By Bernard Moret, Henry D. Shapiro

ISBN-10: 0805380086

ISBN-13: 9780805380088

Ebook by means of Moret, Bernard, Shapiro, Henry D.

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Additional info for Algorithms from P to NP, Vol. I: Design and Efficiency

Example text

5)) has a unique solution. Proof. 1 in [2]. 4) and suppose that opt(P) is not a singleton. 4) is nonempty. 4). 3) and closed. 4) has a solution. 4). Then, any point in the nontrivial segment [ˆx 1 , xˆ 2 ] is also optimal for this problem, so that, by the strict concavity of the log, c B has a constant value over the segment. On the other hand, f is also constant on the segment (which is contained in opt(P)), as well as c(i) for i ∈ / B (which vanishes on the segment). It follows that the Lagrangian has a constant value over a nontrivial segment, a contradiction with its assumed strong convexity.

Let us define j := j w j − µ j e. One has for all indices i: j j µ j + (i) µ j + (i) j j w(i) = and λ = . 6) and dividing by µ j give i∈N j (i) j λˆ (i) µ + j µj λ (i) + i∈B j (i) j c(i) (ˆx ) µ + j µj w ≤m+ (i) o(µ j ) . 1) implies that (i) = o(µ j ). Let (ˆx 0 , λˆ 0 ) be a limit point of {(x j , λ j )}. Taking the limit in the preceding estimate yields λˆ (i) c(i) (ˆx ) + ≤ m. c(i) (ˆx 0 ) (λˆ 0 )(i) i∈N i∈B Necessarily c B (ˆx 0 ) > 0 and (λˆ 0 ) N > 0. Observe now that, by strict complementarity, there are exactly m terms on the left-hand side of the preceding inequality.

2, {x j } and {λ j } are bounded, and by definition of B and N: c(i) (ˆx ) = 0 for i ∈ B, and λˆ (i) = 0 for i ∈ N. Hence j j λˆ N w N + λ B c B (ˆx ) ≤ mµ j + O( Now, using s j = O( tions r j = o(µ j ) and j j j ) + O( s j ). 1), and the assump= o(µ j ), we obtain finally j j λˆ N w N + λ B c B (ˆx ) ≤ mµ j + o(µ j ). 6) We pursue by adapting an idea used by McLinden [13] to give properties of the limit points of the path µ → (ˆx µ , λˆ µ ). Let us define j := j w j − µ j e. One has for all indices i: j j µ j + (i) µ j + (i) j j w(i) = and λ = .

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Algorithms from P to NP, Vol. I: Design and Efficiency by Bernard Moret, Henry D. Shapiro

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