On the Easiest and Hardest Fitness Functions

The hardness of fitness functions is an important research topic in the field of evolutionary computation. In theory, this paper can help with understanding the ability of evolutionary algorithms (EAs). In practice, this paper 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 EA? Which are the hardest? How are these functions constructed? This paper provides theoretical answers to these questions. The easiest and hardest fitness functions are constructed for an elitist (1 + 1) EA to maximize 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-based fitness landscape. This paper also reveals that in a fitness function class, the easiest function to one algorithm may become the hardest to another algorithm, and vice versa.