An Efficient Decomposition Algorithm for Static, Stochastic, Linear and Mixed-Integer Linear Programs with Conditional-Value-at-Risk Constraints

LIMITED DISTRIBUTION NOTICE: This report has been submitted for publication outside of IBM and will probably be copyrighted if accepted for publication. It has been issued as a Research Report for early dissemination of its contents. In view of the transfer of copyright to the outside publisher, its distribution outside of IBM prior to publication should be limited to peer communications and specific requests. After outside publication, requests should be filled only by reprints or legally obtained copies of the article (e.g. , payment of royalties). Copies may be requested from IBM T. J. We present an efficient decomposition algorithm for single-stage, stochastic linear programs, where conditional value at risk (CVaR) appears as a risk measure in multiple constraints. It starts with a well-known nonlinear, convex reformulation of conditional value at risk constraints, and establishes the connection to a combinatorially large polyhedral representation of the convex feasible set induced by the above refor-mulation. The algorithm is developed as a column-generation procedure in the dual space of the resulting polyhedral representation. In the special case, where CVaR appears in the objective function alone, it offers an alternative view into CVarMin, a breakthrough algorithm in the literature, which specialized the L-shaped algorithm of two-stage stochastic programs with recourse for CVaR minimization. CVarMin concentrated on CVaR in the objective function, which led naturally to a two-stage interpretation. The optimization problem with multiple CVaR constraints does not have a natural two-stage interpretation, nevertheless we explicitly extend the polyhedral ideas of CVaR minimization and integrated chance constraints to the case of multiple CVaR constraints. We also present an algorithm that engineers the decomposition scheme to address mixed-integer linear programming problems with multiple CVaR constraints.