Many derivative-free methods for constrained problems are not efficient for minimizing functions on "thin" domains. Other algorithms, like those based on Augmented Lagrangians, deal with thin constraints using penalty-like strategies. When the constraints are computationally inexpensive but highly nonlinear, these methods spend many potentially expensive objective function evaluations motivated by the difficulties of improving feasibility. An algorithm that handles efficiently this case is proposed in this paper. The main iteration is splitted into two steps: restoration and minimization. In the restoration step the aim is to decrease infeasibility without evaluating the objective function. In the minimization step the objective function $f$ is minimized on a relaxed feasible set. A global minimization result will be proved and computational experiments showing the advantages of this approach will be presented.
Department of Applied Mathematics, Institute of Mathematics, Statistics and Scientific Computing, State University of Campinas, Campinas, SP, Brazil (2011). Submitted.