Stochastic optimization problems with CVaR risk measure and their sample average approximation
Considered are the single CVaR and mixed CVaR minimization problems \[ \min_{x\in \mathcal{X}}\mathrm{CVaR}_{\alpha }(x) \eqno(1) \] and \[ \min_{x\in \mathcal{X}}\left\{ \lambda _{1}\mathrm{CVaR}_{\alpha _{1}}(x) +\cdots +\lambda _{J}\mathrm{CVaR}_{\alpha _{J}}(x) \right\} \eqno(2) \] where \(x\in \mathcal{X}\subseteq R^{n}\) is a control vector, \(\mathcal{X}\) denotes the set of feasible decisions, \(\alpha _{i}\in (0,1)\) denote the probability levels and \(\lambda _{i}>0\) represents weights with \(\sum_{i=1}^{J}\lambda _{i}=1,i=1,\dots ,J\). The aim of this paper is the analysis of a Monte Carlo simulation-based sampling approach to problems (1) and (2), called the sample average approximation (SAA) method. The authors show that under certain regularity conditions any accumulation point of the weak GKKT (generalized Karush-Kuhn-Tucker) points produced by the SAA method is almost surely a weak stationary point of the original CVaR or mixed CVaR optimization problems. They show that, as the sample size increases the difference of the optimal value between the SAA problems and the original problem tends to zero with probability approaching to one exponentially fast.
scientific article; zbMATH DE number 5814309
scientific article