Covariance Matrix Adaptation Evolution Strategy for Constrained Optimization Problem
Keiichirou Hoshimura · 2007
Recently engineers in many fields have faced solving complicated optimization problems. The objective functions of such problems are often nondifferentiable, or even if differentiable, their derivatives may not be calculated explicitly. Moreover, the problems are nonconvex in general, and hence, it is difficult to find the global optima. In order to overcome such difficulties, Hansen proposed the Covariance Matrix Adaptation Evolution Strategy (CMA-ES), which is an evolutionary algorithm generating a number of search points by using normal distribution. CMA-ES finds a global (or better local) minimum without using derivatives of the objective functions. However it is applicable only to the unconstrained problem. In this paper we propose three CMA-ES type methods for the constrained optimization problem. These methods are based on the l1-penalty method, and the differences of the methods are generating mechanism of search points. The first method generates search points by standard normal distribution. We note that the method is unapplicable to the problems whose objective functions are not defined out of the feasible region since the method sometimes generates a infeasible points. The second method generates search points by using lognormal distribution and the third method uses the projection onto the feasible region. Therefore the second and third methods always generate search points in the feasible region. We compared these three methods by solving standard test problems. According to the results, the method based on the normal distribution is superior to the other methods for most problems. On the other hand, the method based on the projection showed better performance when many inequality constraints are active at a solution.