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Landscape properties of the very large-scale and the variable neighborhood search metaheuristics for the multidimensional assignment problem

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Abstract

We study the recent metaheuristic search algorithm for the multidimensional assignment problem (MAP) using fitness landscape theory. The analyzed algorithm performs a very large-scale neighborhood search on a set of feasible solutions to the problem. We derive properties of the landscape graphs that represent these very large-scale search algorithms acting on the solutions of the MAP. In particular, we show that the search graph is a generalization of a hypercube. We extend and generalize the original very large-scale neighborhood search to develop the variable neighborhood search. The new search is capable of searching even larger large-scale neighborhoods. We perform numerical analyses of the search graph structures for various problem instances of the MAP and different neighborhood structures of the MAP algorithm based on a very large-scale search. We also investigate the correlation between fitness (i.e., objective values) and distance (i.e., path lengths) of the local minima (i.e., sinks of the landscape). Our results can be used to design improved search-based metaheuristics for the MAP.

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Data availability

Because of the size of the simulated data, it is neither practical nor necessary to share all data that we generated or analyzed during this study. The minimal dataset that would be necessary to interpret, and build upon the findings reported in the article consists of numerous tables and figures, all of which are included in this published article. For purpose of the replication, the authors will make the code used to simulate the data available via GitHub.

References

  1. Agaev, R., Chebotarev, P.: On the spectra of nonsymmetric Laplacian matrices. Linear Algebra Appl. 399, 157–168 (2005)

    Article  MathSciNet  Google Scholar 

  2. Agaev, R.P., Chebotarev, P.Y.: The matrix of maximal outgoing forests of a digraph and its applications. Avtom. Telemekhanika 9, 15–43 (2000)

    MathSciNet  Google Scholar 

  3. Altenberg, L.: Fitness landscapes: NK landscapes. In: Bäck, T, Fogel, D.B., Michalewicz, Z. (eds.) Handbook of Evolutionary Computation. pp 5–10, Oxford University Press (1997)

  4. Bosman, A.S., Engelbrecht, A., Helbig, M.: Visualising basins of attraction for the cross-entropy and the squared error neural network loss functions. Neurocomputing 400, 113–136 (2020)

    Article  Google Scholar 

  5. Burkard, R.E., Çela, E.: Linear assignment problems and extensions. In: Du, D-Z, Pardalos, P.M. (eds.) Handbook of Combinatorial Optimization: Supplement Volume A, 75–149 pp, Springer, US. https://doi.org/10.1007/978-1-4757-3023-4_2 (1999)

  6. Choromanska, A., Henaff, M., Mathieu, M., Ben Arous, G., LeCun, Y.: The loss surfaces of multilayer networks. In: Lebanon, G., Vishwanathan, S.V.N. Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics, 192–204 pp, vol. 38, Proceedings of Machine Learning Research, PMLR, San Diego, California, USA. http://proceedings.mlr.press/v38/choromanska15.pdf (2015)

  7. Choromanska, A., LeCun, Y., Arous, G.B.: Open problem: the landscape of the loss surfaces of multilayer networks. In: Conference on Learning Theory, pp. 1756–1760. PMLR (2015)

  8. Daolio, F., Tomassini, M., Vérel, S., Ochoa, G.: Communities of minima in local optima networks of combinatorial spaces. Phys. A 390(9), 1684–1694 (2011)

    Article  Google Scholar 

  9. Hansen, P., Mladenović, N., Moreno Pérez, J.A.: Variable neighbourhood search: methods and applications. Ann. Oper. Res. 175(1), 367–407 (2010)

    Article  MathSciNet  Google Scholar 

  10. Hordijk, W., Kauffman, S.A.: Correlation analysis of coupled fitness landscapes. Complexity 10(6), 41–49 (2005)

    Article  MathSciNet  Google Scholar 

  11. Jones, T., Forrest, S., et al.: Fitness distance correlation as a measure of problem difficulty for genetic algorithms. In: ICGA, vol. 95, pp. 184–192 (1995)

  12. Kammerdiner, A., Pasiliao, E.: Application of graph-theoretic approaches to the random landscapes of the three-dimensional assignment problem. Optim. Lett. 7(1), 79–87 (2013)

    Article  MathSciNet  Google Scholar 

  13. Kammerdiner, A., Pasiliao, E.: In and out forests on combinatorial landscapes. Eur. J. Oper. Res. 236(1), 78–84 (2014)

