How can surrogates influence the convergence of evolutionary algorithms?

Surrogate-assisted evolutionary algorithms have been widely utilized in science and engineering fields, while rare theoretical results were reported on how surrogates influence the performances of evolutionary algorithms (EAs). This paper focuses on theoretical analysis of a (1+1) surrogate-assisted evolutionary algorithm ((1+1)SAEA), which consists of one individual and pre-evaluates a newly generated candidate using a first-order polynomial model (FOPM) before it is precisely evaluated at each generation. By performing comparisons between a unimodal problem and a multi-modal problem, we rigorously estimate the variation of exploitation ability and exploration ability introduced via the FOPM. Theoretical results show that the FOPM employed to pre-evaluate the candidates sometimes accelerate the convergence of evolutionary algorithms, while sometimes prevents the individuals from converging to the global optimal solution. Thus, appropriate adaptive strategies of candidate generation and surrogate control are needed to accelerate the convergence of the (1+1)EA. Then, the accelerating effect of FOPM decreases monotonically with p, the probability of performing precise function evaluation when a candidate is pre-evaluated worse than the present individual.