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An alternative hypervolume-based selection mechanism for multi-objective evolutionary algorithms

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Abstract

In this paper, we are interested in selection mechanisms based on the hypervolume indicator with a particular emphasis on the mechanism used in an improved version of the S metric selection evolutionary multi-objective algorithm (SMS-EMOA) called iSMS-EMOA, which exploits the locality property of the hypervolume. Here, we propose a new selection scheme which approximates the contribution of solutions to the hypervolume and it is designed to preserve the locality property exploited by iSMS-EMOA. This approach is proposed as an alternative to the use of exact hypervolume calculations and is aimed for solving many-objective optimization problems. The proposed approach is called “approximate version of the improved SMS-EMOA (aviSMS-EMOA)” and is validated using standard test problems (with three or more objectives) and performance indicators taken from the specialized literature. Our preliminary results indicate that our proposed approach is a good alternative to solve many-objective optimization problems, if we consider both quality in the solutions and running time required to obtain them because it outperforms two versions of the original SMS-EMOA that approximate the contributions to the hypervolume, it outperforms MOEA/D using penalty boundary intersection and it is competitive with respect to the original SMS-EMOA in several of the test problems adopted. Also, its computational cost is reasonable (it is slower than MOEA/D, but faster than SMS-EMOA).

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Notes

  1. \(I_H\) cannot be computed exactly in polynomial time in the number of objective functions unless \(P = NP\).

  2. An indicator \(I:\Omega \rightarrow \mathbb {R}\) is Pareto compliant if for all \(\mathcal {A},\mathcal {B} \subseteq \Omega : \mathcal {A} \preceq \mathcal {B} \Rightarrow I(\mathcal {A}) \ge I(\mathcal {B})\) assuming that greater indicator values correspond to higher quality, where \(\mathcal {A}\) and \(\mathcal {B}\) are approximations of the Pareto optimal set, \(\Omega \) is the feasible region and \(\mathcal {A} \preceq B\) means that every point \(\mathbf {b} \in \mathcal {B}\) is weakly dominated by at least one point \(\mathbf {a} \in \mathcal {A}\).

  3. Given a nondominated front of individuals, the hypervolume value for an individual i is equal to the product of the one-dimensional lengths to the next worse objective function value in front for each objective.

  4. MOEA/D is not based on \(I_H\), but in this case we also used the “two set coverage” indicator.

  5. It is important to clarify that aviSMS-EMOA is not 33 times faster than avoSMS-EMOA because avoSMS-EMOA does not always calculate 100 contributions to \(I_H\) (only when after applying Pareto ranking, a single front is obtained). Also, aviSMS-EMOA only uses the selection mechanism based on \(I_H\) and its locality property when, after applying Pareto ranking, we only obtain a single front. Otherwise, both algorithms use the number of solutions that dominate certain solution as suggested by Beume et al. in (2007).

  6. It is important to mention that our aim was to validate the selection mechanism. Therefore, we decided to use the same MOEA in all cases and only changed the selection mechanism. For this reason, we did not use the original HyPE.

  7. We decided to use PBI because the resulting optimal solutions with PBI are normally much better distributed than those obtained by the Tchebycheff approach (Zhang and Li 2007).

  8. We decided to use \(I_{\mathrm{IGD}}\) only in MOPs with three objective functions, because the results of the indicator depend on the reference set that we use and we know that generating a good reference set in MOPs with many objective functions is a difficult task.

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Acknowledgments

The first author acknowledges support from CONACyT for pursuing graduate studies in Computer Science at CINVESTAV-IPN. The second author acknowledges support from a 2014 Cátedra Marcos Moshinsky in Mathematics and from CONACyT Project No. 221551.

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  1. Departamento de Computación, CINVESTAV-IPN, Evolutionary Computation Group, Av. IPN 2508. San Pedro Zacatenco, 07300, Mexico, D.F., Mexico

    Adriana Menchaca-Mendez & Carlos A. Coello Coello

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  1. Adriana Menchaca-Mendez
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  2. Carlos A. Coello Coello
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Correspondence to Adriana Menchaca-Mendez.

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Communicated by V. Loia.

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Menchaca-Mendez, A., Coello Coello, C.A. An alternative hypervolume-based selection mechanism for multi-objective evolutionary algorithms. Soft Comput 21, 861–884 (2017). https://doi.org/10.1007/s00500-015-1819-x

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