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A Linear Decision-Based Approximation Approach to Stochastic Programming

Department of Industrial and Enterprise Systems Engineering, University of Illinois at Urbana–Champaign, Urbana, Illinois 61801
NUS Business School, NUS Risk Management Institute, and Singapore MIT Alliance (SMA), Singapore
Fuqua School of Business, Duke University, Durham, North Carolina 27708
Stern School of Business, New York University, New York, New York 10012
xinchen{at}uiuc.edu
dscsimm{at}nus.edu.sg
psun{at}duke.edu
jzhang{at}stern.nyu.edu

Stochastic optimization, especially multistage models, is well known to be computationally excruciating. Moreover, such models require exact specifications of the probability distributions of the underlying uncertainties, which are often unavailable. In this paper, we propose tractable methods of addressing a general class of multistage stochastic optimization problems, which assume only limited information of the distributions of the underlying uncertainties, such as known mean, support, and covariance. One basic idea of our methods is to approximate the recourse decisions via decision rules. We first examine linear decision rules in detail and show that even for problems with complete recourse, linear decision rules can be inadequate and even lead to infeasible instances. Hence, we propose several new decision rules that improve upon linear decision rules, while keeping the approximate models computationally tractable. Specifically, our approximate models are in the forms of the so-called second-order cone (SOC) programs, which could be solved efficiently both in theory and in practice. We also present computational evidence indicating that our approach is a viable alternative, and possibly advantageous, to existing stochastic optimization solution techniques in solving a two-stage stochastic optimization problem with complete recourse.

Subject classifications: programming; stochastic.
History: Received February 2006; revision received February 2007; accepted March 2007.