A Simple Evolution Strategy to Solve Constrained Optimization Problems

In this paper, we argue that the self-adaptation mechanism of a conventional evolution strategy combined with some (very simple) tournament rules based on feasibility similar to some previous proposals (e.g., [1]) can provide us with a highly competitive evolutionary algorithm for constrained optimization. In our proposal, however, no extra mechanisms are provided to maintain diversity. In order to verify our hypothesis, we performed a small comparative study among five different types of ES: (μ +, λ)-ES with and without correlated mutation and a (μ + 1)-ES using the “1/5-success rule”. The tournament rules adopted in the five types of ES implemented are the following: Between 2 feasible solutions, the one with the highest fitness value wins, if one solution is feasible and the other one is infeasible, the feasible solution wins and if both solutions are infeasible, the one with the lowest sum of constraint violation is preferred. To evaluate the performance of the five types of ES under study, we decided to use ten (out of 13) of the test functions described in [2]. The (μ+1)−ES had the best overall performance (both in terms of the best solution found and in terms of its statiscal measures). The algorithm of the type of ES adopted (due to its simplicity, we decided to call it Simple Evolution Strategy, or SES) is presented in Figure 1. Compared with respect to other state-of-the-art techniques (due to space limitations we only compare with respect to [2]), our algorithm produced very competitive results (See Table 1). Besides being a very simple approach, it is worth reminding that SES does not require any extra parameters (besides those used with an evolution strategy) and the number of fitness function evaluations performed (350,000) is the same used in [2].