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 (in SIAM J. Optim. 10(4):1196–1211, 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 (in SIAM J. Numer. Anal. 23:707–716, 1986). Convergence properties and extensive numerical results are presented. Our results establish the robustness and efficiency of the new algorithms.