Bundle-based decomposition for large-scale convex optimization: Error estimate and application to block-angular linear programs

Medhi, Deepankar

doi:10.1007/bf01581138

Cited by 13 publications

(9 citation statements)

References 11 publications

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“…Several algorithmic approaches have been proposed for MMCF: among them, we just name Column Generation [13,57,8] , Resource Decomposition [28,50,49] , Primal Partitioning [23,18] , specialized Interior Point [62,48,70,19] , Cost Decomposition [23,51,54,61,20,69] , Primal-Dual [67] and ε-approximation [52,34] methods. Unfortunately, the available surveys are either old [5,45] or concentrated on just the "classical" approaches : hence, we refer the interested reader to [25] , where the relevant literature on the subject is surveyed.…”

Section: Given a Directed Graph G(nmentioning

confidence: 99%

“…Many NDO algorithms can be used to maximize ϕ; among them: the (many variants of the) Subgradient method [40,53] , the Cutting Plane approach [44] (that is the dual form of the Dantzig-Wolfe decomposition [21,23,42] ), the Analytic Center Cutting Plane approach [30,55] and various Bundle methods [39,51,54] . The latter can also be shown [26] to be intimately related to seemingly different approaches such as smooth Penalty Function [61] and Augmented Lagrangean [20] algorithms, where ϕ is approximated with a smooth function.…”

Section: Given a Directed Graph G(nmentioning

confidence: 99%

“…The availability of the specialized solver has been crucial for developing an efficient Bundle code: other than supporting the LVG strategy, it is much faster than non-specialized QP codes in solving sequences of (Πβ t ) problems. This has been shown in [24] by comparing it with two standard QP codes (interestingly enough, one of them being exactly the QP solver used for the previous Bundle approach to MMCF [54] ), which have been outperformed by up to two orders of magnitude; since the coordination cost ranges from 1% to 20% of the total running time, a Bundle algorithm using a non-specialized QP solver would rapidly become impractical as the instances size increases.…”

Section: The Cost Decomposition Codementioning

confidence: 99%

“…Decomposition methods have been around for 40 years, and several proposals for the "coordination phase" [22,23,51,54,61,20] can be found in the literature. Our aim is to assess the effectiveness of a Cost Decomposition approach based on a NonDifferentiable Optimization (NDO) algorithm belonging to the class of "Bundle methods" [63,39] .…”

Section: Introductionmentioning

confidence: 99%

“…Our aim is to assess the effectiveness of a Cost Decomposition approach based on a NonDifferentiable Optimization (NDO) algorithm belonging to the class of "Bundle methods" [63,39] . To the best of our knowledge, only one attempt [54] has previously been made to use Bundle methods in this context, and just making use of pre-existent NDO software: here we use a specialized Bundle code for MMCF. Innovative features of our code include a new heuristic for setting the trust region parameter, an efficient specialized Quadratic Program (QP) solver [24] and a Lagrangean variables generation strategy.…”

Section: Introductionmentioning

confidence: 99%

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A Bundle Type Dual-Ascent Approach to Linear Multicommodity Min-Cost Flow Problems

