A Bayesian Approach to Constrained Multi-objective Optimization

This paper addresses the problem of derivative-free multi-objective optimization of real-valued functions under multiple inequality constraints. Both the objective and constraint functions are assumed to be smooth, nonlinear, expensive-to-evaluate functions. As a consequence, the number of evaluations that can be used to carry out the optimization is very limited. The method we propose to overcome this difficulty has its roots in the Bayesian and multi-objective optimization literatures. More specifically, we make use of an extended domination rule taking both constraints and objectives into account under a unified multi-objective framework and propose a generalization of the expected improvement sampling criterion adapted to the problem. A proof of concept on a constrained multi-objective optimization test problem is given as an illustration of the effectiveness of the method.