Abstract
In this paper we focus on linearly constrained black-box optimization problems. We consider to use the covariance matrix adaptation evolution strategy (CMA-ES) to solve linearly constrained problems. A goodness of the CMA-ES is its invariant properties: invariance to increasing transformation of the objective function and invariance to affine transformation of the search space coordinate. These peroperties make the CMA-ES a state-of-the-art search algorithm for ill-conditioned and nonseparable unconstrained problems. When it is applied to a constrained problem, however, it is coupled with a constraint handling method that often breaks the invariance properties that the CMA-ES exhibits. To fully exploit the performance of the CMA-ES in constrained problems, a constraint handling method needs to have all the following invariance: invariance to arbitrary element-wise increasing transformation of the objective and constraint functions, invariance to affine transformation of the search space, and invariance to redundant constraints. In this paper, we propose a novel linear constraint handling technique with the above-mentioned invariance properties for the CMA-ES. The proposed method virtually transforms a constrained problem into an unconstrained one by adaptive penalization. The penalized fitness is defined as a weighted sum of the ranking on the objective and the ranking on the constraint violations measured by the Mahalanobis distance between each candidate solution to its projection onto the boundary of the constraints. Experimental results show that the CMA-ES with the proposed constraint handling exhibits the above-mentioned invariance properties and performs similarly both on constrained problems and their unconstrained counterpart.
