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Unformatted text preview: Lecture 13: Duality and Sensitivity March 5, 2007 Lecture 13 Outline Perturbed Primal Problem Primal Value Function Role in Sensitivity Role in ZeroGap Duality and Problem Reformulations Convex Optimization 1 Lecture 13 Perturbed Problem and Its Dual Primal optimization problem and its dual minimize f ( x ) subject to g ( x ) Ax = b x X maximize q ( , ) subject to We assume that the problem is convex and its optimal value f * is finite Perturbed problem and its dual minimize f ( x ) subject to g ( x ) u Ax = b + v x X maximize q ( , ) u T  v T subject to x is a primal variable, x R n [there are n decision variables] u is a parameter, u R m [there are m inequalities] v is a parameter, v R r [there are r linear equalities] Convex Optimization 2 Lecture 13 Primal Value Function Perturbed problem and its dual minimize f ( x ) subject to g ( x ) u Ax = b + v x X maximize q ( , ) u T  v T subject to Let p ( u, v ) : R m R r 7 R be the function that, to each ( u, v ) , assigns the optimal value of the perturbed problem: p ( u, v ) = inf g ( x ) u Ax = b + v x X f ( x ) for all ( u, v ) R m R r It is referred to as primal value function or primal function Note that p (0 , 0) = f * Primal value function p ( u, v ) is convex In some cases, the primal optimal value p (0 , 0) provides some informa tion about p ( u, v ) Convex Optimization 3 Lecture 13 Examples of Primal Value Functions Example 1 minimize x subject to x dom f = { x R  x } p ( u ) = inf x u x =  u for u + for u < Example 2 minimize e x 1 x 2 subject to x 1 dom f = { ( x 1 , x 2 )  x 1 , x 2 } p ( u ) = inf x 1 u e x 1 x 2 = 1 for u = 0 for u > + for u < Convex Optimization 4 Lecture 13 Global Sensitivity Result Assume that the primal problem has finite optimal value f * , there is no duality gap [i.e., q * = f * ], and its dual has an optimal solution ( * , * ) Consider the perturbed primal problem and its dual Lowerbound property of the dual to the perturbed problem yields: p ( u, v ) q ( * , * ) u T * v T * = p (0 , 0) u T * v T * for all u, v Sensitivity Interpretation * j large: p increases greatly if we tighten constraint j ( u j < ) * j small: p does not decrease much if we loosen constraint j ( u j > ) * i large and positive: p increases greatly if we take v i <...
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This note was uploaded on 08/22/2008 for the course GE 498 AN taught by Professor Angelianedich during the Spring '07 term at University of Illinois at Urbana–Champaign.
 Spring '07
 AngeliaNedich

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