Abstract. Nonmonotone projected gradient techniques are considered for the minimization of differentiable functions on closed convex sets. The classical projected gradient schemes are extended to include a nonmonotone steplength strategy that is based on the Grippo–Lampariello–Lucidi nonmonotone line search. In particular, the nonmonotone strategy is combined with the spectral gradient choice of steplength to accelerate the convergence process. In addition to the classical projected gradient nonlinear path, the feasible spectral projected gradient is used as a search direction to avoid additional trial projections during the one-dimensional search process. Convergence properties and extensive numerical results are presented.

A new method is introduced for large scale convex constrained optimization. The general model algorithm involves, at each iteration, the approximate minimization of a convex quadratic on the feasible set of the original problem and global convergence is obtained by means of nonmonotone line searches. A specific algorithm, the Inexact Spectral Projected Gradient method (ISPG), is implemented using inexact projections computed by Dykstra’s alternating projection method and generates interior iterates. The ISPG method is a generalization of the Spectral Projected Gradient method (SPG), but can be used when projections are difficult to compute. Numerical results for constrained least-squares rectangular matrix problems are presented. Key words: Convex constrained optimization, projected gradient, nonmonotone line search, spectral gradient, Dykstra’s algorithm. AMS Subject Classification: 49M07, 49M10, 65K, 90C06, 90C20. 1