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.Key words. projected gradients, nonmonotone line search, large-scale problems, bound constrained problems, spectral gradient method AMS subject classifications. 49M07, 49M10, 65K, 90C06, 90C20 PII. S10526234973309631. Introduction. We consider the projected gradient method for the minimization of differentiable functions on nonempty closed and convex sets. Over the last few decades, there have been many different variations of the projected gradient method that can be viewed as the constrained extensions of the optimal gradient method for unconstrained minimization. They all have the common property of maintaining feasibility of the iterates by frequently projecting trial steps on the feasible convex set. This process is in general the most expensive part of any projected gradient method. Moreover, even if projecting is inexpensive, as in the box-constrained case, the method is considered to be very slow, as is its analogue, the optimal gradient method (also known as steepest descent), for unconstrained optimization. On the positive side, the projected gradient method is quite simple to implement and very effective for large-scale problems.This state of affairs motivates us to combine the projected gradient method with two recently developed ingredients in optimization. First we extend the typical globalization strategies associated with these methods to the nonmonotone line search schemes developed by Grippo, Lampariello, and Lucidi [17] for Newton's method. Second, we propose to associate the spectral steplength, introduced by Barzilai and Borwein [1] and analyzed by Raydan [26]. This choice of steplength requires little computational work and greatly speeds up the convergence of gradient methods. In fact, while the spectral gradient method appears to be a generalized steepest descent method, it is clear from its derivation that it is related to the quasi-Newton family of methods through an approximated secant equation. The fundamental difference is

The Barzilai and Borwein gradient method for the solution of large scale unconstrained minimization problems is considered. This method requires few storage locations and very inexpensive computations. Furthermore, it does not guarantee descent in the objective function and no line search is required. Recently, the global convergence for the convex quadratic case has been established. However, for the nonquadratic case, the method needs to be incorporated in a globalization scheme. In this work, a nonmonotone line search strategy that guarantees global convergence is combined with the Barzilai and Borwein method. This strategy is based on the nonmonotone line search technique proposed by Grippo, Lampariello, and Lucidi [SIAM J. Numer. Anal., 23 (1986), pp. 707-716]. Numerical results to compare the behavior of this method with recent implementations of the conjugate gradient method are presented. These results indicate that the global Barzilai and Borwein method may allow some significant reduction in the number of line searches and also in the number of gradient evaluations.

Abstract. A fully derivative-free spectral residual method for solving largescale nonlinear systems of equations is presented. It uses in a systematic way the residual vector as a search direction, a spectral steplength that produces a nonmonotone process and a globalization strategy that allows for this nonmonotone behavior. The global convergence analysis of the combined scheme is presented. An extensive set of numerical experiments that indicate that the new combination is competitive and frequently better than well-known Newton-Krylov methods for large-scale problems is also presented.

In a recent paper, Barzila.i and Borwein presented a new choice of steplength for the gradient method. Their choice does not guarantee descent in the objective function and greatly speeds up the convergence of the method. We derive a relationship between the gradient method for minimizing a quadratic function and the shifted power method. This relationship allows us to establish the convergence of the Barzilai and Borwein method when applied to the problem of minimizing any strictly convex quadratic function (Barzilai and Borwein considered only 2-dimensional problems). Our point of view also allows us to explain the improvement obtained by using this new choice of steplength.

Fortran 77 software implementing the SPG method is introduced. 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 user provides objective function and gradient values, and projections onto the feasible set. Some recent numerical tests are reported on very large location problems, indicating that SPG is substantially more efficient than existing general-purpose software on problems for which projections can be computed efficiently.