To ensure empirical evaluation of multi- and many-objective evolutionary algorithms, researchers perform benchmarking across test problems and algorithms. Due to the volume of performance data and the heterogeneity of problem characteristics, analyzing results becomes complex and prone to misinterpretation. Performance profiles have proven effective for visualizing and interpreting such results; however, they do not account for the relative difficulty or importance of individual problems and may overweight easy or less informative cases, potentially obscuring distinctions between algorithm performance. In this work, we address this limitation by extending the classical performance profile approach with a difficulty-aware weighting scheme that emphasizes more challenging problems. Weights can be assigned either a priori, based on problem characteristics such as the number of objectives or decision variables, or a posteriori, based on computational effort. We define and prove key mathematical properties of classical performance profiles, including local and global stability, and show that these properties extend to the proposed weighted formulation. By employing a difficulty-aware weighting scheme, the approach biases aggregation toward higher-dimensional instances, enabling a more discriminative assessment of scalability, robustness, and performance. The advantages of the weighted approach are demonstrated through experiments with algorithms applied to problem sets with numbers of objectives.
Regina C. L. C. de Sousa, Dênis E. C. Vargas, Elizabeth F. Wanner et al.· Annual Conference on Genetic...· 0 citations
We address the problem of achieving convergence and diversity in many-objective problems, focusing on continuous and unconstrained functions. It is known that with increasing numbers of objectives (say from 4 to 20) even modern many-objective Evolutionary Algorithms (EAs) may struggle to converge to, and fully distribute across the Pareto front. This paper presents a general and modular hybrid approach that integrates local search into reference-vector-based Many-Objective Evolutionary Algorithms (MaOEAs), addressing issues such as weakened selection pressure and the increasing complexity of exploring high-dimensional objective spaces. The hybrid approach employs Sequential Quadratic Programming (SQP) guided by achievement scalarizing directions, derived from either the Weighted Achievement Scalarizing Function (W-ASF) or the Penalty-based Boundary Intersection (PBI) schemes, depending on the decomposition strategy of the underlying MaOEA. It is designed to be broadly applicable with limited parameter tuning, facilitating integration with algorithms from the NSGA-III and MOEA-DD families. The effectiveness of the proposed approach is demonstrated through extensive experiments on standard continuous-variable many-objective benchmark problems as well as on representative real-world case studies. Results show that integrating local search significantly enhances performance, while a principled method for setting hybrid parameters ensures robustness and reproducibility. Although limited to an empirical study over a (large) test function suite, these findings highlight the potential of combining mathematical programming techniques with evolutionary algorithms for high-dimensional many-objective optimization problems.
Regina C. L. C. de Sousa, Dênis E. C. Vargas, Elizabeth F. Wanner et al.· Journal of Heuristics· 0 citations