duality_b - EE236A (Fall 2007-08) Lecture 10 Duality (part...

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Unformatted text preview: EE236A (Fall 2007-08) Lecture 10 Duality (part 2) duality in algorithms sensitivity analysis via duality duality for general LPs examples mechanics interpretation circuits interpretation two-person zero-sum 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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duality_b - EE236A (Fall 2007-08) Lecture 10 Duality (part...

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