Optimization Methods for Solving Problems of Transcomputational Complexity

Abstract

This chapter addresses subgradient nonsmooth optimization methods. The generalized gradient descent method known later as the subgradient method is considered. It was used to solve many high-dimensional problems of production and transportation scheduling with decomposition schemes. Subgradient methods with space dilation, whose special cases are the ellipsoid method and r-algorithms, are analyzed. The application of nonsmooth optimization methods to various mathematical programming problems is discussed. The algorithmic scheme of the well-known descent method is considered. This scheme has made it possible to develop a wide range of local search algorithms, which were used in application packages elaborated at the Institute of Cybernetics. The modern method of local type, the probabilistic method of global equilibrium search, is analyzed. It was applied to solve various classes of problems of transcomputational complexity and appeared to be the most efficient discrete optimization technique. The approaches to accelerating the solution of discrete optimization problems are discussed. The stability of vector discrete optimization problems is analyzed. Different types of stability of integer optimization problems and regularization of unstable problems are considered. Some of the chapter deals with stochastic optimization that involves decision making under risk and uncertainty. Stochastic quasigradient methods and their modifications for stochastic programming problems are considered. The solution of stochastic global optimization, stochastic discrete optimization, stochastic minimax, and vector stochastic optimization problems is analyzed. The use of stochastic optimization methods in risk assessment and management is discussed. The chapter reviews the results of the mathematical theory of control of distributed parameter systems and stochastic and discrete systems and addresses the robust stability, determination of invariant sets of dynamic systems, and spacecraft control. The methods of the analysis of dynamic games, including those with groups of participants under various constraints, are considered as the development of motion control methods.