Convergence properties of inexact projected gradient methods

Abstract

In this article, we give a more comprehensive theoretical analysis of the inexact spectral projected gradient (ISPG) method introduced by Birgin et al. (Birgin, E.G., Martínez, J.M. and Raydan, M., 2003, Inexact spectral projected gradient methods on convex set. IMA Journal of Numerical Analysis, 23, 539–559) for the minimization of differentiable functions on closed convex sets. In doing so, we remove the boundedness of level sets of the objective function. Under weaker conditions, we establish the convergence theory. Moreover, we present a new inexact hybrid projection method based on the ISPG method. This new method has an encouraging convergence property which is that the whole sequence of iterates converges to a solution of the problem under no assumptions other than pseudoconvexity and continuous differentiability of f(·).

Keywords:

• Nonmonotone linear search
• Projected gradient
• Convergence

Keywords:

• Mathematics Subject Classifications 2000: 90C30
• 65K05

Acknowledgments

The authors are grateful to the two referees for their detailed comments and suggestions which have improved the presentation of this article.

This research is supported by the National Science Foundation of China under Grant 10571106.