A new proximity metric based on optimality conditions for single and multi-objective optimization: Method and validation

With the widespread application of Evolutionary Algorithms (EAs), their performance needs to be evaluated using more than the usual performance metrics. In the EA literature, various metrics assess the convergence ability of these algorithms. However, many of them require prior knowledge of the Pareto-optimal front. Recently, two Karush–Kuhn–Tucker Proximity Metrics (KKTPMs) have been introduced to measure convergence without needing prior knowledge. One relies on the Augmented Achievement Scalarization Function (AASF) method (AASF-KKTPM), and the other on Benson's method (B-KKTPM). However, both require specific parameters and reference points, making them computationally expensive. In this paper, we introduce a novel version of KKTPM applicable to single-, multi-, and many-objective optimization problems, utilizing the Penalty-based Boundary Intersection (PBI) method (PBI-KKTPM). Additionally, we introduce an approximate approach to reduce the computational burden of solving PBI-KKTPM optimization problems. Through extensive computational experiments across 23 case studies, our proposed metric demonstrates a significant reduction in computational cost, ranging from 20.68% to 60.03% compared to the computational overhead associated with the AASF-KKTPM metric, and from 16.48% to 61.15% compared to the computational overhead associated with the B-KKTPM metric. Noteworthy features of the proposed metric include its independence from knowledge of the true Pareto-optimal front and its applicability as a termination criterion for EAs. Another feature of the proposed metric is its ability to deal with black box problems very efficiently.