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Boosting local search with Lagrangian relaxation

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Abstract

Local search algorithms play an essential role in solving large-scale combinatorial optimization problems. Traditionally, the local search procedure is guided mainly by the objective function of the problem. Hence, the greedy improvement paradigm poses the potential threat of prematurely getting trapped in low quality attraction basins. In this study, we intend to utilize the information extracted from the relaxed problem, to enhance the performance of the local search process. Considering the Lin-Kernighan-based local search (LK-search) for the p-median problem as a case study, we propose the Lagrangian relaxation Assisted Neighborhood Search (LANS). In the proposed algorithm, two new mechanisms, namely the neighborhood reduction and the redundancy detection, are developed. The two mechanisms exploit the information gathered from the relaxed problem, to avoid the search from prematurely targeting low quality directions, and to cut off the non-promising searching procedure, respectively. Extensive numerical results over the benchmark instances demonstrate that LANS performs favorably to LK-search, which is among the state-of-the-art local search algorithms for the p-median problem. Furthermore, by embedding LANS into other heuristics, the best known upper bounds over several benchmark instances could be updated. Besides, run-time distribution analysis is also employed to investigate the reason why LANS works. The findings of this study confirm that the idea of improving local search by leveraging the information induced from relaxed problem is feasible and practical, and might be generalized to a broad class of combinatorial optimization problems.

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Notes

  1. Sometimes the algorithms that combine local search with stochastic perturbations are also loosely denoted as local search. However, in this study, we only refer to local search as those algorithms that traverse neighborhoods deterministically.

  2. In this study, we are mostly interested in the local search procedure, hence, for the constructive heuristic, we simply employ the random initialization (i.e., \(p\) out of \(m\) facilities are selected as medians uniformly at random).

  3. The functionality of the subroutine findBestNeighbor in Line 10 is to find the best pair of variables that leads to the best profit from the candidate sets. In this study, this subroutine is partially based on the implementation of Resende and Werneck (2003).

  4. Note that the variant presented in this paper is slightly different from the original version by Kochetov et al. (2005), in that during the preliminary experiments, we observe that the version presented in Algorithm 1 is generally more effective.

  5. More comprehensive numerical results could be found at http://oscar-lab.org/people/~zren/lans.

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Acknowledgments

The authors would like to thank the anonymous reviewers for their insightful comments and suggestions. This work was supported in part by the Fundamental Research Funds for the Central Universities under Grant DUT13RC(3)53, in part by the New Century Excellent Talents in University under Grant NCET-13-0073, in part by China Postdoctoral Science Foundation under Grant 2014M551083, in part by National Program on Key Basic Research Project under Grant 2013CB035906, and in part by the National Natural Science Foundation of China under Grant 61175062 and Grant 61370144.

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  1. School of Software, Dalian University of Technology, Dalian, China

    Zhilei Ren, He Jiang, Shuwei Zhang, Jingxuan Zhang & Zhongxuan Luo

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  1. Zhilei Ren
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Correspondence to He Jiang.

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Ren, Z., Jiang, H., Zhang, S. et al. Boosting local search with Lagrangian relaxation. J Heuristics 20, 589–615 (2014). https://doi.org/10.1007/s10732-014-9255-0

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