lec7 - IE521 Advanced Optimization Lecture 7 Dr. Zeliha Akc...

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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....
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lec7 - IE521 Advanced Optimization Lecture 7 Dr. Zeliha Akc...

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