Consistency Analysis Of Optimality Conditions For A Class Of Stochastic Semidefinite Programming Problems

Stochastic cone programming is an important model in stochastic programming,which has been applied in finance,engineering and management and has attracted the attention of many scholars.As a special kind of random cone programming problem,stochastic semidefinite programming has important applications in least trace factor analysis,portfolio problem with risk constraint and matrix completion problem,so the study of stochastic semidefinite programming has important theoretical and application value.In the study of stochastic semidefinite programming,an important means is to use the sample-based approximation method to solve the problem,which involves the compatibility of the solution of the approximation problem.In this paper,the compatibility analysis of the stable point conditions of the stochastic semidefinite programming problem is carried out.The research contents include establishing the compatibility theory of the first-order optimality conditions of the sample mean approximation problem,and establishing the compatibility theory of the second-order stability points of the optimality conditions of the sample mean approximation problem.In particular,In the first chapter,we introduce the research background of stochastic programming,deterministic semidefinite programming problem and stochastic semidefinite programming.In chapter 2,we give some basic concepts and results of set-valued analysis and variational analysis,which will be used to establish the compatibility analysis of sample mean approximation problems for stochastic semidefinite programming problems.In chapter 3,we first study the compatibility of sample mean approximate solutions of random generalized equations,and establish the error bounds between the solution set of sample mean approximate problems of random generalized equations and the solution set of true problems.On this basis,we establish the consistency of the first-order optimality conditions of the sample mean approximation problem for stochastic semidefinite programming problems,and establish the error bounds of the optimal solution set for the sample mean approximation problem.In chapter 4,we establish the compatibility of the second-order necessary optimality conditions for the sample mean approximation problems of random semidefinite programming problems under the Robinson constraint specification.In the case of strict complementarity,the compatibility of the second-order sufficient optimality conditions for sample mean approximation problems of stochastic semidefinite programming problems is established.The established compatibility analysis will provide a theoretical support for solving the sample mean approximation problem by using the second-order method.