Hybrid Variable Neighbourhood Decomposition Search for 0-1 Mixed Integer Programming Problem

doi:10.1016/j.endm.2010.05.112

Electronic Notes in Discrete Mathematics

Volume 36, 1 August 2010, Pages 883-890

https://doi.org/10.1016/j.endm.2010.05.112 Get rights and content

Abstract

In this paper we propose new hybrid heuristics for the 0-1 mixed integer programming problem, based on the variable neighbourhood decomposition search principle and on exploiting information obtained from a series of relaxations. In the case of a maximization problem, we add iteratively pseudo-cuts to the problem in order to produce a sequence of lower and upper bounds of the problem, so that integrality gap is reduced. We validate our approaches on the well-known 0-1 multidimensional knapsack problem, in which the general-purpose CPLEX MIP solver is used as a black box for solving subproblems generated during the search process. The results obtained with these methods are comparable with the current state-of-the-art heuristics on a set of large scale instances.

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The last class of heuristics are those that neither require a feasible MIP solution as input nor finish their work after finding the first feasible solution. Prominent members of this group include OCTAhedral Neighbourhood Enumeration (OCTANE) [5], Pivot and Complement [1,7], Pivot and Shift [8], Tabu Search [38], Pivot-Cut-and-Dive [13] and various Matheuristics [29,30,36,45]. The main contributions of this paper are:
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Another stream of research deals with combining VNS with MILP solvers to tackle large-sized general 0–1 MILP instances. For instance, Lazić et al. [47] and Hanafi et al. [48] both hybridize VNDS with a commercial MIP solver to tackle large-sized 0–1 MILP instances. A method to determine an initial feasible solution for large-sized 0–1 MILP instances is proposed by Hanafi et al. [49].
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The results reported in the paper demonstrate the efficiency of the approach. Extended variants of this approach were proposed in Hanafi, Lazic, Mladenovic, Wilbaut, and Crévits (2010). A pseudo-cut is a linear inequality that excludes certain points from being feasible as solutions to the input problem.
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It also provides an efficient Skewed VNS (SVNS) heuristic, and these extensions of VNS were applied (by the researchers) to some classical optimization problems. Later, those researchers used the extensions of VNS to solve 0-1 mixed integer programming (MIP) problems with hybrid algorithms (see for example [27,16,40]). Combining VNS with other heuristic methods generates new variants of VNS.
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Hybrid Variable Neighbourhood Decomposition Search for 0-1 Mixed Integer Programming Problem

Abstract

European Journal of Operational Research

Computers & Operations Research

Computers & Operations Research

European Journal of Operational Research

European Journal of Operational Research

Exploring relaxation induced neighborhoods to improve MIP solutions

Math. Progr.