    Article  MathSciNet  Google Scholar 

  14. Kammerdiner, A., Semenov, A., Pasiliao, E.: Multidimensional assignment problem for multipartite entity resolution (2021)

  15. Kammerdiner, A., Vaughan, C.F.: Very large-scale neighborhood search for the multidimensional assignment problem. In: Butenko, S., Pardalos, P.M., Shylo, V. (eds.) Optimization Methods and Applications. Springer, Cham (2017)

    Google Scholar 

  16. Kammerdiner, A.R.: Multidimensional assignment problem. In: Floudas, C.A., Pardalos, P.M. (eds.) Encyclopedia of Optimization, 2396–2402 pp , Springer US, Boston, MA. https://doi.org/10.1007/978-0-387-74759-0_411 (2009)

  17. Karp, R.: Reducibility among combinatorial problems. Complex Comput Comput 85–104 (1972)

  18. Malan, K.M.: A survey of advances in landscape analysis for optimisation. Algorithms 14(2), 40 (2021)

    Article  MathSciNet  Google Scholar 

  19. Merz, P.: Advanced fitness landscape analysis and the performance of memetic algorithms. Evol. Comput. 12(3), 303–325 (2004)

    Article  MathSciNet  PubMed  Google Scholar 

  20. Ochoa, G., Qu, R., Burke, E.K.: Analyzing the landscape of a graph based hyper-heuristic for timetabling problems. In: Proceedings of the 11th Annual conference on Genetic and Evolutionary Computation, pp. 341–348 (2009)

  21. Pierskalla, W.P.: The tri-substitution method for the three-dimensional assignment problem. CORS J. 5, 71–81 (1967)

    Google Scholar 

  22. Pierskalla, W.P.: Letter to the editor-the multidimensional assignment problem. Oper. Res. 16(2), 422–431 (1968)

    Article  Google Scholar 

  23. Reidys, C.M., Stadler, P.F.: Combinatorial landscapes. SIAM Review 44(1), 3–54 (2002)

    Article  MathSciNet  Google Scholar 

  24. Richter, H.: Coupled map lattices as spatio-temporal fitness functions: Landscape measures and evolutionary optimization. Physica D 237(2), 167–186 (2008)

    Article  MathSciNet  Google Scholar 

  25. Schiavinotto, T., Stützle, T.: A review of metrics on permutations for search landscape analysis. Comput. Oper. Res. 34(10), 3143–3153 (2007)

    Article  Google Scholar 

  26. Silva, R.M., Resende, M.G., Pardalos, P.M.: Finding multiple roots of a box-constrained system of nonlinear equations with a biased random-key genetic algorithm. J. Global Optim. 60(2), 289–306 (2014)

    Article  MathSciNet  Google Scholar 

  27. Stadler, P.F., Schnabl, W.: The landscape of the traveling salesman problem. Phys. Lett. A 161(4), 337–344 (1992)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

A. Kammerdiner was supported by the AFRL (National Research Council Fellowship).

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Authors and Affiliations

  1. The University of Florida Research and Engineering Educational Facility (REEF), Shalimar, FL, USA

    Alexander Semenov

  2. Herbert Wertheim College of Engineering, University of Florida, Gainesville, FL, USA

    Alexander Semenov

  3. Munitions Directorate, Eglin Air Force Base, Eglin, FL, USA

    Alla Kammerdiner & Eduardo L. Pasiliao

Authors
  1. Alla Kammerdiner
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  2. Alexander Semenov
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  3. Eduardo L. Pasiliao
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Correspondence to Alla Kammerdiner.

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Appendix

Appendix

This section supplements the numerical results in Sect. 5.3. Below we present the results of pairwise comparison of three alternative traversal strategies (i.e., BFS, DFS, and BsFS) of the VNS and VLSN algorithms with Random Key GA. To statistically compare the pairs of algorithms, we construct 95% confidence intervals for the differences in the objective values as described in Sect. 5.1.

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Kammerdiner, A., Semenov, A. & Pasiliao, E.L. Landscape properties of the very large-scale and the variable neighborhood search metaheuristics for the multidimensional assignment problem. J Glob Optim 88, 653–683 (2024). https://doi.org/10.1007/s10898-023-01285-w

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