Chapters 7.3-7.4 and 8.1 Sara Gestrelius April 21 th , 2015, Link - - PowerPoint PPT Presentation

Chapters 7.3-7.4 and 8.1

Sara Gestrelius April 21th, 2015, Link¨

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Dual Problem Geometric Solution References

Reference

All today’s material from Cheng et al. [2010] unless specified that from Hillier and Lieberman [2010].

2 2013-08-13

• S. Gestrelius: Chapters 7.3-7.4 and 8.1

Dual Problem Geometric Solution References

Outline

1

The dual problem The primal problem and the dual problem Economic interpretation The dual and the primal: important relationships

2

Geometric Solution (8.1) Example Requirements space

2 2013-08-13

• S. Gestrelius: Chapters 7.3-7.4 and 8.1

Dual Problem Geometric Solution References The primal problem and the dual problem Economic interpretation

Outline

1

The dual problem The primal problem and the dual problem Economic interpretation The dual and the primal: important relationships

2

Geometric Solution (8.1) Example Requirements space

3 2013-08-13

• S. Gestrelius: Chapters 7.3-7.4 and 8.1

Dual Problem Geometric Solution References The primal problem and the dual problem Economic interpretation

The primal and the dual problem

Primal (P) max z =

• j

cjxj (1) s.t.

• j

aijxj ≤ bi (i = 1, 2, ...m) (2) xj ≥ 0 (j = 1, 2, ...n) (3) Primal (P’) max z = cTx (4) s.t. Ax ≤ b (5) x ≥ 0 (6) Dual (D) min w =

• i

biui (7) s.t.

• i

aijui ≤ cj (j = 1, 2, ...n) (8) ui ≥ 0 (i = 1, 2, ...m) (9) Dual (D’) max w = bTu (10) s.t. ATu ≥ c (11) u ≥ 0 (12)

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Dual Problem Geometric Solution References The primal problem and the dual problem Economic interpretation

Augmented matrix form

From: Hillier and Lieberman [2010]

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• S. Gestrelius: Chapters 7.3-7.4 and 8.1

Dual Problem Geometric Solution References The primal problem and the dual problem Economic interpretation

An example

Primal (P) max z = x1 + 2x2 − 8x3 s.t. x1 + 3x2 + 5x3 ≤ 8 2x1 − 5x3 ≤ 7 x1, x2, x3 ≥ 0 Dual (D) min w = 8u1 + 7u2 s.t. u1 + 2u2 ≥ 1 3u1 ≥ 2 5u1 − 5u2 ≥ −8 u1, u2 ≥ 0

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• S. Gestrelius: Chapters 7.3-7.4 and 8.1

Dual Problem Geometric Solution References The primal problem and the dual problem Economic interpretation

Non-standard formulations

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• S. Gestrelius: Chapters 7.3-7.4 and 8.1

Dual Problem Geometric Solution References The primal problem and the dual problem Economic interpretation

Outline

1

The dual problem The primal problem and the dual problem Economic interpretation The dual and the primal: important relationships

2

Geometric Solution (8.1) Example Requirements space

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• S. Gestrelius: Chapters 7.3-7.4 and 8.1

Dual Problem Geometric Solution References The primal problem and the dual problem Economic interpretation

Primal (P) : Maximize profit max z = xbread + 2xcake + 3xmuffin s.t. 2xbread + xcake + 3xmuffin ≤10 amount of flour available 2xcake + 2xmuffin ≤7 amount of eggs available xbread, xcake, xmuffin ≥ 0 Dual (D): Minimize insurance costs min w = 10uflour + 7ueggs s.t. 2uflour ≥1 profit from bread uflour + 2ueggs ≥2 profit from cake 3uflour + 2ueggs ≥3 profit from muffin uflour, ueggs ≥ 0

7 2013-08-13

• S. Gestrelius: Chapters 7.3-7.4 and 8.1

Dual Problem Geometric Solution References The primal problem and the dual problem Economic interpretation

Outline

1

The dual problem The primal problem and the dual problem Economic interpretation The dual and the primal: important relationships

2

Geometric Solution (8.1) Example Requirements space

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• S. Gestrelius: Chapters 7.3-7.4 and 8.1

Dual Problem Geometric Solution References The primal problem and the dual problem Economic interpretation

