Adaptive Entropy-Feedback Optimization: A Diversity-Regulated Metaheuristic with Bias-Controlled Evaluation for Global and Engineering Design Optimization

Population-based metaheuristics balance exploration and exploitation, yet in most algorithms this balance follows a predetermined schedule that is blind to the actual state of the search. This paper introduces the Adaptive Entropy-Feedback Optimizer (AEFO), a metaheuristic in which the exploration–exploitation trade-off is regulated in closed loop by a measured, normalized Shannon entropy of the population, coupled to an elite-guided Lévy differential exploration operator, a per-dimension stochastic over-relaxation exploitation operator with an expected coordinate-wise contraction rate of 1/2 (proved under explicit assumptions), and a stagnation response combining coordinate-wise refinement with opposition-based reinjection. By construction, none of its per-iteration search operators references a fixed point of the search domain, so AEFO avoids the structural center bias documented in several recent metaheuristics. AEFO is evaluated under a strictly function-evaluation-equal protocol against seven baselines spanning canonical (PSO, DE), recent swarm (GWO, WOA), adaptive differential evolution (JADE, L-SHADE), and model-based (CMA-ES) families, on the classical 30-dimensional benchmark suite, a bias-controlled suite with shifted optima (including a multi-shift generalization study with boundary shifts), the CEC2022 suite, scalability tests at D=50 and D=100, and four constrained engineering design problems under two constraint-handling strategies, with Holm-corrected Wilcoxon tests, Cliff’s delta effect sizes, Friedman/Iman–Davenport analysis, and critical-difference diagrams over paired runs. The results delineate AEFO’s position precisely: it significantly outperforms PSO, DE, GWO, and WOA on the bias-controlled suite (Wilcoxon win/tie/loss of 6/0/0, 4/2/0, 6/0/0, and 6/0/0, respectively), while GWO and WOA collapse by many orders of magnitude when optima are shifted, confirmed by center-distance and error-direction diagnostics and by five independent random shifts plus near-boundary shifts per function, and it remains competitive with, but is on average ranked behind, the strong adaptive baselines JADE, L-SHADE, and CMA-ES. On the engineering problems, AEFO’s best designs are within 0.07% of the best-known values for the spring, welded beam, cantilever beam, and discrete-thickness pressure vessel, and within 0.47% for the continuous pressure vessel; every reported AEFO design is strictly feasible after independent 50-digit re-evaluation, although AEFO’s median results remain behind those of the adaptive differential-evolution baselines. A full architectural ablation attributes the largest single contribution to the per-dimension over-relaxation operator and shows that the benefit of measured-entropy control over open-loop scheduling is landscape-dependent. A cost analysis shows that the entropy sensor accounts for about 30% of AEFO’s internal CPU time and that AEFO’s per-evaluation overhead is of the same order as that of differential evolution and more than ten times lower than that of CMA-ES.