This preview shows pages 1–6. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: IE521 Advanced Optimization Lecture 7 Dr. Zeliha Akc ¸a November 2011 1 / 17 Reading I Bertsimas 4.1  4. 2. 2 / 17 Duality Theory: Motivation I Consider the following problem: min x 2 + y 2 s.t. x + y = 1 I Introduce a Lagrange multiplier p , and define L ( x , y , p ) . L ( x , y , p ) = x 2 + y 2 + p ( 1 x y ) I For fixed p , in order to minimize L ( x , y , p ) over x , y , take the derivative and set to zero: I ∂ L /∂ x = and ∂ L /∂ y = ⇒ x = y = p 2 I What is the idea here? 3 / 17 Duality Theory: Motivation I The idea is not to strictly enforce the constraint x + y = 1 . I Associate a Lagrange multiplier, or price , with the constraint. I Allow the constraint to be violated for a price . I When the price is properly chosen, the optimal solution to second problem is equal to the optimal solution to the original problem. ⇒ Under a specific value of p , the presence or absence of the constraint does not affect the optimal cost I In an LP, associate a price with each constraint, I and search for the right price values for which the presence or absence of the constraint does not affect the optimal cost. I These prices can be found by solving a new linear program called dual problem . 4 / 17 Constructing the Dual Problem I Consider an LP in standard form, called as Primal Problem: min c > x s.t. Ax = b x ≥ I Reformulate by relaxing the constraint: min c > x + p > ( b Ax ) s.t. x ≥ I Let g ( p ) be the optimal cost for this new problem.be the optimal cost for this new problem....
View Full
Document
 Fall '11
 ZelihaAkça
 Optimization

Click to edit the document details