Adaptive evolutionary algorithms with linear average convergence rate for optimization problems with Lipschitz continuous functions

Abstract

Evolutionary methods, such as genetic algorithm, evolutionary strategy and differential evolution, are widely employed in the optimization of complex problems across various domains. After convergence is achieved, analyzing the convergence rate becomes essential for assessing the efficiency of the optimization process. This work investigates strategies aimed at improving the average convergence rate in optimization problems characterized by Lipschitz continuous objective functions. Strategies for evolutionary algorithms are proposed, designed to promote a positive-adaptive mutation process and ensure a linear average convergence rate, thereby providing efficient solutions to complex optimization problems. Lower bounds for the average convergence rate are also derived, considering the Lipschitz constant of the objective function and the problem’s dimensionality. To validate the theoretical results, the proposed positive-adaptive mutation strategies are applied to Genetic Algorithm, Evolutionary Strategy, and Differential Evolution in solving various benchmark optimization problems and the practical Economic Dispatch Problem. In all tests, the proposed mutation strategy demonstrated a superior average convergence rate compared to the other strategies used in the comparisons. Furthermore, the results of the Wilcoxon test provide statistical evidence to confirm a significant difference at a significance level of 0,05. When comparing the global mean of the average convergence rates, our strategy achieved an improvement of at least 12% compared to other relevant adaptations from the literature.