We present an algorithm for nonlinear least-squares and nonlinear feasibility problems, i.e. for systems of nonlinear equations and nonlinear inequalities, which depend on the outcome of expensive functions for which derivatives are assumed to be unavailable. Our algorithm combines derivative-free techniques with filter trust-region methods to keep the number of expensive function evaluations low and to obtain a robust method. Under adequate assumptions, we show global convergence to a feasible point. Numerical results indicate a significant reduction in function evaluations compared to other derivative based and derivative-free solvers for nonlinear feasibility problems.
M. Kaiser, K. Klamroth,A. Thekale and Ph. L.Toint, "Solving Structured Nonlinear Least-Squares and Nonlinear Feasibility Problems With Expensive Functions, Technical Report NAXYS-07-2010, University of Namur, Namur, Belgium, 2010.