Adaptive subgradient method for the split quasi-convex feasibility problems

Abstract

In this paper, we consider a type of the celebrated convex feasibility problem, named as split quasi-convex feasibility problem (SQFP). The SQFP is to find a point in a sublevel set of a quasi-convex function in one space and its image under a bounded linear operator is contained in a sublevel set of another quasi-convex function in the image space. We propose a new adaptive subgradient algorithm for solving SQFP problem. We also discuss the convergence analyses for two cases: the first case where the functions are upper semicontinuous in the setting of finite dimensional, and the second case where the functions are weakly continuous in the infinite-dimensional settings. Finally some numerical examples in order to support the convergence results are given.

Keywords:

• Split feasibility problem
• quasi-convex feasibility problem
• adaptive subgradient method
• convergence

Acknowledgements

The authors would like to thank the anonymous referees for their careful reading, and suggestions which allowed us to improve the first version of this paper.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

N. Nimana was supported by the Thailand Research Fund through the Royal Golden Jubilee PhD Program [grant number PHD/0079/2554].