A filter trust-region algorithm for unconstrained optimization with strong global convergence properties

We present a new filter trust-region approach for solving unconstrained nonlinear optimization problems making use of the filter technique introduced by Fletcher and Leyffer to generate non-monotone iterations. We also use the concept of a multidimensional filter used by Gould et al. (SIAM J. Optim. 15(1):17–38, 2004) and introduce a new filter criterion showing good properties. Moreover, we introduce a new technique for reducing the size of the filter. For the algorithm, we present two different convergence analyses. First, we show that at least one of the limit points of the sequence of the iterates is first-order critical. Second, we prove the stronger property that all the limit points are first-order critical for a modified version of our algorithm. We also show that, under suitable conditions, all the limit points are second-order critical. Finally, we compare our algorithm with a natural trust-region algorithm and the filter trust-region algorithm of Gould et al. on the CUTEr unconstrained test problems Gould et al. in ACM Trans. Math. Softw. 29(4):373–394, 2003. Numerical results demonstrate the efficiency and robustness of our proposed algorithms.