Theorem Suppose that the primal problem has a finite optimal solution value Then the shadow prices for the primal problem form an optimal solution to the dual problem (And the objective values are the same)

Davidrsquos Mineral Problem

21

FACTOptimalsimplexmultipliersare shadowprices

Shadowprices

22

Summary

bull The dual problem to a maximizing LP provides upper bounds on the optimal objective function

bull The maximum solution value for the primal is the same as the minimum solution value for the dual

bull The shadow prices are optimal for the dual LP

bull Next alternative optimality conditions

23

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij gecj for j = 1 to n

Optimality Condition 1A solution x is optimalfor the primal problem ifit is a basic feasiblesolution and the tableausatisfies the optimalityconditions

Optimality Condition 2

A solution x is optimal

for the primal problem if

it is feasible and if there is

a feasible solution y for

the dual with

Σj=1m cj xj = Σj=1m yi bi

24

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij ge cj for j = 1 to n

ComplementarySlackness Conditions

ndashSuppose that y is feasible

for the dual and let

1048698cj= cj - Σi=1m1048698yi aij

Suppose x is feasible for

the primal

Theorem (complementary

slackness) x and y are

optimal for the primal and

dual if and only if

cj xj = 0 for all j

25

Complementary Slackness Illustrated

prices

26

Next Lecture on Duality

bull Dual linear programs in general formbull Optimality Conditions in general formbull Illustrating Duality with 2-person 0-sum game theory

27

Coverage for first midterm(Closed book exam)

bull Formulations

bull 2D graphing and

finding opt solution

bull Setting up an LP in

standard form

bull The simplex algorithm

starting with a bfs

bull Phase 1 of the

simplex algorithm

bull Interpreting sensitivity analysis including shadow prices and ranges and reduced costs

bull Pricing out

bull Using tableaus to determine shadow prices reduced costs and ranges

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27

16

Strong Duality Theorem

Theorem If the primal problem has a finiteoptimal solution value then so does the dualproblem and these two values are the same

17

Strong Duality Illustrated

Prices

Optimal dual solution y1 = ndash13 y2 = 53 v = 143

Optimal primal solution x1 = 23 x2 = 0 x3 = 0 x4 = 13 z = 143

18

Strong Duality Illustrated the final tableau

Optimal primal solution x1 = 23 x2 = 0 x3 = 0 x4 = 13 v = 143

Observation

The costcoeffients satisfycomplementaryslackness

19

Summary

bull If we take an LP in standard form (max) we can formulate a dual problem1048698bull Weak Duality Each solution of the dual gives an upper bound on the maximum objective value for the primal

bull Strong duality if there are feasible solutions to the primal and dual then the optimal objective value for both problems is the same

20

Shadow prices solve the dual

bull Theorem Suppose that the primal problem has a finite optimal solution value Then the shadow prices for the primal problem form an optimal solution to the dual problem (And the objective values are the same)

Davidrsquos Mineral Problem

21

FACTOptimalsimplexmultipliersare shadowprices

Shadowprices

22

Summary

bull The dual problem to a maximizing LP provides upper bounds on the optimal objective function

bull The maximum solution value for the primal is the same as the minimum solution value for the dual

bull The shadow prices are optimal for the dual LP

bull Next alternative optimality conditions

23

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij gecj for j = 1 to n

Optimality Condition 1A solution x is optimalfor the primal problem ifit is a basic feasiblesolution and the tableausatisfies the optimalityconditions

Optimality Condition 2

A solution x is optimal

for the primal problem if

it is feasible and if there is

a feasible solution y for

the dual with

Σj=1m cj xj = Σj=1m yi bi

24

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij ge cj for j = 1 to n

ComplementarySlackness Conditions

ndashSuppose that y is feasible

for the dual and let

1048698cj= cj - Σi=1m1048698yi aij

Suppose x is feasible for

the primal

Theorem (complementary

slackness) x and y are

optimal for the primal and

dual if and only if

cj xj = 0 for all j

25

Complementary Slackness Illustrated

prices

26

Next Lecture on Duality

bull Dual linear programs in general formbull Optimality Conditions in general formbull Illustrating Duality with 2-person 0-sum game theory

