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A nonmonotone GRASP

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Abstract

A greedy randomized adaptive search procedure (GRASP) is an iterative multistart metaheuristic for difficult combinatorial optimization problems. Each GRASP iteration consists of two phases: a construction phase, in which a feasible solution is produced, and a local search phase, in which a local optimum in the neighborhood of the constructed solution is sought. Repeated applications of the construction procedure yields different starting solutions for the local search and the best overall solution is kept as the result. The GRASP local search applies iterative improvement until a locally optimal solution is found. During this phase, starting from the current solution an improving neighbor solution is accepted and considered as the new current solution. In this paper, we propose a variant of the GRASP framework that uses a new “nonmonotone” strategy to explore the neighborhood of the current solution. We formally state the convergence of the nonmonotone local search to a locally optimal solution and illustrate the effectiveness of the resulting Nonmonotone GRASP on three classical hard combinatorial optimization problems: the maximum cut problem (MAX-CUT), the weighted maximum satisfiability problem (MAX-SAT), and the quadratic assignment problem (QAP).

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Notes

  1. The QAPLIB - A Quadratic Assignment Problem Library has an online version at http://anjos.mgi.polymtl.ca/qaplib/.

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Authors and Affiliations

  1. Fakultät für Mathematik, TU Dortmund, Dortmund, Germany

    M. De Santis

  2. Department of Mathematics and Applications, University of Napoli FEDERICO II, Napoli, Italy

    P. Festa

  3. Istituto di Analisi dei Sistemi ed Informatica-CNR, Rome, Italy

    G. Liuzzi

  4. Dipartimento di Ingegneria Informatica Automatica e Gestionale, “La Sapienza” University of Rome, Rome, Italy

    S. Lucidi

  5. Dipartimento di Matematica, University of Padova, Padua, Italy

    F. Rinaldi

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  1. M. De Santis
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  2. P. Festa
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  3. G. Liuzzi
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  4. S. Lucidi
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  5. F. Rinaldi
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Corresponding author

Correspondence to P. Festa.

Appendices

Appendix 1: Detailed tables for MAX-CUT, MAX-SAT and QAP

In this appendix, we report the detailed tables related to the comparison between NM-GRASP and the original version of GRASP for MAX-CUT, MAX-SAT and QAP. The first column of the three tables reports the name of the instance. The remaining columns report for each of the two approaches the average CPU time (Time), the average number of iterations (Iter), the average objective function value (Obj) with the standard deviation in brackets, and the best objective function value obtained over the ten runs (Best Obj) with the number of times the best value is obtained in brackets.

Appendix 2: Time to target-plots analysis on MAX-CUT problems

To plot the empirical distribution, we associate with the i-th sorted running time \((t_i)\) a probability \(p_i = (i - \frac{1}{2} )/100\), and plot the points \(z_i = (t_i , p_i )\), for \(i = 1,\ldots ,100\). For the instances g1250.n, G40, sg3dl142000.mc, and toruspm3-15-50 we fixed as target values 2518, 2275, 2379, and 2925, respectively. These values represent a standard target for both heuristics. As we can see in Fig. 9, apart from the instance toruspm3-15-50 where for 3 runs the classical GRASP is better, we can notice that the NM-GRASP is always superior. It is able to reach the target value in less than 100 s CPU time for all the runs, while in several runs the classical GRASP needs more than 1000 s.

Fig. 9
figure 9

TTTplots for the easy targets

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Fig. 10
figure 10

TTTplots for the classical GRASP targets

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Fig. 11
figure 11

TTTplots for the Nonmonotone GRASP targets

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Figure 10 depicts the empirical distributions of the random variable time-to-target-solution-value using as target values 2532, 2293, 2382, and 2932, for the instances g1250.n, G40, sg3dl142000.mc, and toruspm3-15-50, respectively. These values are the best objective function values found by the classical GRASP over 10 runs. As we can see from the plots, also in this case, the NM-GRASP is able to reach the target value in less than 100 s for all the runs. On the other hand, the classical GRASP failed to reach the target solution within the time limit in several runs, especially for instances g1250.n and G40.

By using instances g1250.n, G40, sg3dl142000.mc, and toruspm3-15-50, we plot in Fig. 11 the empirical distributions of the random variable time-to-target-solution-value using as target values 2556, 2362, 2420, and 2980, respectively. These target values are the best cuts found by the NM-GRASP over 10 runs. In this case, the classical GRASP failed to reach the target solution within the time limit for all runs and all instances. On the contrary, the NM-GRASP is able to reach the target solution for all runs for instances g1250.n and sg3dl142000.mc.

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De Santis, M., Festa, P., Liuzzi, G. et al. A nonmonotone GRASP. Math. Prog. Comp. 8, 271–309 (2016). https://doi.org/10.1007/s12532-016-0107-9

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