Оптимизация в конечномерном евклидовом пространстве

Abstract

In this paper, optimization models in Euclidean space are divided into four complexity classes. Ef-fective algorithms have been developed to solve the problems of the first two classes of complexity. These are the primal-dual interior-point methods. Discrete and combinatorial optimization problems of the third complexity class are recommended to be converted to the fourth complexity class with continuous change of variables. Effective algorithms have not been developed for problems of the third and fourth complexity classes, with the exception of a narrow class of problems that are unimodal. The general optimization problem is formulated as a minimum (maximum) objective function in the presence of constraints. The complexity of the problem depends on the structure of the objective function and its feasible region. If the functions that determine the optimization model are quadratic or polynomial, then semidefinite programming can be used to obtain estimates of so-lutions in such problems. Effective methods have been developed for semidefinite optimization problems. Sometimes it’s enough to develop an algorithm without building a mathematical model. We see such an example when sorting an array of numbers. Effective algorithms have been devel-oped to solve this problem. In the work for sorting problems, an optimization model is constructed, and it coincides with the model of the assignment problem. It follows from this that the sorting problem is unimodal. Effective algorithms have not been developed to solve multimodal problems. The paper proposes a simple and effective algorithm for the optimal allocation of resources in mul-tiprocessor systems. This problem is multimodal. In the general case, for solving multimodal prob-lems, a method of exact quadratic regularization is proposed. This method has proven its compara-tive effectiveness in solving many test problems of various dimensions. Keywords: Euclidean space, optimization, unimodal problems, multimodal problems, complexity classes, numerical methods.