Lagrangean decomposition for integer programming : theory and applications

Guignard, Monique; Kim, Siwhan

doi:10.1051/ro/1987210403071

Cited by 47 publications

(17 citation statements)

References 15 publications

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“…Thus, these two parts are somehow complementary to each other. We study this scheme in a general nonconvex setting, thereby putting in perspective the results of [20,21]. Our results there are by no means fundamentally new, they are in fact variations around [26,9,1,28], and many others; we also mention the recent work [12], rather close in spirit to ours.…”

Section: Scope Of the Papersupporting

confidence: 57%

“…The first strategy is adopted in [20,21] for the following specific problem: linear objective f(x) = c x and constraints g(x) = Ax − b; as for X, it has itself two sets of constraints: The first strategy is adopted in [20,21] for the following specific problem: linear objective f(x) = c x and constraints g(x) = Ax − b; as for X, it has itself two sets of constraints:…”

Section: Improving the Duality Gapmentioning

confidence: 99%

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A geometric study of duality gaps, with applications

Lemaréchal¹,

Renaud

2001

Math. Program.

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Lagrangian relaxation is often an efficient tool to solve (large-scale) optimization problems, even nonconvex. However it introduces a duality gap, which should be small for the method to be really efficient.Here we make a geometric study of the duality gap. Given a nonconvex problem, we formulate in a first part a convex problem having the same dual. This formulation involves a convexification in the product of the three spaces containing respectively the variables, the objective and the constraints. We apply our results to several relaxation schemes, especially one called "Lagrangean decomposition" in the combinatorial-optimization community, or "operator splitting" elsewhere. We also study a specific application, highly nonlinear: the unit-commitment problem.

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Section: Scope Of the Papersupporting

confidence: 57%

Section: Improving the Duality Gapmentioning

confidence: 99%

A geometric study of duality gaps, with applications

Lemaréchal¹,

Renaud

2001

Math. Program.

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“…These results have the surprising implication that the bounds from two a priori different primal relaxations of the capacitated plant location problem are actually equal. This means that a simple Lagrangean substitution yields exactly the same strong bound as the computationally more expensive Lagrangean decomposition method introduced by Guignard and Kim (1987).…”

Section: A Survey Of Subsequent Workmentioning

confidence: 92%

“…We will omit discussion of Lagrangean decomposition, a solution technique closely related to Lagrangean relaxation. For an overview of Lagrangean decomposition, see, for example, Guignard and Kim (1987).…”

Section: A Brief Review Of Lagrangean Relaxationmentioning

confidence: 99%

Hub Location Problems: The Location of Interacting Facilities

Kara

Taner

2011

International Series in Operations Research &Amp; Management Science

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), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. PrefaceThis book is the final result of a number of incidents that occurred to us while discussing location issues with colleagues at conferences. Frequently we attended presentations, in which the authors quoted the well-known references that helped to make the discipline what it is today. Upon further inquiry, though, it turned out that some of these colleagues had never actually read the original papers. We then discussed among ourselves which contributions could be credited with shaping the field. And, lo and behold, we found that we, too, had neglected to read some of the papers that form the foundation of our science. Whether it was laziness or other things that got in the way, it had become clear that something had to be done. Our first thought was to collect the original contributions (once we could agree on what they were) and reprint them. When discussing this possibility with a publisher, we immediately ran into a roadblock in the form of copyright. While this appeared to have stopped our enthusiastic effort dead in its tracks, we kept on collecting and reading what we considered original contributions.This went on until we met Camille Price, who suggested that, rather than reprinting the original contributions, we should invite some of the leaders in our field and ask them to describe the original contribution, explain and interpret it, and comment on the impact that it had to the field. This was, of course, an excellent idea, and the response by our colleagues to our pertinent requests was equally enthusiastic. What you hold in your hands is the result of this effort.In other words, the purpose of this book is to provide easy access to the main contributions to location theory. The book is organized as follows. The introductory chapter provides an overview of some of the many facets of location analysis. This is followed by contributions in the main three fields of inquiry: minisum, minimax, and covering problems. The next chapters are part of an ever-growing list of nonstandard location models: models including competitive components, those that locate undesirable facilities, those with probabilistic features, and those that allow interactions between facilities. The following chapters discuss solution techniques: after a discussion of exact and heuristic techniques, we devote an entire chapter to Weiszfeld's method, and another to Lagrangean techniques. The last chapters of this book deal with the spheres of influence that the facilities generate and that attract viii customers to them, an...

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“…For Lagrangean decomposition, see Jornsten and Nasberg [59] and Guignard and Kim [53]. For cut-and-price, recent papers include Fukasawa et al [41] on vehicle routing and Ochoa et al [86] on capacitated spanning trees.…”

Section: Hybrid Algorithms and Stronger Dual Boundsmentioning

confidence: 99%

Reformulation and Decomposition of Integer Programs

Vanderbeck

Wolsey

2009

50 Years of Integer Programming 1958-2008

110

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In this survey we examine ways to reformulate integer and mixed integer programs. Typically, but not exclusively, one reformulates so as to obtain stronger linear programming relaxations, and hence better bounds for use in a branch-and-bound based algorithm. First we cover in detail reformulations based on decomposition, such as Lagrangean relaxation, Dantzig-Wolfe column generation and the resulting branch-and-price algorithms. This is followed by an examination of Benders' type algorithms based on pro jection. Finally we discuss in detail extended formulations involving additional variables that are based on problem structure. These can often be used to provide strengthened a priori formulations. Reformulations obtained by adding cutting planes in the original variables are not treated here.

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Lagrangean decomposition for integer programming : theory and applications

Cited by 47 publications

References 15 publications

A geometric study of duality gaps, with applications

A geometric study of duality gaps, with applications

Hub Location Problems: The Location of Interacting Facilities

Reformulation and Decomposition of Integer Programs

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