<i>L</i>-Shaped Linear Programs with Applications to Optimal Control and Stochastic Programming

Slyke, Richard M. Van; Wets, Roger J.-B.

doi:10.1137/0117061

Cited by 1,017 publications

(449 citation statements)

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“…We will solve problem SEF using the L-shaped method with integer first-stage variables (Van Slyke andWets (1969), Wollmer (1980)). The L-shaped method is an iterative procedure closely related to Benders' decomposition (1962) that divides the extensive form given by (2) into |Ξ| + 1 problems: a first-stage master program, and one second-stage subproblem for each scenario.…”

Section: An L-shaped Approachmentioning

confidence: 99%

A stochastic integer programming approach to solving a synchronous optical network ring design problem

2004

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We develop stochastic integer programming techniques tailored toward solving a Synchronous Optical Network (SONET) ring design problem with uncertain demands.Our approach is based on an L-shaped algorithm, whose (integer) master program prescribes a candidate network design, and whose (continuous) subproblems relay information regarding potential shortage penalty costs to the ring design decisions. This naive implementation performs very poorly due to two major problems: (1) the weakness of the master problem relaxations, and (2) the limited information passed to the master problem by the optimality cuts. Accordingly, we enforce certain necessary conditions regarding shortage penalty contributions to the objective function within the master problem, along with a corresponding set of valid inequalities that improve the solvability of the master problem. We also detail how a nonlinear reformulation of the model can be used to capture an exponential number of optimality cuts generated by the linear model. We augment these techniques with a powerful upper-bounding heuristic to further accelerate the convergence of the algorithm, and demonstrate the effectiveness of our methodologies on a test bed of randomly generated stochastic SONET instances.

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Section: An L-shaped Approachmentioning

confidence: 99%

A stochastic integer programming approach to solving a synchronous optical network ring design problem

2004

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“…When the stochastic uncertainty follows a discrete distribution with a small number of scenarios, the problem can be rewritten deterministically and solved using the simplex or interior point methods. A common approach for this is Benders' algorithm (see Benders 1962, Van Slyke andWets 1969), which uses a planecutting technique also known as the L-shaped method. With a discrete source of uncertainty, this has the advantage of scalability.…”

Section: Simulation-based Stochastic Programmingmentioning

confidence: 99%

Augmented Markov Chain Monte Carlo Simulation for Two-Stage Stochastic Programs with Recourse

2014

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I n this paper, we develop a simulation-based approach for two-stage stochastic programs with recourse.We construct an augmented probability model with stochastic shocks and decision variables. Simulating from the augmented probability model solves for the expected recourse function and the optimal first-stage decision. Markov chain Monte Carlo methods, together with ergodic averaging, provide a framework to compute the optimal solution. We illustrate our methodology via the two-stage newsvendor problem with unimodal and bimodal continuous uncertainty. Finally, we present performance comparisons of our algorithm and the sample average approximation method.

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“…In [11] it was shown how to modify a Nested Benders Decomposition [1,12,14] of problem (1.11) to exploit the recombining property (1.13) of the process ξ.…”

Section: Solution Algorithmmentioning

confidence: 99%

Optimization of Dispersed Energy Supply —Stochastic Programming with Recombining Scenario Trees

Epe

Küchler

Römisch

et al. 2009

Optimization in the Energy Industry

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L-Shaped Linear Programs with Applications to Optimal Control and Stochastic Programming

Cited by 1,017 publications

References 20 publications

A stochastic integer programming approach to solving a synchronous optical network ring design problem

A stochastic integer programming approach to solving a synchronous optical network ring design problem

Augmented Markov Chain Monte Carlo Simulation for Two-Stage Stochastic Programs with Recourse

Optimization of Dispersed Energy Supply —Stochastic Programming with Recombining Scenario Trees

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