1 15.053 Tuesday, March 5 Duality – The art of obtaining bounds – weak and strong duality...
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Transcript of 1 15.053 Tuesday, March 5 Duality – The art of obtaining bounds – weak and strong duality...
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
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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
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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
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