Asymptotic Analysis For A Stochastic Semidefinite Programming

Stochastic programming,which reflects the potential uncertainty of artificial intelligence,logistics management,financial engineering and other practical problems,has become one of the most potential directions in the field of mathematical optimization.In recent years,with the application of stochastic programming models in more fields,new stochastic optimization models are emerging.Recently,as a new stochastic programming model,stochastic semi-definite cone programming has attracted more and more attention,which provides a unified research form for minimum trace factor analysis and portfolio problem with risk constraints.The theoretical premise of solving stochastic semidefinite programming with expected value by approximation method is the asymptotic analysis of stochastic semidefinite programming,so it is important to establish the asymptotic analysis of stochastic semidefinite programming.In this paper,the estimator of the sample average approximation problem for stochastic semidefinite programming is established based on the asymptotic analysis of the convergence by distribution.In the first chapter,we introduce the research background,including the research background of stochastic programming,the research background of deterministic semi-definite cone programming and the research background of stochastic semi-definite cone programming.Secondly,in Chapter 2,the asymptotic analysis theory based on distributed convergence is established for the approximate estimation of the sample mean of stochastic semidefinite programming.In this paper,we obtain the asymptotic normality conditions of the sample mean approximation estimates for stochastic semidefinite programming problems,and give the concrete forms of the correlation covariance matrix.The established results extend the asymptotic analysis of the existing stochastic programming problems with equality and inequality constraints to the stochastic programming problems with semi-definite cone constraints.Thirdly,in Chapter 3,the method to estimate the confidence interval of the real stationary point of stochastic semidefinite programming is given by using the results in Chapter 2.Finally,in Chapter 4,the results established in the first two chapters are applied to the minimum trace factor analysis and the portfolio selection problem with risk constraints,by numerical experiments,the confidence region of the real stationary point of stochastic semidefinite programming problem is obtained.