An interior point method for nonlinear constrained derivative-free optimization

In this paper, we consider constrained optimization problems where both the objective and constraint functions are of the black-box type. Furthermore, we assume that the nonlinear inequality constraints are non-relaxable, i.e. the values of the objective function and constraints cannot be computed outside of the feasible region. This situation happens frequently in practice especially in the black-box setting where function values are typically computed by means of complex simulation programs which may fail to execute if the considered point is outside of the feasible region. For such problems, we propose a new derivative-free optimization method which is based on the use of a merit function that handles inequality constraints by means of a log-barrier approach and equality constraints by means of an exterior penalty approach. We prove the convergence of the proposed method to KKT stationary points of the problem under standard assumptions (we do not require any convexity assumption). Furthermore, we also carry out a preliminary numerical experience on standard test problems and comparison with state-of-the-art solvers showing the efficiency of the proposed method.