Asymptotic Analysis Of Stochastic Second Order Cone Programming Problems

Stochastic programming is an important branch of mathematical programming,which seeks the optimal solution of a problem under uncertain conditions by establishing a mathematical model.With the continuous deepening of research on stochastic programming problems,new models of stochastic cone programming problems are constantly emerging,which have wide applications in practical problems such as transportation,financial engineering,and optimal location selection.The stochastic second-order cone programming problem studied in this paper is a new type of stochastic programming problem,which includes the constrained optimization problem of stochastic inequality as its special case.Due to the difficulty in accurately calculating expected value functions in stochastic second-order cone programming problems,approximation methods such as sample average approximation(SAA)are often used for research Based on the perturbation analysis of deterministic second-order cone programming problems and the basic theory of stochastic programming problems,this paper intends to conduct asymptotic analysis on the sample average approximation problem of stochastic second-order cone programming problems to provide theoretical guarantee for establishing the confidence interval of the true solution of stochastic second-order cone programming problems.The specific research contents are as follows:Based on the perturbation analysis of deterministic second-order cone programming problems and the basic theory of stochastic programming problems,this paper intends to conduct an asymptotic analysis of the sample average approximation problem of stochastic second-order cone programming problems,providing a theoretical guarantee for establishing the confidence interval of the true solution of stochastic second-order cone programming.The specific research content is as follows:The first chapter mainly introduces the research background of stochastic programming problems and stochastic second-order cone programming problems,gives the purpose and significance of this study,lists the research content and chapter arrangement,and gives a symbolic explanation.In Chapter 2,the sample average approximation problem corresponding to the random second-order cone programming problem is given.Under the given assumptions,the first order optimality condition and the compatibility theory of the optimal solution set for the sample average approximation problem of the random second-order cone programming problem are established.That is,conditions are given to ensure that the stable point of the sample average approximation problem and the convergence point of the optimal solution set sequence are also the stable point and the optimal solution set of the true problem with probability 1,respectively.In Chapter 3,based on strict complementarity conditions and constrained nondegenerate conditions,a compatibility theory of second-order optimality conditions for sample average approximation problems of stochastic second-order cone programming problems is established.That is,conditions are given to ensure that the aggregation points of the sequence of points satisfying the second-order optimality conditions of the sample average approximation problem also satisfy the second-order optimality conditions of the true problem with probability 1.In Chapter 4,under Fritz-John condition,the optimality function for stochastic secondorder cone programming is proposed.The consistency and asymptotic normality of the optimality function for the sample average approximation problem of stochastic second-order cone programming with respect to the optimality function of the true problem are studied,providing theoretical guarantees for establishing confidence regions.In chapter 5,provides a summary and outlook,summarizing the main content of the previous text and providing prospects for future research.