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Unformatted text preview: EE236A (Fall 200708) Lecture 10 Duality (part 2) duality in algorithms sensitivity analysis via duality duality for general LPs examples mechanics interpretation circuits interpretation twoperson zerosum games 101 Duality in algorithms many algorithms produce at iteration k a primal feasible x ( k ) and a dual feasible z ( k ) with c T x ( k ) + b T z ( k ) as k hence at iteration k we know p bracketleftbig b T z ( k ) , c T x ( k ) bracketrightbig useful for stopping criteria algorithms that use dual solution are often more efficient Duality (part 2) 102 Nonheuristic stopping criteria (absolute error) c T x ( k ) p is less than if c T x ( k ) ) + b T z ( k ) < (relative error) ( c T x ( k ) p ) /  p  is less than if b T z ( k ) > & c T x ( k ) + b T z ( k ) b T z ( k ) or c T x ( k ) < & c T x ( k ) ) + b T z ( k ) c T x ( k ) Duality (part 2) 103 target value is achievable ( p ) if c T x ( k ) ) target value is unachievable ( p > ) if b T z ( k ) > Duality (part 2) 104 Sensitivity analysis via duality perturbed problem: minimize c T x subject to Ax b + d A R m n ; d R m given; optimal value p ( ) global sensitivity result : if z is (any) dual optimal solution for the unperturbed problem, then for all p ( ) p  d T z proof. z is dual feasible for all ; by weak duality, p ( )  ( b + d ) T z = p  d T z Duality (part 2) 105 interpretation p ( ) p p  d T z d T z > : < increases p d T z > and large: < greatly increases p d T z > and small: > does not decrease p too much d T z < : > increases p d T z < and large: > greatly increases p d T z < and small: > does not decrease p too much Duality (part 2) 106 Local sensitivity analysis assumption: there is a nondegenerate optimal vertex x , i.e. , x is an optimal vertex : rank A = n , where A = bracketleftbig a i 1 a i 2 a i K bracketrightbig T and I = { i 1 , . . . , i K } is the set of active constraints at x x is nondegenerate: A R n n w.l.o.g. we assume I = { 1 , 2 , . . . , n } consequence: dual optimal z is unique proof: by complementary slackness, z i = 0 for i > n by dual feasibility, summationdisplay i...
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 Spring '07
 Dr.Vandenber

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