A Summary and Illustration of Disjunctive Decomposition with Set Convexification

2010

Operations Research

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This paper introduces disjunctive decomposition for two-stage mixed 0-1 stochastic integer programs (SIPs) with random recourse. Disjunctive decomposition allows for cutting planes based on disjunctive programming to be generated for each scenario subproblem under a temporal decomposition setting of the SIP problem. A new class of valid inequalities for mixed 0-1 SIP with random recourse is presented. In particular, valid inequalities that allow for sharing cut coefficients among scenario subproblems for SIP with random recourse but deterministic technology matrix and righthand side vector are obtained. The valid inequalities are used to derive a disjunctive decomposition method whose derivation has been motivated by real-life stochastic server location problems with random recourse, which find many applications in operations research. Computational results with large-scale instances to demonstrate the potential of the method are reported.

Section: Related Workmentioning

confidence: 99%

Disjunctive Decomposition for Two-Stage Stochastic Mixed-Binary Programs with Random Recourse

2010

Operations Research

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“…The D 2 algorithm is derived in [17] and illustrated in [16]. In the D 2 algorithm the scenario subproblem LP relaxation takes the following form:…”

Section: The D 2 Algorithmmentioning

confidence: 99%

A comparative study of decomposition algorithms for stochastic combinatorial optimization

Sen

2007

Comput Optim Appl

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This paper presents comparative computational results using three decomposition algorithms on a battery of instances drawn from two different applications. In order to preserve the commonalities among the algorithms in our experiments, we have designed a testbed which is used to study instances arising in server location under uncertainty and strategic supply chain planning under uncertainty. Insights related to alternative implementation issues leading to more efficient implementations, benchmarks for serial processing, and scalability of the methods are also presented. The computational experience demonstrates the promising potential of the disjunctive decomposition (D 2 ) approach towards solving several large-scale problem instances from the two application areas. Furthermore, the study shows that convergence of the D 2 methods for stochastic combinatorial optimization (SCO) is in fact attainable since the methods scale well with the number of scenarios.

“…Several surveys on algorithms for SMIP (Schultz et al, 1996, Klein Haneveld and van der Vlerk, 1999, Louveaux and Schultz, 2003, Schultz, 2003, Sen, 2005) and a few textbooks are available (Birge and Louveaux, 1997, Ruszczyn'ski and Shapiro, 2003, Shapiro et al, 2009. Closely related cutting plane methods for SMIP include the method by Carøe and Tind (1997), Carøe (1998) and the disjunctive decomposition (D2) method (Sen et al, 2002, Sen andHigle, 2005). Both methods are based on disjunctive programming (Balas, 1975, Blair and Jeroslow, 1978, Sherali and Shetty, 1980 and provide a rather general setting for the study of the convex hull of feasible points of SMIP problems under the two-stage setting.…”

Section: Introductionmentioning

confidence: 99%

Fenchel decomposition for stochastic mixed-integer programming

2011

J Glob Optim

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This paper introduces a new cutting plane method for two-stage stochastic mixed-integer programming (SMIP) called Fenchel decomposition (FD). FD uses a class of valid inequalities termed, FD cuts, which are derived based on Fenchel cutting planes from integer programming. First, we derive FD cuts based on both the first and second-stage variables, and devise an FD algorithm for SMIP and establish finite convergence for binary first-stage. Second, we derive FD cuts based on the second-stage variables only and use an idea from disjunctive programming to lift the cuts to the higher dimension space including the first-stage variables. We then devise an alternative algorithm (FD-L algorithm) based on the lifted FD cuts. Finally, we report on computational results based on several test instances from the literature. The results are promising and show that both algorithms can outperform a standard direct solver and a disjunctive decomposition algorithm on some large-scale instances. Furthermore, the FD-L algorithm provides better performance than the FD algorithm in general.