Minimax Distributionally Robust Optimization Problem Based On Hellinger Divergence Function

Many practical problems with important values can be modeled as minimax distributionally robust optimization problem,which often exist uncertainty distribution.The key to solving problems is to specify ambiguous set for distribution.So construction of ambiguity set is paid more attention.One of representative approaches is described as follows.A distribution P0 of random vectors(called nominal distribution)is obtained by statistical fitting for data or information.An ambiguous set consists of distributions whose distance from the nominal distribution0 P is not more than a certain positive constant.In this paper,distance between two distributions is defined according to Hellinger divergence function.Furthermore,ambiguity set for distribution is specified.An equivalent form of the 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 Hellinger divergence function.Firstly,distance between two distributions is defined based on the Hellinger 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 prove existence of solution of the inner maximization problem.Consequently,a 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.