Searching for a practical evidence of the No Free Lunch theorems
This provides incremental evidence for the No Free Lunch theorems by generating specific test cases, relevant for researchers in optimization and evolutionary algorithms.
The paper tackles the problem of finding test functions where Random Search outperforms other evolutionary algorithms, as a practical demonstration of the No Free Lunch theorems, and evolves such functions using genetic programming and binary representations, showing effectiveness in experiments.
According to the No Free Lunch (NFL) theorems all black-box algorithms perform equally well when compared over the entire set of optimization problems. An important problem related to NFL is finding a test problem for which a given algorithm is better than another given algorithm. Of high interest is finding a function for which Random Search is better than another standard evolutionary algorithm. In this paper, we propose an evolutionary approach for solving this problem: we will evolve test functions for which a given algorithm A is better than another given algorithm B. Two ways for representing the evolved functions are employed: as GP trees and as binary strings. Several numerical experiments involving NFL-style Evolutionary Algorithms for function optimization are performed. The results show the effectiveness of the proposed approach. Several test functions for which Random Search performs better than all other considered algorithms have been evolved.