By Freeadman M.H., Teichner P.
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Additional resources for 4-Manifold topology II: Dwyers filtration and surgery kernels
1 several methods for solving dual problems based on evaluating the dual function D are studied. e. 8) can be solved fast. 7) is block-separable. 8) decomposes into several subproblems, which typically can be solved relatively fast. 4 Dual-equivalent convex relaxations In the following, several convex relaxations are studied that are equivalent to a related dual problem. Furthermore, it is shown that dual relaxations are stronger than convex underestimating-relaxations. t. h0 (x) hi (x) ≤ 0, i = 1, .
It was presented in (Everett, 1963) for resource allocation problems. The reader is referred to (Geoﬀrion, 1974) and (Lemaréchal, 2001) for an introduction into this ﬁeld. A comprehensive overview on duality theory is given in (Rockafellar and Wets, 1997). The presented theory forms the background for the computation of a relaxation, which is the main tool in relaxation-based MINLP solution methods (see Chapters 11, 12 and 13). 1 Convexiﬁcation of sets and functions The following deﬁnitions and results on the convexiﬁcation of sets and functions will be used in the subsequent sections.
T. Ê f (x) Ax + b ≤ 0 xJk ∈ conv(Gk ), k = 1, . . 12) Ê where f : n → is a nonlinear convex function that is not block-separable. t. f (x) Ax + b ≤ 0 aT xJk ≤ a, x ∈ [x, x] (a, a) ∈ Nkj , k = 1, . . 9). 7. t. 14) k = 1, . . , p by producing inner and outer approximations of conv(Gk ). This technique has three main advantages: (i) It is possible to ﬁx Lagrangian subproblems that are ‘explored’; (ii) It is possible to work with near-optimal solutions of Lagrangian subproblems; (iii) It is easy to update relaxations after branching operations in branch-cut-and-price algorithms.