Сonvex relaxation in multiextremal problems
Journal Title: Комп’ютерне моделювання: аналіз, управління, оптимізація - Year 2018, Vol 1, Issue 1
Abstract
In this paper multiextremal problems in a finite-dimensional Euclidean space are considered. Such problems arise in the mathematical modeling of complex systems in the economy, finance, management, technological processes, transport, design, computer science and other fields. It is known that this class of problems contains discrete problems, the problems of solving nonlinear equations and other applied problems To solve multiextremal problems branch and boundary methods, semidefinite programming, duality methods, genetic and evolutionary methods, and many others are used nowadays. These methods are used to solve applied problems and to solve numerous test problems. Numerous experiments show that only for some test problems the existing methods allow finding optimal solutions. These methods, however, do not guarantee the best solutions in applied problems. Convex relaxation of multiextremal problems allows one to transform them to convex one-extremal problems. For some multiextremal problems such a transformation will be exact. For semi-definite optimization, this will be when the found matrix has rank one. For dual relaxation, it will happen when the duality gap is zero. In the general case, convex relaxation allows one to obtain only an estimate of the optimal solution. Various types of convex relaxation are used: semidefinite, Lagrange duality, exact quadratic regularization, reformulation-linearization technique. The most effective method of convex relaxation is the exact quadratic regularization. With its use, we obtain the widest class of multiextremal problems, for which convex relaxation will be exact. This is achieved by transformation of the original problem and displacement of the space. The efficiency of exact quadratic regularization is confirmed by numerous experiments for solving testing and applied multiextremal problems.
Authors and Affiliations
А. И. Косолап
Keywords
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