A computational study of a solver system for processing two-stage stochastic LPs with enhanced Benders decomposition

Zverovich, Victor; Fábián, Csaba I.; Ellison, Eldon F. D.; Mitra, Gautam

doi:10.1007/s12532-012-0038-z

Cited by 48 publications

(41 citation statements)

References 31 publications

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“…The two-stage stochastic linear programming problem (TSSP) deals with uncertainty through the analysis of possible outcomes (scenarios). Several solution methodologies based on column generation and cutting plane methods have been proposed to deal with this class of problems [8,12,61,66,72]. The TSSP can be posed as two interconnected problems.…”

Section: Two-stage Stochastic Programming Problem (Tssp)mentioning

confidence: 99%

“…Also, we describe the subproblem which turns out to be the dual of the recourse problem (59)- (61) and decomposable by scenario. As in [72], we are interested in the aggregated version of the master problem. The problem decomposes naturally in scenarios having (68) as the linking constraints.…”

Section: Decomposition and Column Generation Formulationmentioning

confidence: 99%

“…Notice that the same instance name may lead to more than one instance by varying the number of scenarios. From the last two columns of the table, we see that instances with a large number of scenarios have very large-scale corresponding DEPs and hence some of them challenge the state-of-the-art implementations of simplex-type methods or interior point methods [72]. On the other hand, these instances can be effectively solved by using a decomposition technique.…”

Section: Computational Experimentsmentioning

confidence: 99%

“…The remaining columns in the table show the relative performance of these methods in relation to PDCGM, i.e., the ratio between the values in columns 5 to 8 and the corresponding values in columns 3 and 4. We have taken the iteration numbers exactly as they are presented in [72], while the CPU times have been scaled according to the benchmark available at https: //www.cpubenchmark.net/singleThread.html. The computer used in [72] was a Linux PC with an Intel Core i5 2.4 GHz CPU and 6.0 GB of memory, and has a score of 1355 whereas the machine we used to run the PDCGM experiments has a score of 1602.…”

Section: Computational Experimentsmentioning

confidence: 99%

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Large-scale optimization with the primal-dual column generation method

2015

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The primal-dual column generation method (PDCGM) is a general-purpose column generation technique that relies on the primal-dual interior point method to solve the restricted master problems. The use of this interior point method variant allows to obtain suboptimal and well-centered dual solutions which naturally stabilizes the column generation. As recently presented in the literature, reductions in the number of calls to the oracle and in the CPU times are typically observed when compared to the standard column generation, which relies on extreme optimal dual solutions. However, these results are based on relatively small problems obtained from linear relaxations of combinatorial applications. In this paper, we investigate the behaviour of the PDCGM in a broader context, namely when solving large-scale convex optimization problems. We have selected applications that arise in important real-life contexts such as data analysis (multiple kernel learning problem), decision-making under uncertainty (two-stage stochastic programming problems) and telecommunication and transportation networks (multicommodity network flow problem). In the numerical experiments, we use publicly available benchmark instances to compare the performance of the PDCGM against recent results for different methods presented in the literature, which were the best available results to date. The analysis of these results suggests that the PDCGM offers an attractive alternative over specialized methods since it remains competitive in terms of number of iterations and CPU times even for large-scale optimization problems.

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Section: Two-stage Stochastic Programming Problem (Tssp)mentioning

confidence: 99%

Section: Decomposition and Column Generation Formulationmentioning

confidence: 99%

Section: Computational Experimentsmentioning

confidence: 99%

Section: Computational Experimentsmentioning

confidence: 99%

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Large-scale optimization with the primal-dual column generation method

2015

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“…An overview of different Benders decomposition variations and their performance was published in the computational study [18].…”

Section: A Literature Reviewmentioning

confidence: 99%

Large-scale power system planning using enhanced Benders decomposition

Skar

Doorman

Tomasgård

2014

2014 Power Systems Computation Conference

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An enhanced Benders decomposition algorithm for two-stage stochastic LPs is presented and applied to a largescale dynamic generation and transmission expansion planning model for the European power system. The improved algorithm is a variation of the traditional multi-cut Benders decomposition algorithm where the scenario aggregation used for the optimality cuts is reduced at a given error threshold. Experimental results show that this technique improves convergence and reduces computation time. An analysis using the planning model to compute an optimal development of the European power sector under a global climate policy is also discussed.Index Terms-Benders decomposition, investments under uncertainty, large-scale power system planning

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Massively parallelizable proximal algorithms for large‐scale stochastic optimal control problems

Sampathirao,

Patrinos,

Bemporad

et al. 2023

Optim Control Appl Methods

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Scenario‐based stochastic optimal control problems suffer from the curse of dimensionality as they can easily grow to six and seven figure sizes. First‐order methods are suitable as they can deal with such large‐scale problems, but may perform poorly and fail to converge within a reasonable number of iterations. To achieve a fast rate of convergence and high solution speeds, in this article, we propose the use of two proximal quasi‐Newtonian limited‐memory algorithms—minfbe applied to the dual problem and the Newton‐type alternating minimization algorithm (nama)—which can be massively parallelized on lockstep hardware such as graphics processing units. In particular, we use minfbe and nama to solve scenario‐based stochastic optimal control problems with affine dynamics, convex quadratic cost functions (with the stage cost functions being strongly convex in the control variable) and joint state‐input convex constraints. We demonstrate the performance of these methods, in terms of convergence speed and parallelizability, on large‐scale problems involving millions of variables.

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A computational study of a solver system for processing two-stage stochastic LPs with enhanced Benders decomposition

Cited by 48 publications

References 31 publications

Large-scale optimization with the primal-dual column generation method

Large-scale optimization with the primal-dual column generation method

Large-scale power system planning using enhanced Benders decomposition

Massively parallelizable proximal algorithms for large‐scale stochastic optimal control problems

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