On the Easiest and Hardest Fitness Functions

Abstract: The hardness of fitness functions is an important research topic in the field of evolutionary computation. In theory, the study can help understanding the ability of evolutionary algorithms. In practice, the study may provide a guideline to the design of benchmarks. The aim of this paper is to answer the following research questions: Given a fitness function class, which functions are the easiest with respect to an evolutionary algorithm? Which are the hardest? How are these functions constructed? The paper provides theoretical answers to these questions. The easiest and hardest fitness functions are constructed for an elitist (1+1) evolutionary algorithm to maximise a class of fitness functions with the same optima. It is demonstrated that the unimodal functions are the easiest and deceptive functions are the hardest in terms of the time-fitness landscape. The paper also reveals that the easiest fitness function to one algorithm may become the hardest to another algorithm, and vice versa.