27

Coverage for first midterm(Closed book exam)

bull Formulations

bull 2D graphing and

finding opt solution

bull Setting up an LP in

standard form

bull The simplex algorithm

starting with a bfs

bull Phase 1 of the

simplex algorithm

bull Interpreting sensitivity analysis including shadow prices and ranges and reduced costs

bull Pricing out

bull Using tableaus to determine shadow prices reduced costs and ranges

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27

17

Strong Duality Illustrated

Prices

Optimal dual solution y1 = ndash13 y2 = 53 v = 143

Optimal primal solution x1 = 23 x2 = 0 x3 = 0 x4 = 13 z = 143

18

Strong Duality Illustrated the final tableau

Optimal primal solution x1 = 23 x2 = 0 x3 = 0 x4 = 13 v = 143

Observation

The costcoeffients satisfycomplementaryslackness

19

Summary

bull If we take an LP in standard form (max) we can formulate a dual problem1048698bull Weak Duality Each solution of the dual gives an upper bound on the maximum objective value for the primal

bull Strong duality if there are feasible solutions to the primal and dual then the optimal objective value for both problems is the same

20

Shadow prices solve the dual

bull Theorem Suppose that the primal problem has a finite optimal solution value Then the shadow prices for the primal problem form an optimal solution to the dual problem (And the objective values are the same)

Davidrsquos Mineral Problem

21

FACTOptimalsimplexmultipliersare shadowprices

Shadowprices

22

Summary

bull The dual problem to a maximizing LP provides upper bounds on the optimal objective function

bull The maximum solution value for the primal is the same as the minimum solution value for the dual

bull The shadow prices are optimal for the dual LP

bull Next alternative optimality conditions

23

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij gecj for j = 1 to n

Optimality Condition 1A solution x is optimalfor the primal problem ifit is a basic feasiblesolution and the tableausatisfies the optimalityconditions

Optimality Condition 2

A solution x is optimal

for the primal problem if

it is feasible and if there is

a feasible solution y for

the dual with

Σj=1m cj xj = Σj=1m yi bi

24

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij ge cj for j = 1 to n

ComplementarySlackness Conditions

ndashSuppose that y is feasible

for the dual and let

1048698cj= cj - Σi=1m1048698yi aij

Suppose x is feasible for

the primal

Theorem (complementary

slackness) x and y are

optimal for the primal and

dual if and only if

cj xj = 0 for all j

25

Complementary Slackness Illustrated

prices

26

Next Lecture on Duality

bull Dual linear programs in general formbull Optimality Conditions in general formbull Illustrating Duality with 2-person 0-sum game theory

27

Coverage for first midterm(Closed book exam)

bull Formulations

bull 2D graphing and

finding opt solution

bull Setting up an LP in

standard form

bull The simplex algorithm

starting with a bfs

bull Phase 1 of the

simplex algorithm

bull Interpreting sensitivity analysis including shadow prices and ranges and reduced costs

bull Pricing out

bull Using tableaus to determine shadow prices reduced costs and ranges

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27

18

Strong Duality Illustrated the final tableau

Optimal primal solution x1 = 23 x2 = 0 x3 = 0 x4 = 13 v = 143

Observation

The costcoeffients satisfycomplementaryslackness

19

Summary

bull If we take an LP in standard form (max) we can formulate a dual problem1048698bull Weak Duality Each solution of the dual gives an upper bound on the maximum objective value for the primal

bull Strong duality if there are feasible solutions to the primal and dual then the optimal objective value for both problems is the same

20

Shadow prices solve the dual

bull Theorem Suppose that the primal problem has a finite optimal solution value Then the shadow prices for the primal problem form an optimal solution to the dual problem (And the objective values are the same)

Davidrsquos Mineral Problem

21

FACTOptimalsimplexmultipliersare shadowprices

Shadowprices

22

Summary

bull The dual problem to a maximizing LP provides upper bounds on the optimal objective function

bull The maximum solution value for the primal is the same as the minimum solution value for the dual

bull The shadow prices are optimal for the dual LP

bull Next alternative optimality conditions

23

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij gecj for j = 1 to n

Optimality Condition 1A solution x is optimalfor the primal problem ifit is a basic feasiblesolution and the tableausatisfies the optimalityconditions

