Nonmonotone Spectral Projected Gradient Methods on Convex Sets

TLDR

Nonmonotone projected gradient methods are studied for minimizing differentiable functions over closed convex sets. The authors extend classical projected gradient schemes by incorporating a Grippo–Lampariello–Lucidi nonmonotone line search and a spectral gradient steplength, using a feasible spectral projected gradient direction to reduce trial projections. The method converges and shows strong performance in extensive numerical experiments.

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.