Robust two-stage stochastic linear optimization with risk aversion

We study a two-stage stochastic linear optimization problem where the recourse function is risk-averse rather than risk neutral. In particular, we consider the mean-conditional value-at-risk objective function in the second stage. The model is robust in the sense that the distribution of the underlying random variable is assumed to belong to a certain family of distributions rather than to be exactly known. We start from analyzing a simple case where uncertainty arises only in the objective function, and then explore the general case where uncertainty also arises in the constraints. We show that the former problem is equivalent to a semidefinite program and the latter problem is generally NP-hard. Applications to two-stage portfolio optimization, material order problems, stochastic production-transportation problem and single facility minimax distance problem are considered. Numerical results show that the proposed robust risk-averse two-stage stochastic programming model can effectively control the risk with solutions of acceptable good quality.