Abstract
Reference sets generated with uniformly distributed weight vectors on a unit simplex are widely used by several multi-objective evolutionary algorithms (MOEAs). They have been employed to tackle multi-objective optimization problems (MOPs) with four or more objective functions, i.e., the so-called many-objective optimization problems. These MOEAs have shown a good performance on MOPs with regular Pareto front shapes, i.e., simplex-like shapes. However, it has been observed that in many cases, their performance degrades on MOPs with irregular Pareto front shapes. In this paper, we designed a new selection mechanism that aims to promote a Pareto front shape invariant performance of MOEAs that use weight vector-based reference sets. The newly proposed selection mechanism takes advantage of weight vector-based reference sets and seven pair-potential functions. It was embedded into the non-dominated sorting genetic algorithm III (NSGA-III) to increase its performance on MOPs with different Pareto front geometries. We use the DTLZ and DTLZ\(^{-1}\) test problems to perform an empirical study about the usage of these pair-potential functions for this selection mechanism. Our experimental results show that the pair-potential functions can enhance the distribution of solutions obtained by weight vector-based MOEAs on MOPs with irregular Pareto front shapes. Also, the proposed selection mechanism permits maintaining the good performance of these MOEAs on MOPs with regular Pareto front shapes.
The first author acknowledges support given by Tecnológico de Monterrey and Consejo Nacional de Ciencia y Tecnología (CONACYT) to pursue graduate studies under the number CVU 859751.
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Notes
- 1.
Let \(\vec {x},\vec {y} \in \mathbb {R}^m\). It is said that \(\vec {x}\) Pareto-dominates \(\vec {y}\) (denoted as \(\vec {x} \prec \vec {y}\)) if \(f_{i}(\vec {x}) \le f_{i}(\vec {y}), \forall i \in \{ 1, 2, \dots , m \}\) and \(\exists j \in \{ 1, 2, \dots , m \}\) such that \(f_{j}(\vec {x}) < f_{j}(\vec {y})\).
- 2.
It is said that a vector \(\vec w \in \mathbb {R}^{m}\) is a convex weight vector if \(\sum _{i=1}^{m}w_{i}=1\) and \(w_{i} \ge 0, \forall i \in \{ 1, 2, \dots , m \}\).
- 3.
An approximation set \(\mathcal {A}\) is a set of objective vectors that aims to approximate the Pareto front such that \(\forall \vec {a_{i}}, \vec {a_{j}} \in \mathcal {A} \mid \vec {a_{i}} \nprec \vec {a_{j}}\) and \(\vec {a_{j}} \nprec \vec {a_{i}}\).
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Márquez-Vega, L.A., Falcón-Cardona, J.G., Covantes Osuna, E. (2021). Towards a Pareto Front Shape Invariant Multi-Objective Evolutionary Algorithm Using Pair-Potential Functions. In: Batyrshin, I., Gelbukh, A., Sidorov, G. (eds) Advances in Computational Intelligence. MICAI 2021. Lecture Notes in Computer Science(), vol 13067. Springer, Cham. https://doi.org/10.1007/978-3-030-89817-5_28
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