Abstract
The use of multi-objective evolutionary algorithms (MOEAs) that employ a set of convex weight vectors as search directions, as a reference set or as part of a quality indicator has been widely extended. However, a recent study indicates that these MOEAs do not perform very well when tackling multi-objective optimization problem (MOPs), having different Pareto front geometries. Hence, it is necessary to propose MOEAs whose good performance is not strongly depending on certain Pareto front shapes. In this paper, we propose a Pareto-front shape invariant MOEA that combines the individual effect of two indicator-based density estimators. We selected the weakly Pareto-compliant IGD\(^+\) indicator to promote convergence and the Riesz s-energy indicator that leads to uniformly distributed point sets for the large class of rectifiable d-dimensional manifolds. Our proposed approach, called CRI-EMOA, is compared with respect to MOEAs that adopt convex weight vectors (NSGA-III, MOEA/D and MOMBI2) as well as to MOEAs not using this set of vectors (\(\varDelta _p\)-MOEA and GDE-MOEA) on MOPs belonging to the test suites DTLZ, DTLZ\(^{-1}\), WFG and WFG\(^{-1}\). Our experimental results show that CRI-EMOA outperforms the considered MOEAs, regarding the hypervolume indicator and the Solow-Polasky indicator, on most of the test problems and that its performance does not depend on the Pareto front shape of the problems.
The first author acknowledges support from CONACyT and CINVESTAV-IPN to pursue graduate studies in Computer Science. The second author gratefully acknowledges support from CONACyT grant no. 2016-01-1920 (Investigación en Fronteras de la Ciencia 2016).
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Notes
- 1.
A unary indicator I is a function that assigns a real value to set of points \(\mathcal {A}=\{\varvec{a}^1, \dots , \varvec{a}^N\}\), where \(\varvec{a}^i \in \mathbb {R}^m\).
- 2.
Given \(\varvec{u}, \varvec{v} \in \mathbb {R}^m\), \(\varvec{u}\) Pareto dominates \(\varvec{v}\) (denoted as \(\varvec{u} \prec \varvec{v}\)) if and only if \(\forall i=1, \dots , m, u_i \le v_i\) and there exists at least an index \(j \in \{1, \dots , m\} \, : \, u_j < v_j\).
- 3.
Let \(\mathcal {A}\) and \(\mathcal {B}\) be two non-empty sets of m-dimensional vectors and let I be a unary indicator. I is Pareto-compliant if and only if \(\mathcal {A}\) dominates \(\mathcal {B}\) implies \(I(\mathcal {A}) > I(\mathcal {B})\) (assuming maximization of I).
- 4.
\(\beta \) is a standardized measure of dispersion that shows the extent of variability to the mean of the population.
- 5.
The source code of CRI-EMOA is available at http://computacion.cs.cinvestav.mx/~jfalcon/CRI-EMOA.html.
- 6.
We used the implementation available at: http://web.ntnu.edu.tw/~tcchiang/publications/nsga3cpp/nsga3cpp.htm.
- 7.
We used the implementation available at: http://dces.essex.ac.uk/staff/zhang/webofmoead.htm.
- 8.
We used the implementation available at http://computacion.cs.cinvestav.mx/~rhernandez/.
- 9.
The source code of \(\varDelta _p\)-MOEA and GDE-MOEA was provided by its author, Adriana Menchaca Méndez.
- 10.
The Solow-Polasky indicator requires a parameter \(\theta \) that was set to 10.
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Falcón-Cardona, J.G., Coello Coello, C.A., Emmerich, M. (2019). CRI-EMOA: A Pareto-Front Shape Invariant Evolutionary Multi-objective Algorithm. In: Deb, K., et al. Evolutionary Multi-Criterion Optimization. EMO 2019. Lecture Notes in Computer Science(), vol 11411. Springer, Cham. https://doi.org/10.1007/978-3-030-12598-1_25
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