A rank-conditioned α-stable perturbation for differential evolution: Separating perturbation frequency and tail heaviness.

Heavy-tailed perturbations are commonly used in differential evolution (DE) to restore diversity and mitigate premature convergence. Adaptive Cauchy perturbation refines this strategy by varying perturbation probabilities according to dimension-wise convergence, but it retains a fixed Cauchy law and cannot adjust perturbation severity across individuals. This paper proposes rank-conditioned α-stable perturbation, a two-level adaptive operator that decouples perturbation activation from tail-shape control. Dimension-wise convergence indicators determine coordinate-specific perturbation probabilities, while each individual's fitness rank determines the stability index of a symmetric α-stable kernel. Thus, higher-ranked individuals receive lighter-tailed perturbations for local refinement, whereas lower-ranked individuals receive heavier-tailed perturbations for occasional long-range exploration. Unlike fixed-kernel heavy-tailed DE operators, the proposed operator induces a per-individual exploration-exploitation trade-off within each generation. It is embedded into L-SRTDE, the winner of the IEEE CEC 2024 competition, and evaluated on the IEEE CEC 2017 benchmark suite across multiple dimensions. Results show the best Friedman rankings among the compared methods and favorable Wilcoxon outcomes against recent DE variants. Controlled comparisons with conventional Cauchy and adaptive dimension-wise Cauchy perturbations show that rank-guided tail-shape adaptation complements probability-based activation, with modest overhead and stable performance across α-stable parameter settings.