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L13_sens - Lecture 13 Duality and Sensitivity March 5 2007...

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Lecture 13: Duality and Sensitivity March 5, 2007
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Lecture 13 Outline Perturbed Primal Problem Primal Value Function Role in Sensitivity Role in Zero-Gap Duality and Problem Reformulations Convex Optimization 1
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Lecture 13 Perturbed Problem and Its Dual Primal optimization problem and its dual minimize f ( x ) subject to g ( x ) 0 Ax = b x X maximize q ( μ, λ ) subject to μ 0 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 ( μ, λ ) - u T μ - v T λ subject to μ 0 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
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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 ( μ, λ ) - u T μ - v T λ subject to μ 0 Let p ( u, v ) : R m × R r 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
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Lecture 13 Examples of Primal Value Functions Example 1 minimize - x subject to x 0 dom f = { x R | x 0 } p ( u ) = inf x u - x = - u for u 0 + for u < 0 Example 2 minimize e - x 1 x 2 subject to x 1 0 dom f = { ( x 1 , x 2 ) | x 1 0 , x 2 0 } p ( u ) = inf x 1 u e - x 1 x 2 = 1 for u = 0 0 for u > 0 + for u < 0 Convex Optimization 4
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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 ( μ * , λ * ) Consider the perturbed primal problem and its dual Lower-bound property of the dual to the perturbed problem yields: p ( u, v ) q ( μ * , λ * ) - u T μ * - v T λ * = p (0 , 0) - u T μ * - v T λ * for all u, v Sensitivity Interpretation μ * j large: p increases greatly if we tighten constraint j ( u j < 0 ) μ * j small: p does not decrease much if we loosen constraint j ( u j > 0 ) λ * i large and positive: p increases greatly if we take v i < 0 λ * i large and negative: p increases greatly if we take v i > 0 λ * i small and positive: p does not decrease much if we take v i > 0 λ * i small and negative: p
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