Frangioni

Gallo

1999

INFORMS Journal on Computing

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We present a Cost Decomposition approach for the linear Multicommodity Min-Cost Flow problem, where the mutual capacity constraints are dualized and the resulting Lagrangean Dual is solved with a dual-ascent algorithm belonging to the class of Bundle methods. Although decomposition approaches to block-structured Linear Programs have been reported not to be competitive with general-purpose software, our extensive computational comparison shows that, when carefully implemented, a decomposition algorithm can outperform several other approaches, especially on problems where the number of commodities is "large" with respect to the size of the graph. Our specialized Bundle algorithm is characterized by a new heuristic for the trust region parameter handling, and embeds a specialized Quadratic Program solver that allows the efficient implementation of strategies for reducing the number of active Lagrangean variables. We also exploit the structural properties of the single-commodity Min-Cost Flow subproblems to reduce the overall computational cost. The proposed approach can be easily extended to handle variants of the problem. Keywords: Multicommodity Flows, Bundle methods, Decomposition Methods, Lagrangean Dual IntroductionThe Multicommodity Min-Cost Flow problem (MMCF), i.e. the problem of shipping flows of different nature (commodities) at minimal cost on a network, where different commodities compete for the resources represented by the arc capacities, has been widely addressed in the literature since it models a wide variety of transportation and scheduling problems [28,58,68,3,2,6,11] . MMCF is a structured Linear Program (LP), but the instances arising from practical applications are often huge and the usual solution techniques are not efficient enough: this is especially true if the solution is required to be integral, since then the problem is NP-hard and most of the solution techniques From an algorithmic viewpoint, MMCF has motivated many important ideas that have later found broader application: examples are the column generation approach [22] and the Dantzig-Wolfe decomposition algorithm [21] . This "pushing" effect is still continuing, as demonstrated by a number of interesting recent developments [59,31,32,33] .MMCF problems also arise in finding approximate solutions to several hard graph problems [10,43] . Recently, some ε-approximation approaches have been developed for the problem [36,60,64] , making MMCF one of the few LPs for which approximations algorithms of practical interest are known [52,34] .

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Section: Given a Directed Graph G(nmentioning

confidence: 99%

Section: Given a Directed Graph G(nmentioning

confidence: 99%

Section: The Cost Decomposition Codementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Bundle Type Dual-Ascent Approach to Linear Multicommodity Min-Cost Flow Problems

Frangioni

Gallo

1999

INFORMS Journal on Computing

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Decomposing linear programs for parallel solution

Pınar

Çatalyürek

Aykanat

et al. 1996

Lecture Notes in Computer Science

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Coarse grain parallelism inherent in the solution of Linear Programming (LP) problems with block angular constraint matrices has been exploited in recent research works. However, these approaches su er from unscalability and load imbalance since they exploit only the existing block angular structure of the LP constraint matrix. In this paper, we consider decomposing LP constraint matrices to obtain block angular structures with speci ed number of blocks for scalable parallelization. We propose hypergraph models to represent LP constraint matrices for decomposition. In these models, the decomposition problem reduces to the well-known hypergraph partitioning problem. A Kernighan-Lin based multiway hypergraph partitioning heuristic is implemented for experimenting with the performance of the proposed hypergraph models on the decomposition of the LP problems selected from NETLIB suite. Initial results are promising and justify further research on other hypergraph partitioning heuristics for decomposing large LP problems.

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An effective model to decompose Linear Programs for parallel solution

Pınar

Aykanat

1996

Applied Parallel Computing Industrial Computation and Optimization

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Although inherent parallelism in the solution of block angulax Linear Programming (LP) problems has been exploited in many research works, the literature that addresses decomposing constraint matrices into block angular form for parallel solution is very rare and recent. We have previously proposed hypergraph models, which reduced the problem to the hypergraph partitioning problem. However, the quality of the results reported were limited due to the hypergraph partitioning tools we have used. Very recently, multilevel graph partitioning heuristics have been proposed leading to very successful graph partitioning tools; Chaco and Metis. In this paper, we propose an effective graph model to decompose matrices into block angular form, which reduces the problem to the well-known graph partitioning by vertex separator problem. We have experimented the validity of our proposed model with various LP problems selected from NETLIB and other sources. The results are very attractive both in terms of solution quality and running times.

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Bundle-based decomposition for large-scale convex optimization: Error estimate and application to block-angular linear programs

Cited by 13 publications

References 11 publications

A Bundle Type Dual-Ascent Approach to Linear Multicommodity Min-Cost Flow Problems

A Bundle Type Dual-Ascent Approach to Linear Multicommodity Min-Cost Flow Problems

Decomposing linear programs for parallel solution

An effective model to decompose Linear Programs for parallel solution

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