Quantile normalization of optimization functions

This study investigates two critical methodological paradoxes inherent in benchmarking nature-inspired optimization algorithms: (1) the isomorphism paradox, where performance evaluations are biased by reliance on raw objective function values, and (2) the independence of irrelevant alternatives (IIA) paradox, wherein algorithm rankings are inconsistently altered by the inclusion or exclusion of competitors in benchmark suites. We demonstrate that these paradoxes constitute a foundational dilemma: value-based evaluation strategies risk isomorphism bias, while rank-based methods are prone to IIA-driven instability.To address this challenge, we propose a quantile normalization framework grounded in dual analytical perspectives that transform raw objective values into their cumulative distribution function (CDF)-equivalent ranks. This method resolves isomorphism bias by decoupling performance assessment from absolute function scales and ensures rank consistency across benchmark configurations, thereby mitigating IIA inconsistencies.We further formalize the isomorphism paradox through mathematical analysis, highlight the broader implications of this approach, underscoring its foundational role in enabling robust, scale-invariant, and rank-stable benchmarking practices. The framework’s applicability extends to performance evaluation across diverse optimization domains, offering a principled foundation for future algorithmic comparisons.