The value function of an integer program

“…Note that, for each t and s, Q s t (x) is the value function of an integer program and is known to be piece-wise constant over certain cones in the space of x with discontinuities along the boundaries of these cones [5]. Existing branch and bound methods for stochastic integer programs attempt to partition the space of first stage variables into (hyper)rectangular cells.…”

Section: A Decomposition Based Branch and Bound Algorithmmentioning

confidence: 99%

Dynamic Capacity Acquisition and Assignment under Uncertainty

Ahmed

Garcia

2003

Annals of Operations Research

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Given a set of m resources and n tasks, the dynamic capacity acquisition and assignment problem seeks a minimum cost schedule of capacity acquisitions for the resources and the assignment of resources to tasks, over a given planning horizon of T periods. This problem arises, for example, in the integrated planning of locations and capacities of distribution centers (DCs), and the assignment of customers to the DCs, in supply chain applications. We consider the dynamic capacity acquisition and assignment problem in an environment where the assignment costs and the processing requirements for the tasks are uncertain. Using a scenario based approach, we develop a stochastic integer programming model for this problem. The highly non-convex nature of this model prevents the application of standard stochastic programming decomposition algorithms. We use a recently developed decomposition based branch-andbound strategy for the problem. Encouraging preliminary computational results are provided.

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“…Thus Benders' decomposition (Benders, 1962) is applicable in this case (Wollmer, 1980). Otherwise, when the second-stage variables involve integrality restrictions, f (ω, x) is lower semicontinuous with respect to x (Blair and Jeroslow, 1982), and is generally nonconvex (Schultz, 1993). The focus of this paper is on SP2 with x ∈ vert(conv(X)) such as binary first-stage, and mixed-integer second-stage.…”

Section: Introductionmentioning

confidence: 99%

Fenchel decomposition for stochastic mixed-integer programming

Ntaimo

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.

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The value function of an integer program

Cited by 118 publications

References 12 publications

Convex analysis in groups and semigroups: a sampler

Convex analysis in groups and semigroups: a sampler

Dynamic Capacity Acquisition and Assignment under Uncertainty

Fenchel decomposition for stochastic mixed-integer programming

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