Logarithmic Barrier Decomposition Methods for Semi‐infinite Programming

Abstract

A computational study of some logarithmic barrier decomposition algorithms for semi‐infinite programming is presented in this paper. The conceptual algorithm is a straightforward adaptation of the logarithmic barrier cutting plane algorithm which was presented recently by den Hartog et al. (Annals of Operations Research, 58, 69–98, 1995), to solve semi‐infinite programming problems. Usually decomposition (cutting plane methods) use cutting planes to improve the localization of the given problem. In this paper we propose an extension which uses linear cuts to solve large scale, difficult real world problems. This algorithm uses both static and (doubly) dynamic enumeration of the parameter space and allows for multiple cuts to be simultaneously added for larger/difficult problems. The algorithm is implemented both on sequential and parallel computers. Implementation issues and parallelization strategies are discussed and encouraging computational results are presented.

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“…Using recent results in the mathematical programming literature (e.g. Mosheyev & Zibulevsky, 2000;Kaliski et al, 1997) we claim that it is possible to generalize it to the infinite case. Computationally both algorithms find a provably optimal solution in a small number of iterations.…”

“…In Kaliski et al (1997) a similar barrier algorithm using the log-barrier has been used (cf also Mosheyev & Zibulevsky, 2000). It is future work to rigorously prove that Algorithm 2 also converges to the optimal solution when the hypothesis space is infinite.…”

Improving the efficiency of DC global optimization methods by improving the DC representation of the objective function

“…The algorithm we will use to solve the semi-infinite quadratic programming problem (5) is an adaptation of the semi-infinite linear programming methods described in [6] and [8], which use interior point techniques. The algorithm needs an initial feasible point at which all the constraints are satisfied as strict inequalities; in the next section we explain how such a point can be obtained.…”

Applying Global Optimization to a Problem in Short-Term Hydrothermal Scheduling

“…The We have obtained the least deviation decomposition of each power hydrogeneration function at each reservoir by using the algorithms described in the articles Kaliski et al (1997) and Zhi-Quan et al (1999). These results are not explicity indicated in this paper and we only use them to obtain our computational results.…”

“…In section 5, by using the concept of Least Deviation Decomposition (see Luc et al (1999)), a semi-infinite programming problem has been formulated to calculate the optimal d.c. representation of a polynomial. In order to obtain more efficient implementations we have obtained the least deviation decomposition of each power hydrogeneration function (see (15.1)) following the algorithms described in Kaliski et al (1997) and Zhi-Quan et al (1999). These results are not explicity indicated in this paper and we only use them to obtain our computational results.…”