Obtaining lower bounds from the progressive hedging algorithm for stochastic mixed-integer programs

Gade, Dinakar; Hackebeil, Gabriel A.; Ryan, Sarah M.; Watson, Jean‐Paul; Wets, Roger J.-B.; Woodruff, David L.

doi:10.1007/s10107-016-1000-z

Cited by 150 publications

(130 citation statements)

References 40 publications

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“…However, the allocation of scenarios was random since no criterion concerning which scenarios would be grouped into the same bundle was given. Gade et al incorporated scenario bundling into the PHA for two-stage SMIPs and present a method to compute lower bounds for the purpose of assessing the quality of the solutions generated by the PHA [9]. In their work, all bundles are of the same size and there are no common scenarios between any two bundles.…”

Section: Literature Reviewmentioning

confidence: 99%

“…The performance of the PHA can be enhanced by replacing the preceding scenario decomposition with bundle decomposition [5,9], where individual scenarios are combined into bundles and the extensive form is decomposed using scenario bundles into multi-scenario subproblems. In contrast to scenario decomposition, the bundle version of the PHA solves smaller numbers of larger subproblems [7].…”

Section: Introductionmentioning

confidence: 99%

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Fuzzy C-means-based scenario bundling for stochastic service network design

Jiang

Bai

Landa-Silva

et al. 2017

2017 IEEE Symposium Series on Computational Intelligence (SSCI)

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Abstract-Stochastic service network designs with uncertain demand represented by a set of scenarios can be modelled as a large-scale two-stage stochastic mixed-integer program (SMIP). The progressive hedging algorithm (PHA) is a decomposition method for solving the resulting SMIP. The computational performance of the PHA can be greatly enhanced by decomposing according to scenario bundles instead of individual scenarios. At the heart of bundle-based decomposition is the method for grouping the scenarios into bundles. In this paper, we present a fuzzy c-means-based scenario bundling method to address this problem. Rather than full membership of a bundle, which is typically the case in existing scenario bundling strategies such as k-means, a scenario has partial membership in each of the bundles and can be assigned to more than one bundle in our method. Since the multiple bundle membership of a scenario induces overlap between the bundles, we empirically investigate whether and how the amount of overlap controlled by a fuzzy exponent would affect the performance of the PHA. Experimental results for a less-than-truckload transportation network optimization problem show that the number of iterations required by the PHA to achieve convergence reduces dramatically with large fuzzy exponents, whereas the computation time increases significantly. Experimental studies were conducted to find out a good fuzzy exponent to strike a trade-off between the solution quality and the computational time.

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Section: Literature Reviewmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Fuzzy C-means-based scenario bundling for stochastic service network design

Jiang

Bai

Landa-Silva

et al. 2017

2017 IEEE Symposium Series on Computational Intelligence (SSCI)

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“…The basic PH algorithm for two-stage stochastic mixed-integer programs proceeds as follows [8]: While convergence is not guaranteed for mixed-integer problems, computational studies have shown that the PH algorithm can find high-quality solutions within a reasonable number of iterations [25]. The PH algorithm also applies to multi-stage stochastic programs with discrete variables in any stage.…”

Section: Progressive Hedgingmentioning

confidence: 99%

“…We now demonstrate how PH and DD can be integrated through their lower bounds. We first review the lower bounding technique for the PH algorithm proposed by Gade et al [8] and recall equivalence between the best lower bounds obtained by the PH algorithm and the Lagrangian dual from the DD algorithm. Finally, we establish relationships between PH weights and DD multipliers.…”

Section: Integration Of Ph and Ddmentioning

confidence: 99%

“…Originally proposed by Rockafellar and Wets [19] for stochastic programs with only continuous variables, progressive hedging (PH) has been successfully applied by Listes and Dekker [17], Fan and Liu [6], Watson and Woodruff [25], and many others as a heuristic to solve stochastic mixed-integer programs. To assess the quality of the solutions generated by PH relative to the optimal solution, Gade et al [8] presented a lower bounding technique for the PH algorithm and showed that the best possible lower bound obtained from PH is as tight as the lower bound obtained using DD.…”

Section: Introductionmentioning

confidence: 99%

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Integration of progressive hedging and dual decomposition in stochastic integer programs

Guo

Hackebeil

Ryan

et al. 2015

Operations Research Letters

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We present a method for integrating the Progressive Hedging (PH) algorithm and the Dual Decomposition (DD) algorithm of Carøe and Schultz for stochastic mixed-integer programs. Based on the correspondence between lower bounds obtained with PH and DD, a method to transform weights from PH to Lagrange multipliers in DD is found. Fast progress in early iterations of PH speeds up convergence of DD to an exact solution. We report computational results on server location and unit commitment instances. KeywordsStochastic programming, Mixed-integer programming, Progressive hedging, Dual decomposition, Lower bounding Disciplines Industrial Engineering | Systems EngineeringComments NOTICE: this is the author's version of a work that was accepted for publication in Operation Research Letters. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Operations Research Letters, [v.43, iss.3,(2015) AbstractWe present a method for integrating the Progressive Hedging (PH) algorithm and the Dual Decomposition (DD) algorithm of Carøe and Schultz for stochastic mixed-integer programs. Based on the correspondence between lower bounds obtained with PH and DD, a method to transform weights from PH to Lagrange multipliers in DD is found. Fast progress in early iterations of PH speeds up convergence of DD to an exact solution. We report computational results on server location and unit commitment instances.

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A flexible design framework for process systems under demand‐side management

et al. 2020

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Demand response (DR) has been an appealing strategy for both residential and commercial/industrial customers of electricity due to the increasing penetration of renewable energy resources into the power grid and the deregulation of the energy markets. For industrial DR participants, the management of energy consumption along with the satisfaction of production expectations is generally referred to as demand‐side management. Recent research in this area has focused mostly on process scheduling and capacity planning, especially for energy‐intensive processes. This article presents a new direction through an optimization‐based process design paradigm that allows for the potential reconfiguration of the process flow sheet in real time. Based on a case study of pump network design, a network superstructure is first defined, followed by a two‐stage stochastic optimization model. The method of progressive hedging is used to solve the resulting mixed‐integer stochastic programming problem. The results demonstrate that, under various scenarios, the reconfigurable design offers significant benefits compared to a fixed network structure in terms of total design cost and expected operating cost.

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Obtaining lower bounds from the progressive hedging algorithm for stochastic mixed-integer programs

Cited by 150 publications

References 40 publications

Fuzzy C-means-based scenario bundling for stochastic service network design

Fuzzy C-means-based scenario bundling for stochastic service network design

Integration of progressive hedging and dual decomposition in stochastic integer programs

A flexible design framework for process systems under demand‐side management

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