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Transfer weight functions for injecting problem information in the multi-objective CMA-ES

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Abstract

The covariance matrix adaptation evolution strategy (CMA-ES) is one of the state-of-the-art evolutionary algorithms for optimization problems with continuous representation. It has been extensively applied to single-objective optimization problems, and different variants of CMA-ES have also been proposed for multi-objective optimization problems (MOPs). When applied to MOPs, the traditional steps of CMA-ES have to be modified to accommodate for multiple objectives. This fact is particularly evident when the number of objectives is higher than 3 and, with a high probability, all the solutions produced become non-dominated. An open question is to what extent information about the objective values of the non-dominated solutions can be injected in the CMA-ES model for a more effective search. In this paper, we investigate this general question using several metrics that describe the quality of the solutions already evaluated, different transfer weight functions, and a set of difficult benchmark instances including many-objective problems. We introduce a number of new strategies that modify how the probabilistic model is learned in CMA-ES. By conducting an exhaustive empirical analysis on two difficult benchmarks of many-objective functions we show that the proposed strategies to infuse information about the quality indicators into the learned models can achieve consistent improvements in the quality of the Pareto fronts obtained and enhance the convergence rate of the algorithm. Moreover, we conducted a comparison with a state-of-the-art algorithm from the literature, and achieved competitive results in problems with irregular Pareto fronts.

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Notes

  1. A newly generated solution is considered successful if the offspring is better than the parent according to the dominance ranks or a secondary measure, in the original paper they used both crowding distance and contributing hypervolume.

  2. In MOEA/D and MOEA/D-CMA a subproblem is considered a neighbor of itself, hence when updating the solutions of the neighbors it updates its own solution.

  3. An external archive or repository is used to store a predefined number of non-dominated solutions, when the archive is full and a new non-dominated solution is found, it is temporarily added and the solution that has the smallest crowding distance is removed.

  4. When a Pareto set approximation dominates another, the indicator value of the former will be greater than the latter.

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Authors and Affiliations

  1. Computer Science Department, Federal University of Paraná, Avenida Coronel Francisco Heráclito dos Santos, 210, Curitiba, 81531-970, Brazil

    Olacir R. Castro Jr. & Aurora Pozo

  2. Intelligent Systems Group, Department of Computer Science and Artificial Intelligence, University of the Basque Country (UPV/EHU), Paseo Manuel de Lardizábal 1, 20080, San Sebastián, Donostia, Spain

    Jose A. Lozano & Roberto Santana

Authors
  1. Olacir R. Castro Jr.
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  2. Aurora Pozo
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  3. Jose A. Lozano
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  4. Roberto Santana
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Correspondence to Olacir R. Castro Jr..

Additional information

This work was supported by CNPq, National Council for Scientific and Technological Development—Brazil (Productivity Grant Nos. 305986/2012-0 and Program Science Without Borders Nos. 200040/2015-4) and by IT-609-13 program (Basque Government) and TIN2013-41272P (Spanish Ministry of Science and Innovation).

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R. Castro, O., Pozo, A., Lozano, J.A. et al. Transfer weight functions for injecting problem information in the multi-objective CMA-ES. Memetic Comp. 9, 153–180 (2017). https://doi.org/10.1007/s12293-016-0202-5

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