Non Delayed Relax-and-Cut Algorithms

Lucena, Abílio

doi:10.1007/s10479-005-3977-1

Cited by 33 publications

(38 citation statements)

References 39 publications

(59 reference statements)

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“…In addition, we do not add new entries to the Lagrangian vector u k every time new cuts are added to the pool, but only every 50 subgradient iterations, so as to let the subgradient method stabilize somehow before dealing with new Lagrangian components. In this view, our implementation is between the socalled delayed and non-delayed relax-and-cut methods [27].…”

Section: Subgradient Optimizationmentioning

confidence: 99%

“…It was introduced independently by Lucena [26] and by Escudero, Guignard and Malik [19]-who actually proposed the relaxand-cut name; see Lucena [27] for a survey of the technique and of its applications. Very recently, Lucena [28] applied a relax-and-cut approach to the solution of hard single 0-1 knapsack problems, where fractional Gomory cuts were used, for the first time, in a Lagrangian framework.…”

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confidence: 99%

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A relax-and-cut framework for Gomory mixed-integer cuts

Fischetti

Salvagnin

2011

Math. Prog. Comp.

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Gomory Mixed-Integer Cuts (GMICs) are widely used in modern branch-and-cut codes for the solution of Mixed-Integer Programs. Typically, GMICs are iteratively generated from the optimal basis of the current Linear Programming (LP) relaxation, and immediately added to the LP before the next round of cuts is generated. Unfortunately, this approach is prone to instability. In this paper we analyze a different scheme for the generation of rank-1 GMICs read from a basis of the original LP-the one before the addition of any cut. We adopt a relax-and-cut approach where the generated GMICs are not added to the current LP, but immediately relaxed in a Lagrangian fashion. Various elaborations of the basic idea are presented, that lead to very fastyet accurate-variants of the basic scheme. Very encouraging computational results are presented, with a comparison with alternative techniques from the literature also aimed at improving the GMIC quality. We also show how our method can be integrated with other cut generators, and successfully used in a cut-and-branch enumerative framework.

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Section: Subgradient Optimizationmentioning

confidence: 99%

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confidence: 99%

A relax-and-cut framework for Gomory mixed-integer cuts

Fischetti

Salvagnin

2011

Math. Prog. Comp.

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“…Following Lucena (2004), we use the term Relax-and-Cut to refer not only to the algorithm in Escudero et al (1994) but to a much larger class of algorithms which includes that one. According to that guideline, a Relax-and-Cut algorithm must be Lagrangian based.…”

Section: Introductionmentioning

confidence: 99%

“…Following the nomenclature introduced in Lucena (2004), Delayed Relax-and-Cut (DRC) algorithms (Lucena, 2004) then proceed by solving the corresponding Lagrangian Dual Problem. Valid constraints which violate the solution to that problem are then identified and may be either dualized or else kept.…”

Section: Introductionmentioning

confidence: 99%

“…Differently from DRC, violated inequalities are attempted to be identified (and are dualized, in case of success) for every Lagrangian Relaxation Problem eventually solved. That variant of Relax-and-Cut is called Non Delayed Relax-and-Cut (NDRC) in Lucena (2004). NDRC was first proposed and successfully used for the Steiner Problem in Graphs in Lucena (1992;.…”

Section: Introductionmentioning

confidence: 99%

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Lagrangian Relax-and-Cut Algorithms

Lucena

Handbook of Optimization in Telecommunications

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Attempts to allow exponentially many inequalities to be candidates to Lagrangian dualization date from the early 1980s. In the literature, the term Relax-and-Cut is being used to denote the whole class of Lagrangian Relaxation algorithms where Lagrangian bounds are attempted to be improved by dynamically strengthening relaxations with the introduction of valid constraints (possibly selected from families with exponentially many constraints). In this chapter, Relax-and-Cut algorithms are reviewed in their two flavors. Additionally, a general framework to obtain feasible integral solutions (that benefit from Lagrangian bounds) is also presented. Finally, the use of Relax-andCut is demonstrated through an application to a hard-to-solve instance of the Knapsack Problem. For that application, Gomory cuts are used, for the first time, within a Lagrangian relaxation framework.

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Decomposition Methods for Integer Programming

Ralphs

Galati

2011

Wiley Encyclopedia of Operations Research and Management Science

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This article reviews both traditional and integrated decomposition methods for solving mixed‐integer linear programs. These methods attempt to exploit tractable substructures of the problem in order to obtain improved solution procedures. The goal is to derive improved methods of bounding the optimal solution value, which can then be used to drive a branch‐and‐bound algorithm. Such methods are the preferred solution approaches for a wide range of important models. To expose the desired substructure, a common approach is to relax a set of complicating constraints. This is the approach taken by the Dantzig–Wolfe decomposition, Lagrangian relaxation, and cutting‐plane methods. Substructure can also be exposed by relaxing the values of a set of variables, that is, considering restrictions of the original problem. This is the approach taken by Benders' decomposition. This article reviews decomposition methodologies based on relaxation of constraints and examines how they are used to solve mixed‐integer linear programs.

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Non Delayed Relax-and-Cut Algorithms

Cited by 33 publications

References 39 publications

A relax-and-cut framework for Gomory mixed-integer cuts

A relax-and-cut framework for Gomory mixed-integer cuts

Lagrangian Relax-and-Cut Algorithms

Decomposition Methods for Integer Programming

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