Maximum Dispersion, Maximum Concentration: Enhancing the Quality of MOP Solutions

TL;DR

The paper addresses the quality of MOP solutions by coupling convergence within a predefined Region of Interest in the objective space with maximal dispersion in the decision space. It introduces the $\mathcal{C}$-DWU framework, which integrates a preference cone $\mathcal{C}$ into the Dominance-Weighted Uniformity approach and uses penalization terms $P_{\alpha,\theta}(\phi)$ and $P_{\beta,\theta}(\phi)$ to steer search toward ROI while preserving dispersion in the Pareto-Optimal set. Experiments on WFG4, WFG9, and DTLZ2 across varying decision-space dimensions demonstrate that ROI-focused convergence can be achieved while dispersion in the decision space is enhanced, mitigating bias from clustering. The work offers a practical, adjustable mechanism to guide MOEAs toward diverse, ROI-aligned Pareto sets for large-scale MOPs, with clear benefits for decision-makers seeking robust and actionable solution sets.

Abstract

Multi-objective optimization problems (MOPs) often require a trade-off between conflicting objectives, maximizing diversity and convergence in the objective space. This study presents an approach to improve the quality of MOP solutions by optimizing the dispersion in the decision space and the convergence in a specific region of the objective space. Our approach defines a Region of Interest (ROI) based on a cone representing the decision maker's preferences in the objective space, while enhancing the dispersion of solutions in the decision space using a uniformity measure. Combining solution concentration in the objective space with dispersion in the decision space intensifies the search for Pareto-optimal solutions while increasing solution diversity. When combined, these characteristics improve the quality of solutions and avoid the bias caused by clustering solutions in a specific region of the decision space. Preliminary experiments suggest that this method enhances multi-objective optimization by generating solutions that effectively balance dispersion and concentration, thereby mitigating bias in the decision space.

Figures (6)

• Figure 1: Preference cone $\mathcal{C}$ with its axis $\mathbf{v}$ and opening angle $\theta$, objective space $F(\Omega)$, Pareto Front ($PF$) and ROI in bi-dimensional space.
• Figure 2: Penalization of solutions located outside the preference cone $\mathcal{C}$
• Figure 3: Symbolic application of the proposed $\mathcal{C}\hbox{-}w_d$. An example to find a representative subset $R$, with $|R| = 4$, of a set $\mathcal{P}$, where $|\mathcal{P}| = 5$. Figure (a) shows the solution in the decision space not selected by $\mathcal{C}\hbox{-}w_d$ and $w_d$.
• Figure 4: Box plot of the IDG and uniformity measure value in 10 runs of the $\mathcal{C}$-NSGAII and $\mathcal{C}$-DWU algorithms.
• Figure 5: Solutions obtained in the objective space for the two-objective problem WFG4.