The weak and strong duality properties

Weak duality property: If x is a feasible solution for the primal problem and u is a feasible solution for the dual problem, then cx ≤ yb Strong duality property: If x∗ is an optimal solution for the primal problem and u∗ is an optimal solution for the dual problem, then cx∗ = u ∗ b From: Hillier and Lieberman [2010]

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• S. Gestrelius: Chapters 7.3-7.4 and 8.1

Dual Problem Geometric Solution References The primal problem and the dual problem Economic interpretation

Graphical representation

From: Hillier and Lieberman [2010]

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• S. Gestrelius: Chapters 7.3-7.4 and 8.1

Dual Problem Geometric Solution References The primal problem and the dual problem Economic interpretation

Duality theorem

The following are the only possible relationships between the primal and the dual problems.

1 If one problem has feasible solutions and a bounded

• bjective function (and os has an optimal solution), then so

does the other problem, so both the weak and the strong duality properties are applicable.

2 If one problem has feasible solutions and an unbounded

• bjective function (and so no optimal solution), then the
• ther problem has no feasible solutions.

3 If one problem has no feasible solutions, then the othe

problem has either no feasible solutions or an unbounded

• bjective function.

From: Hillier and Lieberman [2010]

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• S. Gestrelius: Chapters 7.3-7.4 and 8.1

Dual Problem Geometric Solution References The primal problem and the dual problem Economic interpretation

Complementary slackness theorem

Primal (P) max z =

• j

cjxj s.t.

• j

aijxj + xsi = bi (i = 1, 2, ...m) xj ≥ 0 (j = 1, 2, ...n) Dual (D) min w =

• i

biui s.t.

• i

aijui − usj = cj (j = 1, 2, ...n) ui ≥ 0 (i = 1, 2, ...m)

Then, x ∗si u∗i = 0 i = 1, 2...m. u ∗sj x∗j = 0 j = 1, 2...n.

11 2013-08-13

• S. Gestrelius: Chapters 7.3-7.4 and 8.1

Dual Problem Geometric Solution References Example Requirements space

Outline

1

The dual problem The primal problem and the dual problem Economic interpretation The dual and the primal: important relationships

2

Geometric Solution (8.1) Example Requirements space

12 2013-08-13

• S. Gestrelius: Chapters 7.3-7.4 and 8.1

Dual Problem Geometric Solution References Example Requirements space

Example

max z = x1 + 3x2 s.t. x1 + x2 ≤ 3 x1 − x2 ≥ 1 x1, x2 ≥ 0

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Dual Problem Geometric Solution References Example Requirements space

Example: Solution space

max z = x1 + 3x2 s.t. x1 + x2 ≤ 3 x1 − x2 ≥ 1 x1, x2 ≥ 0

1

• 1
• 2

2 3 4 5

• 1

1 2 3 4

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• S. Gestrelius: Chapters 7.3-7.4 and 8.1

Dual Problem Geometric Solution References Example Requirements space

Example: Solution space

max z = x1 + 3x2 s.t. x1 + x2 ≤ 3 x1 − x2 ≥ 1 x1, x2 ≥ 0

1

• 1
• 2

2 3 4 5

• 1

1 2 3 4

14 2013-08-13

• S. Gestrelius: Chapters 7.3-7.4 and 8.1

Dual Problem Geometric Solution References Example Requirements space

Example: Solution space

max z = x1 + 3x2 s.t. x1 + x2 ≤ 3 x1 − x2 ≥ 1 x1, x2 ≥ 0

1

• 1
• 2

2 3 4 5

• 1

1 2 3 4

15 2013-08-13

• S. Gestrelius: Chapters 7.3-7.4 and 8.1

Dual Problem Geometric Solution References Example Requirements space

Example: Solution space

max z = x1 + 3x2 s.t. x1 + x2 ≤ 3 x1 − x2 ≥ 1 x1, x2 ≥ 0

1

• 1
• 2

2 3 4 5

• 1

1 2 3 4

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Dual Problem Geometric Solution References Example Requirements space

Example: Objective function

max z = x1 + 3x2 s.t. x1 + x2 ≤ 3 x1 − x2 ≥ 1 x1, x2 ≥ 0

1

• 1
• 2

2 3 4 5

• 1

1 2 3 4

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Dual Problem Geometric Solution References Example Requirements space

Example: Objective function

max z = x1 + 3x2 s.t. x1 + x2 ≤ 3 x1 − x2 ≥ 1 x1, x2 ≥ 0

1

• 1
• 2

2 3 4 5

• 1

1 2 3 4

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Dual Problem Geometric Solution References Example Requirements space

Example: Optimal Solution

max z = x1 + 3x2 s.t. x1 + x2 ≤ 3 x1 − x2 ≥ 1 x1, x2 ≥ 0

1

• 1
• 2

2 3 4 5

• 1

1 2 3 4

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Dual Problem Geometric Solution References Example Requirements space

Solution space

1 An LP with a bounded feasible region always has a finite

• ptimal solution.

