Lagrangian Cardinality Cuts and Variable Fixing for Capacitated Network Design

Sellmann, Meinolf; Kliewer, Georg; Koberstein, Achim

doi:10.1007/3-540-45749-6_73

Cited by 19 publications

(18 citation statements)

References 7 publications

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“…Following the work in [8] where local reasoning was combined with strengthened lower bounds by adding cuts in a Lagrangian fashion, the idea of CP-based Lagrangian Relaxation was first formalized in [23] and later used in [24,22,25] for the automatic recording problem that consists in a conjunction of a Knapsack and a weighted stable set problem. While numerous practical applications of this procedure have shown that CP-based Lagrangian Relaxation can boost the practical performance of solution approaches (see [25,26,19] for examples), there are many outstanding theoretical questions: Why should one perform filtering for sub-optimal Lagrangian multipliers? What are the best penalties for filtering?…”

Section: Introductionmentioning

confidence: 99%

Theoretical Foundations of CP-Based Lagrangian Relaxation

Sellmann

2004

Principles and Practice of Constraint Programming – CP 2004

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Abstract. CP-based Lagrangian Relaxation allows us to reason on local substructures while maintaining a global view on an entire optimization problem. While the idea of cost-based filtering with respect to systematically changing objective functions has been around for more than three years now, so far some important observations have not been explained. In this paper, we prove a simple theorem that explains a variety of effects that are encountered in practice, the most counter-intuitive being the fact that suboptimal Lagrangian multipliers can have stronger filtering abilities than optimal ones.

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Section: Introductionmentioning

confidence: 99%

Theoretical Foundations of CP-Based Lagrangian Relaxation

Sellmann

2004

Principles and Practice of Constraint Programming – CP 2004

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“…In [HY00], a Lagrangianrelaxation based branch-and-bound algorithm is presented. In [SKK02a] we have presented Lagrangian cardinality cuts and coupled variable fixing algorithms in a similar algorithmic approach. Recently, several valid inequalities for the network design problem were discussed in [CCG03].…”

Section: Related Workmentioning

confidence: 98%

“…The description of these system components lies beyond the scope of this paper. Details can be found in [SKK02a,SKK02b]. The focus here is on the relax-and-cut algorithm and the implemented valid inequalities.…”

Section: System Architecturementioning

confidence: 99%

Relax-and-Cut for Capacitated Network Design

Kliewer

Timajev

2005

Algorithms – ESA 2005

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Abstract. We present an evaluation of a Lagrangean-based branch-andbound algorithm with additional valid inequalities for the capacitated network design problem. The focus is on two types of valid inequalities, the cover inequalities and local cuts. We show how these inequalities can be considered in a Lagrangean relaxation without destroying the computationally simple structure of the subproblems. We present an extensive computational study on a large set of benchmark data. The results show that the presented algorithm outperforms many other exact and heuristical solvers in terms of running time and solution quality.

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“…This idea is also used successfully in Holmberg and Yuan (2000), Sellmann et al (2002) and Gomes and Sellmann (2004).…”

Section: Remarkmentioning

confidence: 99%

Alternating control tree search for knapsack/covering problems

et al. 2008

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The Multidimensional Knapsack/Covering Problem (KCP) is a 0-1 Integer Programming Problem containing both knapsack and weighted covering constraints, subsuming the well-known Multidimensional Knapsack Problem (MKP) and the Generalized (weighted) Covering Problem. We propose an Alternating Control Tree Search (ACT) method for these problems that iteratively transfers control between the following three components: (1) ACT-1, a process that solves an LP relaxation of the current form of the KCP. (2) ACT-2, a method that partitions the variables according to 0, 1, and fractional values to create sub-problems that can be solved with relatively high efficiency. (3) ACT-3, an updating procedure that adjoins inequalities to produce successively more constrained versions of KCP, and in conjunction with the solution processes of ACT-1 and ACT-2, ensures finite convergence to optimality. The ACT method can also be used as a heuristic approach using early termination rules. Computational results show that the ACT-framework successfully enhances the performance of three widely different heuristics for the KCP. Our ACT-method involving scatter search performs better than any other known method on a large set of KCP-instances from the literature. The ACT-based methods are also found to be highly effective on the MKP.

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Lagrangian Cardinality Cuts and Variable Fixing for Capacitated Network Design

Cited by 19 publications

References 7 publications

Theoretical Foundations of CP-Based Lagrangian Relaxation

Theoretical Foundations of CP-Based Lagrangian Relaxation

Relax-and-Cut for Capacitated Network Design

Alternating control tree search for knapsack/covering problems

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