Stochastic Programming: Convex Approximation and Modified Linear Decision Rule

Stochastic optimization, especially multistage models, is well known to be computationally excruciating. In this paper, we introduce the concept of semi-complete recourse in the context of stochastic programming as a less restrictive condition compared to complete recourse and propose methods for approximating multistage stochastic programs with risk constraints and semi-complete recourse. We examine subtle issues associated with chance constraints in stochastic optimization with recourse, which motivates our adoption of a class of coherent risk measure to limit constraint violations. We show that methodologies in robust optimization could approximate models that satisfy these risk constraints. We also examine linear decision rule in greater details and show that even on problems with complete recourse, linear decision rule can be inadequate and even lead to infeasible instances. Hence, we propose a new modified linear decision rule for semi-complete recourse variables. We also introduce “computationally friendly” models in the form of second order cone (SOC) program, which could be solved efficiently both in theory and in practice. As a refinement of these decision rules, we introduce a segregated decision rule that could potentially improve the approximation. An attractive aspect of our proposal is the scalability to multistage stochastic programs. ∗Department of Mechanical and Industrial Engineering, University of Illinois at Urbana-Champaign. Email: [email protected] †NUS Business School, National University of Singapore. Email: [email protected]. The research of the author is supported by NUS academic research grant R-314-000-066-122. ‡Fuqua School of Business, Duke University Box 90120, Durham, NC 27708, USA. Email: [email protected] §Stern School of Business, New York University, New York, NY, 10012. Email: [email protected]