duality_b - EE236A(Fall 2007-08 Lecture 10 Duality(part 2...

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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 10–1 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) 10–2 Nonheuristic stopping criteria • (absolute error) c T x ( k )- p ⋆ is less than ǫ if c T x ( k ) ) + b T z ( k ) < ǫ • (relative error) ( c T x ( k )- p ⋆ ) / | p ⋆ | is less than ǫ if- b T z ( k ) > & c T x ( k ) + b T z ( k )- b T z ( k ) ≤ ǫ or c T x ( k ) < & c T x ( k ) ) + b T z ( k )- c T x ( k ) ≤ ǫ Duality (part 2) 10–3 • target value ℓ is achievable ( p ⋆ ≤ ℓ ) if c T x ( k ) ) ≤ ℓ • target value ℓ is unachievable ( p ⋆ > ℓ ) if- b T z ( k ) > ℓ Duality (part 2) 10–4 Sensitivity analysis via duality perturbed problem: minimize c T x subject to Ax ≤ b + ǫd A ∈ R m × n ; d ∈ R m given; optimal value p ⋆ ( ǫ ) global sensitivity result : if z ⋆ is (any) dual optimal solution for the unperturbed problem, then for all ǫ p ⋆ ( ǫ ) ≥ p ⋆- ǫd T z ⋆ proof. z ⋆ is dual feasible for all ǫ ; by weak duality, p ⋆ ( ǫ ) ≥ - ( b + ǫd ) T z ⋆ = p ⋆- ǫd T z ⋆ Duality (part 2) 10–5 interpretation ǫ p ⋆ ( ǫ ) p ⋆ p ⋆- ǫd T z ⋆ • d T z ⋆ > : ǫ < increases p ⋆ • d T z ⋆ > and large: ǫ < greatly increases p ⋆ • d T z ⋆ > and small: ǫ > does not decrease p ⋆ too much • d T z ⋆ < : ǫ > increases p ⋆ • d T z ⋆ < and large: ǫ > greatly increases p ⋆ • d T z ⋆ < and small: ǫ > does not decrease p ⋆ too much Duality (part 2) 10–6 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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This note was uploaded on 09/26/2009 for the course CAAM 236 taught by Professor Dr.vandenber during the Spring '07 term at Monmouth IL.

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duality_b - EE236A(Fall 2007-08 Lecture 10 Duality(part 2...

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