2 The optimal solution of a bounded LP

, if unique, will occur at

• ne and only one extreme point of P.

3 If a bounded LP has two extreme points optimal (hence,

alternative optima), then there are an infinite number of

• ptimal points expressed by the line segment between

them.

1

• 1
• 2

2 3 4 5

• 1

1 2 3 4

20 2013-08-13

• S. Gestrelius: Chapters 7.3-7.4 and 8.1

Dual Problem Geometric Solution References Example Requirements space

Outline

1

The dual problem The primal problem and the dual problem Economic interpretation The dual and the primal: important relationships

2

Geometric Solution (8.1) Example Requirements space

21 2013-08-13

• S. Gestrelius: Chapters 7.3-7.4 and 8.1

Dual Problem Geometric Solution References Example Requirements space

Convex cone

Definition: A convex cone is a convex set with the additional property that λx ∈ C for each x ∈ C and λ ≥ 0.

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Dual Problem Geometric Solution References Example Requirements space

Equality constraints: feasibility

Case A: Equality constraints (Ax=b, x ≥ 0) Let A = (a1, a1, a2....an), then the LP is feasible if b is in the convex cone generated by {a1, a2....an}. 3x1 + 2x2 + x3 = 1 −x1 + x2 + 2x4 = 3 x1, x2, x3 ≥ 0

1

• 1
• 2

2 3 4 5

• 1

1 2 3 4 a1 a3 a2 a4 1

• 1
• 2

2 3 4 5

• 1

1 2 3 4 a1 a3 a2 a4 b

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• S. Gestrelius: Chapters 7.3-7.4 and 8.1

Dual Problem Geometric Solution References Example Requirements space

Equality constraints: feasibility

Case A: Equality constraints (Ax=b, x ≥ 0) Let A = (a1, a1, a2....an), then the LP is feasible if b is in the convex cone generated by {a1, a2....an}. 3x1 + 2x2 + x3 = −2 −x1 + x2 + 2x4 = 2 x1, x2, x3 ≥ 0

1

• 1
• 2

2 3 4 5

• 1

1 2 3 4 a1 a3 a2 a4 1

• 1
• 2

2 3 4 5

• 1

1 2 3 4 a1 a3 a2 a4 b

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• S. Gestrelius: Chapters 7.3-7.4 and 8.1

Dual Problem Geometric Solution References Example Requirements space

Equality constraints: bounded objective function

max z = −x1 − 2x2 s.t. x1 + 3x2 + 2x3 = 3 x1, x2, x3 ≥ 0 − x1 − 2x2 =z x1 + 3x2 + 2x3 =3 x1, x2, x3 ≥ 0

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Dual Problem Geometric Solution References Example Requirements space

Equality constraints: unbounded objective function

max z = −x1 − 2x2 s.t. x1 + 3x2 − 2x3 = 3 x1, x2, x3 ≥ 0 − x1 − 2x2 =z x1 + 3x2 − 2x3 =3 x1, x2, x3 ≥ 0

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Dual Problem Geometric Solution References Example Requirements space

Inquality constraints

Case B: Inequality constraints (Ax, x ≥ 0) As noted above, the requirement space Ax : x ≥ 0 is the convex cone generated by generated by {a1, a2....an}. If a feasible solution exists, this requirement space in Em must overlap the collection of vectors that are less than

• r equal to the requirement vector b (another convex

cone).

26 2013-08-13

• S. Gestrelius: Chapters 7.3-7.4 and 8.1

References

Der-San Cheng, Robert G. Batson, and Yu Dang. Applied Integer Programming Modeling and Solution. John Wiley Sons, 2010. Frederik S. Hillier and Gerald J. Lieberman. Introduction to Operations Research, Ninth Edition, International Edition. McGraw- Hill, 2010.