Augmented Lagrangian Algorithms based on the Spectral Projected Gradient for solving nonlinear programming problems

The Spectral Projected Gradient method ({\tt SPG}) is an algorithm for large-scale bound-constrained optimization introduced recently by Birgin, Mart\'{\i}nez and Raydan. It is based on Raydan's unconstrained generalization of the Barzilai-Borwein method for quadratics. The {\tt SPG} algorithm turned out to be surprisingly effective for solving many large-scale minimization problems with box constraints. Therefore, it is natural to test its performance for solving the subproblems that appear in nonlinear programming methods based on augmented Lagrangians. In this work, augmented Lagrangian methods which use {\tt SPG} as underlying convex-constraint solver are introduced ({\tt ALSPG}), and the methods are tested in two sets of problems. First, a meaningful subset of large-scale nonlinearly constrained problems of the {\tt CUTE} collection is solved and compared with the performance of {\tt LANCELOT}.Second, a family of localization problems in the minimax formulation is solved against the package {\tt FFSQP}.