Minimax Distributionally Robust Optimization Problem Based On ?~2-Divergence Function

Many practical problems with important values can be modeled as mathematical programs,which often exist uncertain parameters.They are often transformed into expected value optimization problems,which exist uncertain probability distribution.Therefore,the key to solving this kind of problems is to construct the uncertain set for probability distribution.So construction of uncertain set is paid more attention.One of representative approaches is described as follows.A distribution P0 of uncertain parameters(called nominal distribution)is obtained by statistical fitting for data or information.An ?-neighborhood of 0P is built by means of divergence for ?>0.The distributions within the neighborhood form a set called uncertain set for distribution.This paper aims at studying the method for solving minimax distributionally robust optimization problem.Uncertain set for distribution is specified based on the?~2- divergence function.An equivalent form of minimax distributionally robust optimization problem is established.Sample average approximation(SAA)method is applied to solve the equivalent problem.The main contents of this paper are as follows.Chapter 1 reviews the research background of the minimax distributionally robust optimization problem and introduces some preliminary knowledge involved in the research.Chapter 2 establishes a deterministic equivalent problem of the minimax distributionally robust optimization problem based on?~2- divergence function.Firstly,?~2- divergence distance is defined based on the?~2- divergence function.Furthermore,the uncertain set for distribution is specified.Secondly,using the change of measure technique,it converts the inner maximization problem of minimax distributionally robust optimization problem to a convex optimization problem with respect to likelihood ratio(L(?)).Finally,Lagrange duality theory for convex optimization problems is applied to solve the inner problem of Lagrange dual problem.Existence of solution of the inner maximization problem is proved.Consequently,an equivalent problem of the minimax distributionally robust optimization problem is built.Chapter 3 solves the equivalent problem by using the sample average approximation(SAA)method.A sample average approximation function of the expected value function is constructed,and corresponding sample average approximation problem is established.It proves that under certain conditions,optimal value,optimal solution set of sample average approximation problem converge to the counterparts of equivalent problem respectively with probability 1 when the sample size is large enough.Chapter 4 illustrates a numerical example.Results obtained are applied to a minimax distributionally robust optimization problem to show the feasibility of the proposed method in this paper.