In this work we explicit a derivative-free trust-region algorithm for unconstrained optimization based on the paper (Computational Optimization and Applications 53: 527-555, 2012) proposed by Powell. The objective function is approximated by quadratic models obtained by polynomial interpolation. The number of points of the interpolation set is fixed. In each iteration only one interpolation point can be replaced and consequently the objective function is evaluated only once per iteration. The method involves two types of steps: trust-region and alternative steps. In a trust-region step the model is minimized in the hope that the reduction achieved by the model is inherited by the objective function. An alternative step aims to keep the good position of the interpolation points. We present the method in an algorithmic scheme and we discuss in details the results related to the global convergence of the algorithm, proving that, under reasonable hypotheses, any accumulation point of the sequence generated by the algorithm is stationary.
Technical Report, Federal University of Paraná, Brazil, March, 2014.