Global convergence of general derivative-free trust-region algorithms to first- and second-order critical points

In this paper we prove global convergence for first- and second-order stationary points of a class of derivative-free trust-region methods for unconstrained optimization. These methods are based on the sequential minimization of quadratic (or linear) models built from evaluating the objective function at sample sets. The derivative-free models are required to satisfy Taylor-type bounds, but, apart from that, the analysis is independent of the sampling techniques. A number of new issues are addressed, including global convergence when acceptance of iterates is based on simple decrease of the objective function, trust-region radius maintenance at the criticality step, and global convergence for second-order critical points. © 2009 Society for Industrial and Applied Mathematics.