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IMA Journal of Management Mathematics Advance Access published online on June 25, 2009
IMA Journal of Management Mathematics, doi:10.1093/imaman/dpp001
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© The authors 2009. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Stochastic scenario decomposition for multistage stochastic programs

Julia L. Higle

IWSE Department, The Ohio State University, Columbus, OH 43210, USA

Brenda Rayco

Modular Mining Systems, Inc., 3289 E. Hemisphere Loop, Tucson, AZ 85706-5028, USA

Suvrajeet Sen{dagger}

IWSE Department, The Ohio State University, Columbus, OH 43210, USA

{dagger} Email: sen.22{at}osu.edu

Received on 4 January 2007. Accepted on 3 April 2009.

Over the past decade, several stochastic approaches have been proposed for two-stage stochastic programs. Many of these algorithms have attractive computational as well as conceptual properties (e.g. convergence with probability one). This paper expands the realm of such approaches to multistage convex stochastic programming problems. We present a stochastic scenario decomposition (SSD) algorithm which is a statistically motivated cutting plane algorithm for the solution of multistage stochastic programs. The method is based on solving a dual problem in which the variables correspond to multipliers associated with the non-anticipativity constraints of the primal problem. Our analytical results verify conditions under which SSD identifies an optimal solution asymptotically. We also overcome some computational hurdles resulting from increases in the column dimension of the SSD master problem. We propose a variable aggregation scheme that allows us to solve much smaller master programs without sacrificing solution quality. Our computational results demonstrate the effectiveness of this aggregation scheme in solving the SSD master program.

Keywords: stochastic programming; decomposition; duality


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