A Feasible Incremental Algorithm For Solving Inequality Constrained Minimax Problems

Minimax optimization problem is a kind of special and important non-smooth optimization problem. On one hand, many practical problems can be abstracted as a minimax optimization problem, such as ones in the fields of the optimal control, economic and financial, energy and environment etc. On the other hand, the processing methods for many branches of mathemati-cal programming itself (such as robust optimization and stochastic program-ming, etc.) can be reduced to solving a class of minimax problem. Therefore, studying highly efficient and stable algorithms of the problem has important theoretical significance and extensive application value. However, at present the research achievements for minimax problem mainly focus on the smooth type of component functions, while for the nonsmooth one which is few.Aiming at the nonsmooth component functions, the thesis puts forward a feasible incremental algorithm for solving inequality constrained minimax problem by combining the ideas of feasible direction method and bundle method, employing the incremental technique, proximal control technique together with subgradient aggregation technique. The main characteristics of the algorithm are as follows:(1) The algorithm does not need to assume that the component functions of the problem are continuously differentiable (i.e., without the assumption of the smoothness) by using the subgradients of functions and the idea of bundle method;(2) In the main iterative process, by using the incremental technique, only one component function value and one subgradient are needed to computed to generated the search direction sub-problem, thus the computational cost is greatly reduced;(3) The algorithm can ensure the feasibility of the serious iteration points and a certain "decsent property" of the objective function;(4) The algorithm aggregates the subgra-dients in bundle set by introducing the subgradient aggregation technique, which avoids the numerical calculation and storage difficulties by controling the number of elements in bundle set no more than two;(5) Global con-vergence of the algorithm is proven and preliminary numerical experiments show that this algorithm is effective.