一类CVaR约束优化问题的渐近分析

Asymptotic analysis for a class of CVaR constrained optimization problems

CVaR约束优化问题在经济学、金融工程和管理科学中有重要应用.在利用样本均值近似方法求解CVaR约束优化问题时,一个重要的问题是能否利用样本均值近似CVaR约束优化问题的解来构造真问题的最优解的置信域.求置信域的理论基础就涉及随机优化问题的渐近分析.首先,针对约束的非凸性,给出了CVaR约束优化问题的一个近似问题,然后利用样本均值近似方法研究这个近似问题,给出这个近似问题样本均值近似解的渐近分析,确保CVaR约束优化问题的样本均值近似估计值依分布收敛于服从多元正态分布的变量,为进一步给出这类问题真实解的置信域提供理论基础.

CVaR constrained optimization has important applications in economics,financial engineering and management science.When using the sample average approximation method to solve the CVaR constrained optimization problem,an important problem is whether we can use the sample average approximation to construct the confidence region of the optimal solution of the real problem.The theoretical basis of finding the confidence region involves the asymptotic analysis of stochastic optimization problems.In this paper,we first give an approximate problem of CVaR constrained optimization problem for nonconvex constraints.Then we study the approximate problem by using the sample average approximation method,and give the asymptotic analysis of the approximate solution of the sample average of the approximate problem,which ensures that the approximate estimate of the sample average of CVaR constrained optimization problem converges to the multivariate normal distribution in distribution,which provide the theoretical basis for estimating confidence region of the real solution of this problem.