Optimality Condition 2

A solution x is optimal

for the primal problem if

it is feasible and if there is

a feasible solution y for

the dual with

Σj=1m cj xj = Σj=1m yi bi

24

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij ge cj for j = 1 to n

ComplementarySlackness Conditions

ndashSuppose that y is feasible

for the dual and let

1048698cj= cj - Σi=1m1048698yi aij

Suppose x is feasible for

the primal

Theorem (complementary

slackness) x and y are

optimal for the primal and

dual if and only if

cj xj = 0 for all j

25

Complementary Slackness Illustrated

prices

26

Next Lecture on Duality

bull Dual linear programs in general formbull Optimality Conditions in general formbull Illustrating Duality with 2-person 0-sum game theory

27

Coverage for first midterm(Closed book exam)

bull Formulations

bull 2D graphing and

finding opt solution

bull Setting up an LP in

standard form

bull The simplex algorithm

starting with a bfs

bull Phase 1 of the

simplex algorithm

bull Interpreting sensitivity analysis including shadow prices and ranges and reduced costs

bull Pricing out

bull Using tableaus to determine shadow prices reduced costs and ranges

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27

19

Summary

bull If we take an LP in standard form (max) we can formulate a dual problem1048698bull Weak Duality Each solution of the dual gives an upper bound on the maximum objective value for the primal

bull Strong duality if there are feasible solutions to the primal and dual then the optimal objective value for both problems is the same

20

Shadow prices solve the dual

bull Theorem Suppose that the primal problem has a finite optimal solution value Then the shadow prices for the primal problem form an optimal solution to the dual problem (And the objective values are the same)

Davidrsquos Mineral Problem

21

FACTOptimalsimplexmultipliersare shadowprices

Shadowprices

22

Summary

bull The dual problem to a maximizing LP provides upper bounds on the optimal objective function

bull The maximum solution value for the primal is the same as the minimum solution value for the dual

bull The shadow prices are optimal for the dual LP

bull Next alternative optimality conditions

23

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij gecj for j = 1 to n

Optimality Condition 1A solution x is optimalfor the primal problem ifit is a basic feasiblesolution and the tableausatisfies the optimalityconditions

Optimality Condition 2

A solution x is optimal

for the primal problem if

it is feasible and if there is

a feasible solution y for

the dual with

Σj=1m cj xj = Σj=1m yi bi

24

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij ge cj for j = 1 to n

ComplementarySlackness Conditions

ndashSuppose that y is feasible

for the dual and let

1048698cj= cj - Σi=1m1048698yi aij

Suppose x is feasible for

the primal

Theorem (complementary

slackness) x and y are

optimal for the primal and

dual if and only if

cj xj = 0 for all j

25

Complementary Slackness Illustrated

prices

26

Next Lecture on Duality

bull Dual linear programs in general formbull Optimality Conditions in general formbull Illustrating Duality with 2-person 0-sum game theory

27

Coverage for first midterm(Closed book exam)

bull Formulations

bull 2D graphing and

finding opt solution

bull Setting up an LP in

standard form

bull The simplex algorithm

starting with a bfs

bull Phase 1 of the

simplex algorithm

bull Interpreting sensitivity analysis including shadow prices and ranges and reduced costs

bull Pricing out

bull Using tableaus to determine shadow prices reduced costs and ranges

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27

20

Shadow prices solve the dual

bull Theorem Suppose that the primal problem has a finite optimal solution value Then the shadow prices for the primal problem form an optimal solution to the dual problem (And the objective values are the same)

Davidrsquos Mineral Problem

21

FACTOptimalsimplexmultipliersare shadowprices

Shadowprices

22

Summary

bull The dual problem to a maximizing LP provides upper bounds on the optimal objective function

bull The maximum solution value for the primal is the same as the minimum solution value for the dual

bull The shadow prices are optimal for the dual LP

bull Next alternative optimality conditions

23

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij gecj for j = 1 to n

Optimality Condition 1A solution x is optimalfor the primal problem ifit is a basic feasiblesolution and the tableausatisfies the optimalityconditions

