Study On Models And Stability Of Distributionally Robust Optimization

In practice,there are not only decision variables but also parameters in optimization models and solutions to a optimization model depend on parameter estimations.Because the information of the parameter vector is not sufficient or the data of the parameter vector is incomplete,or the parameter vectors are random,it is usually very difficult to estimate these parameters.To address the problem,scholars proposed and developed robust optimization method.The research on distributionally robust optimization has important theoretical and practical significance.The main results of this dissertation are summarized as follows:1.Chapter 3 studies the distributionally robust optimization problems under the first-order and the second-order moment uncertainty.Deterministic equivalent optimization reformulation-s for the distributionally robust optimization problems are established when the support set is a general set.Tractable equivalent optimization reformulations for the distributionally robust op-timization model are developed when the support set is the whole space and when it is a convex polyhedral set.Two methods are used to solve these two models.One is Gurobi,which solves the second-order conic optimization reformulations to the two distributionally robust problems.The other is a smoothing Newton method.Furthermore,when the parameters for characterizing the first-order moment and the second-order moment are perturbed,the stability of the distribu-tionally robust optimization problem is studied for the case when the support set being the whole space.2.Chapter 4 is devoted to the study of the the distributionally robust optimization prob-lem in factor models.Based on the factor models and the distributionally robust investment problems,we study distributionally robust mean-variance portfolio selection,Sharp ratio,and VaR and CVaR problems.By the Lagrange duality theory,we obtain the equivalent semidefinite reformulations for the above distributionally robust investment problems.3.Chapter 5 studies the stability analysis of solutions of a class of stochastic generalized equations via Brouwer’s fixed point theorem.The Hausdorff distance between the solution sets corresponding to different probability measures is estimated in terms of psudo-metric of proba-bility measures.The stability results obtained are finally applied to a stochastic conic program-ming.