Stability Of Two Stage Stochastic Programs With Quadratic Recourse

Two stage stochastic programs are stochastic programming problems whose first stage deci-sion can be corrected by the optimal decision of the second stage problem.This type of problems has wide applications in resource allocation,finance and economy etc.In practice,it is difficult to obtain the true probability measure of random variable and approximate problem is often con-sidered,so asymptotic theory plays an important role in designing algorithm for these models.In addition,stochastic quadratic programming can be applied to many problems,especially in economic and finance.But the existing stability results of two stage stochastic programming are mainly about the linear recourse case.There are much works to do for nonlinear recourse.Therefore,stability analysis of two stage stochastic programs with quadratic recourse is very important and meaningful,and the results generalize the corresponding results of linear models.This dissertation is mainly devoted to the study of two stage stochastic programs with quadratic recourse and its application in dominance-constrained optimization problems,and the research of two stage distributionally robust risk optimization problem.The main results of this dissertation can be summarized as follows:1.In Chapter 3,Hadamard directional differentiability of the optimal value function of con-vex quadratic programming(QP)problems is considered.First when all parameters in the problem are perturbed,based on the continuity of the feasible set mapping,we establish the upper semi-continuity of the optimal solution mappings and local uniformly boundedness of the level sets of the QP problem and the restricted Wolfe dual problem,respectively.In addition,we characterize the QP problem as an equivalent min-max optimization problem over two compact convex sets,and this structure enables us to demonstrate the Lipschitz continuity and the Hadamard directional differentiability of the optimal value function.2.In Chapter 4,quantitative stability of two stage stochastic programs with quadratic recourse is considered.First,we establish the Lipschitz continuity of the feasible set mapping of the restricted Wolfe dual problem under the Hausdorff distance.Since the objective func-tion of the two stage stochastic programs mainly consists of the optimal value function of the QP problem,by using the Lipschitz continuity of such dual feasible set mapping,we prove the local Lipschitz continuity of the objective function of two stage stochastic programs.Then,we introduce Fortet—Mourier metric of probability measures.By uti-lizing the relationship between this metric and the minimal information metric and based on the established stability results of stochastic programs w.r.t.probability measure un-der minimal information measure,we derive Lipschitz continuity of the optimal values of the two stage stochastic programs with quadratic recourse w.r.t.probability measure under Fortet-Mourier metric and the upper semi-continuity of the optimal solution sets.Finally,the obtained results are applied to study the asymptotic behavior of the empirical approximation of the model.3.In Chapter 5,quantitative stability of optimization problem with kth order stochastic dom-inance constraints induced by quadratic recourse and its corresponding distributionally robust problem are considered.First,different from the problems discussed in Chapter 4,we consider the QP problems with bounded feasible set and semi-definite positive matrix in the objective function which can be perturbed arbitrarily.By using the Lipschitz conti-nuity of the original feasible set mapping,we demonstrate the local Lipschitz continuity of the objective function of the two stage stochastic programs with quadratic recourse.Then,consider the set of all functions satisfying local Lipschitz continuity and upper bound con-ditions,we define a pseudo-metric of probability measures applicable to the dominance-constrained optimization problems and prove the Lipschitz continuity of the feasible set mapping w.r.t.the probability measure in terms of the pseudo-metric.Base on this re-sult,we study the Lipschitz continuity of the optimal value function w.r.t.the probability measure and the upper semi-continuity of the optimal solution mapping of the problem.Finally,by utilizing the relationship between such pseudo-metric and the total variation metric,based on the continuity of the parameterized ambiguity set under total variation metric,we establish the Holder continuity of the ambiguity set under the pseudo-metric.Furthermore,we demonstrate the quantitative stability results of the feasible set mapping,the optimal value function and the optimal solution mapping of the distributionally robust counterpart.4.In Chapter 6,quantitative stability of two stage distributionally robust risk optimization(DRRO)problem with linear semi-definite recourse is considered.We first construct a parameterized ambiguity set with ζ-ball structure and establish the error bound result of the ambiguity set under the total variation metric.Then we carry out the quantitative stability analysis of the optimal value function and the optimal solution mapping of DRRO problem.Finally,when the objective function is induced by two stage stochastic programs with full random linear semi-definite recourse,we demonstrate the local Lipschitz continuity of the objective function of the two stage stochastic programs and apply the established stability results to this example.