Optimality Condition 2

A solution x is optimal

for the primal problem if

it is feasible and if there is

a feasible solution y for

the dual with

Σj=1m cj xj = Σj=1m yi bi

24

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij ge cj for j = 1 to n

ComplementarySlackness Conditions

ndashSuppose that y is feasible

for the dual and let

1048698cj= cj - Σi=1m1048698yi aij

Suppose x is feasible for

the primal

Theorem (complementary

slackness) x and y are

optimal for the primal and

dual if and only if

cj xj = 0 for all j

25

Complementary Slackness Illustrated

prices

26

Next Lecture on Duality

bull Dual linear programs in general formbull Optimality Conditions in general formbull Illustrating Duality with 2-person 0-sum game theory

27

Coverage for first midterm(Closed book exam)

bull Formulations

bull 2D graphing and

finding opt solution

bull Setting up an LP in

standard form

bull The simplex algorithm

starting with a bfs

bull Phase 1 of the

simplex algorithm

bull Interpreting sensitivity analysis including shadow prices and ranges and reduced costs

bull Pricing out

bull Using tableaus to determine shadow prices reduced costs and ranges

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27

21

FACTOptimalsimplexmultipliersare shadowprices

Shadowprices

22

Summary

bull The dual problem to a maximizing LP provides upper bounds on the optimal objective function

bull The maximum solution value for the primal is the same as the minimum solution value for the dual

bull The shadow prices are optimal for the dual LP

bull Next alternative optimality conditions

23

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij gecj for j = 1 to n

Optimality Condition 1A solution x is optimalfor the primal problem ifit is a basic feasiblesolution and the tableausatisfies the optimalityconditions

Optimality Condition 2

A solution x is optimal

for the primal problem if

it is feasible and if there is

a feasible solution y for

the dual with

Σj=1m cj xj = Σj=1m yi bi

24

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij ge cj for j = 1 to n

ComplementarySlackness Conditions

ndashSuppose that y is feasible

for the dual and let

1048698cj= cj - Σi=1m1048698yi aij

Suppose x is feasible for

the primal

Theorem (complementary

slackness) x and y are

optimal for the primal and

dual if and only if

cj xj = 0 for all j

25

Complementary Slackness Illustrated

prices

26

Next Lecture on Duality

bull Dual linear programs in general formbull Optimality Conditions in general formbull Illustrating Duality with 2-person 0-sum game theory

27

Coverage for first midterm(Closed book exam)

bull Formulations

bull 2D graphing and

finding opt solution

bull Setting up an LP in

standard form

bull The simplex algorithm

starting with a bfs

bull Phase 1 of the

simplex algorithm

bull Interpreting sensitivity analysis including shadow prices and ranges and reduced costs

bull Pricing out

bull Using tableaus to determine shadow prices reduced costs and ranges

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27

22

Summary

bull The dual problem to a maximizing LP provides upper bounds on the optimal objective function

bull The maximum solution value for the primal is the same as the minimum solution value for the dual

bull The shadow prices are optimal for the dual LP

bull Next alternative optimality conditions

23

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij gecj for j = 1 to n

Optimality Condition 1A solution x is optimalfor the primal problem ifit is a basic feasiblesolution and the tableausatisfies the optimalityconditions

Optimality Condition 2

A solution x is optimal

for the primal problem if

it is feasible and if there is

a feasible solution y for

the dual with

Σj=1m cj xj = Σj=1m yi bi

24

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij ge cj for j = 1 to n

ComplementarySlackness Conditions

ndashSuppose that y is feasible

for the dual and let

1048698cj= cj - Σi=1m1048698yi aij

Suppose x is feasible for

the primal

Theorem (complementary

slackness) x and y are

optimal for the primal and

dual if and only if

cj xj = 0 for all j

25

Complementary Slackness Illustrated

prices

26

Next Lecture on Duality

bull Dual linear programs in general formbull Optimality Conditions in general formbull Illustrating Duality with 2-person 0-sum game theory

27

Coverage for first midterm(Closed book exam)

bull Formulations

bull 2D graphing and

finding opt solution

bull Setting up an LP in

standard form

bull The simplex algorithm

starting with a bfs

bull Phase 1 of the

simplex algorithm

bull Interpreting sensitivity analysis including shadow prices and ranges and reduced costs

bull Pricing out

bull Using tableaus to determine shadow prices reduced costs and ranges

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27

23

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij gecj for j = 1 to n

Optimality Condition 1A solution x is optimalfor the primal problem ifit is a basic feasiblesolution and the tableausatisfies the optimalityconditions

