Read e-book online Nonlinear Functional Analysis and Its Applications III: PDF

By Eberhard Zeidler

ISBN-10: 038790915X

ISBN-13: 9780387909158

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Extra resources for Nonlinear Functional Analysis and Its Applications III: Variational Methods and Optimization

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17). 17)). Proof. 17). 17), the statement is proven. 17) is not known in advance. 17) by shifting constraints to the set G. 14. 17) by shifting the inequality constraint gm (x) ≤ 0 to G = {x ∈ G | gk (x) ≤ 0}. Let (D ) be the dual to (P ). 17)) if gk is convex and a constraint qualification holds. Proof. (i) Let L (·; µ) be the Lagrangian to (P ). 17)). (ii) If gm is convex, it follows from strong duality that the above inequality is an equality. In the same way, it can be proven that shifting equality constraints to G may reduce the duality gap.

Then xij = xi xj if and only if xij ≥ 0, xij ≥ xi + xj − 1, xij ≤ xi and xij ≤ xj . t. Ê where A, X = i,j i = 1, . . , m 1≤i

4). 4. t. sin(x1 ) + x2 2x1 + 3x2 ≤ 0 x1 + x22 ≤ 0. 18 Chapter 2. t. 6 Other formulations Apart from the reformulations described in this chapter, there are also other interesting reformulations. 1. A MIQQP containing only binary variables can be transformed into a MIP by the following procedure. Let xi , xj ∈ {0, 1}. Then xij = xi xj if and only if xij ≥ 0, xij ≥ xi + xj − 1, xij ≤ xi and xij ≤ xj . t. Ê where A, X = i,j i = 1, . . , m 1≤i

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Nonlinear Functional Analysis and Its Applications III: Variational Methods and Optimization by Eberhard Zeidler


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