Chapters 7.3-7.4 and 8.1 Sara Gestrelius April 21 th , 2015, Link¨ oping
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 The dual problem 1 The primal problem and the dual problem Economic interpretation The dual and the primal: important relationships Geometric Solution (8.1) 2 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 The dual problem 1 The primal problem and the dual problem Economic interpretation The dual and the primal: important relationships Geometric Solution (8.1) 2 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) Dual (D) � � max z = c j x j (1) min w = b i u i (7) j i s.t. s.t. � � a ij u i ≤ c j ( j = 1 , 2 , ... n ) (8) a ij x j ≤ b i ( i = 1 , 2 , ... m ) (2) i j u i ≥ 0 ( i = 1 , 2 , ... m ) (9) x j ≥ 0 ( j = 1 , 2 , ... n ) (3) Dual (D’) Primal (P’) max w = b T u max z = c T x (10) (4) s.t. s.t. A T u ≥ c Ax ≤ b (5) (11) x ≥ 0 (6) u ≥ 0 (12) 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 Augmented matrix form From: Hillier and Lieberman [2010] 4 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 An example Primal (P) Dual (D) max z = x 1 + 2 x 2 − 8 x 3 min w = 8 u 1 + 7 u 2 s.t. s.t. x 1 + 3 x 2 + 5 x 3 ≤ 8 u 1 + 2 u 2 ≥ 1 2 x 1 − 5 x 3 ≤ 7 3 u 1 ≥ 2 x 1 , x 2 , x 3 ≥ 0 5 u 1 − 5 u 2 ≥ − 8 u 1 , u 2 ≥ 0 5 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 Non-standard formulations 6 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 The dual problem 1 The primal problem and the dual problem Economic interpretation The dual and the primal: important relationships Geometric Solution (8.1) 2 Example Requirements space 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 Primal (P) : Maximize profit max z = x bread + 2 x cake + 3 x muffin s.t. 2 x bread + x cake + 3 x muffin ≤ 10 amount of flour available 2 x cake + 2 x muffin ≤ 7 amount of eggs available x bread , x cake , x muffin ≥ 0 Dual (D): Minimize insurance costs min w = 10 u flour + 7 u eggs s.t. 2 u flour ≥ 1 profit from bread u flour + 2 u eggs ≥ 2 profit from cake 3 u flour + 2 u eggs ≥ 3 profit from muffin u flour , u eggs ≥ 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 The dual problem 1 The primal problem and the dual problem Economic interpretation The dual and the primal: important relationships Geometric Solution (8.1) 2 Example Requirements space 8 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 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] 8 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 Graphical representation 9 2013-08-13 S. Gestrelius: Chapters 7.3-7.4 and 8.1 From: Hillier and Lieberman [2010]
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 objective 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 objective function (and so no optimal solution ), then the other 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 objective function. From: Hillier and Lieberman [2010] 10 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 Complementary slackness theorem Primal (P) Dual (D) � � max z = c j x j min w = b i u i j i s.t. s.t. � � a ij x j + x s i = b i ( i = 1 , 2 , ... m ) a ij u i − u s j = c j ( j = 1 , 2 , ... n ) j i ( i = 1 , 2 , ... m ) u i ≥ 0 x j ≥ 0 ( j = 1 , 2 , ... n ) Then, x ∗ s i u ∗ i = 0 i = 1 , 2 ... m . u ∗ s j 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 The dual problem 1 The primal problem and the dual problem Economic interpretation The dual and the primal: important relationships Geometric Solution (8.1) 2 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 = x 1 + 3 x 2 s.t. x 1 + x 2 ≤ 3 x 1 − x 2 ≥ 1 x 1 , x 2 ≥ 0 12 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 = x 1 + 3 x 2 s.t. x 1 + x 2 ≤ 3 x 1 − x 2 ≥ 1 x 1 , x 2 ≥ 0 4 3 2 1 -2 -1 1 2 3 4 5 -1 13 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 = x 1 + 3 x 2 s.t. x 1 + x 2 ≤ 3 x 1 − x 2 ≥ 1 x 1 , x 2 ≥ 0 4 3 2 1 -2 -1 1 2 3 4 5 -1 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 = x 1 + 3 x 2 s.t. x 1 + x 2 ≤ 3 x 1 − x 2 ≥ 1 x 1 , x 2 ≥ 0 4 3 2 1 -2 -1 1 2 3 4 5 -1 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 = x 1 + 3 x 2 s.t. x 1 + x 2 ≤ 3 x 1 − x 2 ≥ 1 x 1 , x 2 ≥ 0 4 3 2 1 -2 -1 1 2 3 4 5 -1 16 2013-08-13 S. Gestrelius: Chapters 7.3-7.4 and 8.1
Dual Problem Geometric Solution References Example Requirements space Example: Objective function max z = x 1 + 3x 2 s.t. x 1 + x 2 ≤ 3 x 1 − x 2 ≥ 1 x 1 , x 2 ≥ 0 4 3 2 1 -2 -1 1 2 3 4 5 -1 17 2013-08-13 S. Gestrelius: Chapters 7.3-7.4 and 8.1
Dual Problem Geometric Solution References Example Requirements space Example: Objective function max z = x 1 + 3x 2 s.t. x 1 + x 2 ≤ 3 x 1 − x 2 ≥ 1 x 1 , x 2 ≥ 0 4 3 2 1 -2 -1 1 2 3 4 5 -1 18 2013-08-13 S. Gestrelius: Chapters 7.3-7.4 and 8.1
Dual Problem Geometric Solution References Example Requirements space Example: Optimal Solution max z = x 1 + 3x 2 s.t. x 1 + x 2 ≤ 3 x 1 − x 2 ≥ 1 x 1 , x 2 ≥ 0 4 3 2 1 -2 -1 1 2 3 4 5 -1 19 2013-08-13 S. Gestrelius: Chapters 7.3-7.4 and 8.1
Dual Problem Geometric Solution References Example Requirements space Solution space 1 An LP with a bounded feasible region always has a finite optimal solution. 2 The optimal solution of a bounded LP , if unique, will occur at one 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 optimal points expressed by the line segment between them. 4 3 2 1 -2 -1 1 2 3 4 5 -1 20 2013-08-13 S. Gestrelius: Chapters 7.3-7.4 and 8.1
Dual Problem Geometric Solution References Example Requirements space Outline The dual problem 1 The primal problem and the dual problem Economic interpretation The dual and the primal: important relationships Geometric Solution (8.1) 2 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 λ x ∈ C for each x ∈ C and λ ≥ 0. 21 2013-08-13 S. Gestrelius: Chapters 7.3-7.4 and 8.1
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