Asymptotic Analysis And Confidence Optimal Value Estimation For Two Kinds Of Stochastic Programming Problems

Stochastic optimization problems such as stochastic inequality constraint problems and stochastic second-order cone programming problems have been widely used in many fields such as finance,engineering and management.However,the study of the asymptotic convergence rate and the confidence bounds of the optimal value for the sample average approximation of the stochastic second-order cone programming problems have not appeared in the previous papers.On the basis of the previous research,we extend the previous results to the stochastic programming with multiple inequality constraints and the stochastic second-order cone programming,and extend the existing research results.The main contents of this paper are as follows:Firstly,the research background of stochastic programming,the method of sample average approximation,the research progress of asymptotic analysis and confidence estimation of stochastic programming are introduced.Secondly,the sample average approximation problem for stochastic programming with multiple inequality constraints is analyzed asymptotically.Based on the large deviation theory,the asymptotic property of the feasible region of the sample average approximation problem for stochastic programming with multiple inequality constraints is established,and then the asymptotic property of the optimal value of the problem is studied.Such asymptotic property theory includes the analysis of the convergence rate of the optimal value,the estimation of the sample size and the method of estimating the confidence upper and lower bounds of the true optimal value.Thirdly,a class of stochastic second-order cone programming problems can be transformed into a stochastic programming problem with multiple inequality constraints.Therefore,based on the study of stochastic programming with multiple inequality constraints,the asymptotic analysis of the sample average approximation problem for a class of stochastic second-order cone programming problems is carried out,including the asymptotic property of the feasible region,the analysis of the convergence rate of the optimal value,the estimation of the sample size and the method of estimating the confidence upper and lower bounds of the true optimal value.Finally,the method of estimating the confidence upper and lower bounds of the true optimal value is applied to two portfolio problems,and the nondifferentiable constraint function of the original problem is solved by introducing smoothing parameters and other steps.The feasibility of the estimation method is illustrated by numerical experiments.