Monotone and nonmonotone trust‐region‐based algorithms for large scale unconstrained optimization problems

Two trust regions algorithms for unconstrained nonlinear optimization problems are presented: a monotone and a nonmonotone one. Both of them solve the trust region subproblem by the spectral projected gradient (SPG) method proposed by Birgin, Martínez and Raydan (2000). SPG is a nonmonotone projected gradient algorithm for solving large‐scale convex‐constrained optimization problems. It combines the classical projected gradient method with the spectral gradient choice of steplength and a nonmonotone line search strategy. The simplicity (only requires matrix‐vector products, and one projection per iteration) and rapid convergence of this scheme fits nicely with globalization techniques based on the trust region philosophy, for large‐scale problems. In the nonmonotone algorithm the trial step is evaluated by acceptance via a rule which can be considered a generalization of the well known fraction of Cauchy decrease condition and a generalization of the nonmonotone line search proposed by Grippo, Lampariello and Lucidi (1986). Convergence properties and extensive numerical results are presented. Our results establish the robustness and efficiency of the new algorithms.