A gradient projection method for solving split equality and split feasibility problems in Hilbert spaces

Abstract

In this paper, we first study in a Hilbertian framework the weak convergence of a general Gradient Projection Algorithm for minimizing a convex function of class over a convex constraint set. The way of selecting the stepsizes corresponds to the one used by López et al. for the particular case of the Split Feasibility Problem. This choice allows us to avoid the computation of operator norms. Afterwards, a relaxed version of the Gradient Projection Algorithm is considered where the feasible set is approximated by half-spaces making the projections explicit. Finally, to get the strong convergence, each step of the general Gradient Projection Method is combined with a viscosity step. This is done by adapting Halpern’s algorithm to our problem. The general scheme is then applied to the Split Equality Problem, and also to the Split Feasibility Problem.

Keywords:

• gradient projection method
• split equality problem
• split feasibility problem
• relaxed algorithms
• strong convergence

AMS Subject Classifications:

• 65K05
• 90C25
• 47H10

Acknowledgements

The authors would like to thank the three referees for their comments and suggestions on improving an earlier version of this paper. In particular, they greatly appreciated the suggestion of one of the referees to use a recent result due to He and Yang [22] for getting a more succinct proof of Theorem 5.1.

Notes

This research is funded by the Department of Science and Technology at Ho Chi Minh City, Vietnam. Support provided by the Institute for Computational Science and Technology at Ho Chi Minh City (ICST) is gratefully acknowledged.