A Scenario Decomposition Algorithm for Stochastic Programming Problems with a Class of Downside Risk Measures

We present an efficient scenario decomposition algorithm for solving large-scale convex stochastic programming problems that involve a particular class of downside risk measures. The considered risk functionals encompass coherent and convex measures of risk that can be represented as an infimal convolution of a convex certainty equivalent, and include well-known measures, such as conditional value-at-risk, as special cases. The resulting structure of the feasible set is then exploited via iterative solving of relaxed problems, and it is shown that the number of iterations is bounded by a parameter that depends on the problem size. The computational performance of the developed scenario decomposition method is illustrated on portfolio optimization problems involving two families of nonlinear measures of risk, the higher-moment coherent risk measures and log-exponential convex risk measures. It is demonstrated that for large-scale nonlinear problems the proposed approach can provide up to an order of magnitude of improvement in computational time in comparison to state-of-the-art solvers, such as CPLEX, Gurobi, and MOSEK.