Optimality Condition 2

A solution x is optimal

for the primal problem if

it is feasible and if there is

a feasible solution y for

the dual with

Σj=1m cj xj = Σj=1m yi bi

24

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij ge cj for j = 1 to n

ComplementarySlackness Conditions

ndashSuppose that y is feasible

for the dual and let

1048698cj= cj - Σi=1m1048698yi aij

Suppose x is feasible for

the primal

Theorem (complementary

slackness) x and y are

optimal for the primal and

dual if and only if

cj xj = 0 for all j

25

Complementary Slackness Illustrated

prices

26

Next Lecture on Duality

bull Dual linear programs in general formbull Optimality Conditions in general formbull Illustrating Duality with 2-person 0-sum game theory

27

Coverage for first midterm(Closed book exam)

bull Formulations

bull 2D graphing and

finding opt solution

bull Setting up an LP in

standard form

bull The simplex algorithm

starting with a bfs

bull Phase 1 of the

simplex algorithm

bull Interpreting sensitivity analysis including shadow prices and ranges and reduced costs

bull Pricing out

bull Using tableaus to determine shadow prices reduced costs and ranges

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27

24

Primal Problem

max Σj=1m cj xj

st Σj=1n aij xj = bi for i = 1 to m

xj ge 0 for all j = 1 to n

Dual Problem

min Σj=1m yi bi

st Σi=1m yi aij ge cj for j = 1 to n

ComplementarySlackness Conditions

ndashSuppose that y is feasible

for the dual and let

1048698cj= cj - Σi=1m1048698yi aij

Suppose x is feasible for

the primal

Theorem (complementary

slackness) x and y are

optimal for the primal and

dual if and only if

cj xj = 0 for all j

25

Complementary Slackness Illustrated

prices

26

Next Lecture on Duality

bull Dual linear programs in general formbull Optimality Conditions in general formbull Illustrating Duality with 2-person 0-sum game theory

27

Coverage for first midterm(Closed book exam)

bull Formulations

bull 2D graphing and

finding opt solution

bull Setting up an LP in

standard form

bull The simplex algorithm

starting with a bfs

bull Phase 1 of the

simplex algorithm

bull Interpreting sensitivity analysis including shadow prices and ranges and reduced costs

bull Pricing out

bull Using tableaus to determine shadow prices reduced costs and ranges

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27

25

Complementary Slackness Illustrated

prices

26

Next Lecture on Duality

bull Dual linear programs in general formbull Optimality Conditions in general formbull Illustrating Duality with 2-person 0-sum game theory

27

Coverage for first midterm(Closed book exam)

bull Formulations

bull 2D graphing and

finding opt solution

bull Setting up an LP in

standard form

bull The simplex algorithm

starting with a bfs

bull Phase 1 of the

simplex algorithm

bull Interpreting sensitivity analysis including shadow prices and ranges and reduced costs

bull Pricing out

bull Using tableaus to determine shadow prices reduced costs and ranges

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27

26

Next Lecture on Duality

bull Dual linear programs in general formbull Optimality Conditions in general formbull Illustrating Duality with 2-person 0-sum game theory

27

Coverage for first midterm(Closed book exam)

bull Formulations

bull 2D graphing and

finding opt solution

bull Setting up an LP in

standard form

bull The simplex algorithm

starting with a bfs

bull Phase 1 of the

simplex algorithm

bull Interpreting sensitivity analysis including shadow prices and ranges and reduced costs

bull Pricing out

bull Using tableaus to determine shadow prices reduced costs and ranges

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27

Coverage for first midterm(Closed book exam)

bull Formulations

bull 2D graphing and

finding opt solution

bull Setting up an LP in

standard form

bull The simplex algorithm

starting with a bfs

bull Phase 1 of the

simplex algorithm

bull Interpreting sensitivity analysis including shadow prices and ranges and reduced costs

bull Pricing out

bull Using tableaus to determine shadow prices reduced costs and ranges

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
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