Solving Stochastic Programming Problems with Risk Measures by Progressive Hedging

The progressive hedging algorithm for stochastic programming problems in single or multiple stages is a decomposition method which, in each iteration, solves a separate subproblem with modified costs for each scenario. The decomposition exploits the separability of objective functions formulated in terms of expected costs, but nowadays expected costs are not the only objectives of interest. Minimization of risk measures for cost, such as conditional value-at-risk, can be important as well, but their lack of separability presents a hurdle. Here it is shown how the progressive hedging algorithm can nonetheless be applied to solve many such problems through the introduction of additional variables which, like the given decision variables, get updated through aggregation of the independent computations for the various scenarios.