Conditional Value at Risk (CVaR) has been recently used to approximate a chance constraint. In this paper, we study the convergence of stationary points when sample average approximation (SAA) method is applied to a CVaR approximated joint chance constrained stochastic minimization problem. Specifically, we prove, under some moderate conditions, that optimal solutions and stationary points obtained from solving sample average approximated problems converge with probability one (w.p.1) to their true counterparts. Moreover, by exploiting the recent results on large deviation of random functions [28] and sensitivity results for generalized equations [31], we derive exponential rate of convergence of stationary points and give an estimate of sample size. The discussion is extended to the case when CVaR approximation is replaced by a DC-approximation [14]. Some preliminary numerical test results are reported.
Citation
School of Mathematics, University of Southampton, Jan 2012.