Empirical Evaluation of No Free Lunch Violations in Permutation-Based Optimization

A new research paper by Grzegorz Sroka, titled 'Empirical Evaluation of No Free Lunch Violations in Permutation-Based Optimization,' provides empirical evidence that the foundational No Free Lunch (NFL) theorem for optimization does not hold in practical benchmarking scenarios. The NFL theorem states that, when averaging over all possible problems, all optimization algorithms perform equally. Sroka's work investigates when this theoretical averaging ceases to reflect real-world benchmarking results, focusing on iterative search with sampling without replacement. The core finding is that the way a benchmark is constructed—specifically through algebraic reformulations like sums and differences of base functions—can induce statistically meaningful and structured departures from NFL predictions.

Technically, the study constructed benchmarks by algebraically recombining binary objective functions and analyzed function-algorithm relations using correlation structures, hierarchical clustering, and PCA. A one-way ANOVA with Tukey contrasts confirmed that these reformulations cause significant shifts in performance patterns. While a uniformly sampled baseline aligned with NFL symmetry, the modified benchmarks produced stable algorithm re-rankings and coherent clusters. Monte Carlo experiments indicated these 'order effects' persist in larger search spaces and depend on the function class. The results fundamentally challenge the intuition that 'no algorithm is best overall' in practice, showing that benchmark design itself creates a local structure where some algorithms consistently outperform others. This has direct implications for evolutionary computation, statistical procedures based on permutation tests, and motivates more nuanced, representation-aware algorithm selection.