Incremental Quasi-Subgradient Method for Minimizing Sum of Geodesic Quasi-Convex Functions on Riemannian Manifolds with Applications

Abstract

To find the optimal solution of a sum of geodesic quasi-convex functions, we introduce a new path incremental quasi-subgradient method in the setting of a Riemannian manifold whose sectional curvature is nonnegative. To study the convergence analysis of the proposed algorithm, some auxiliary results related to geodesic quasi-convex functions and an existence result for a Greenberg-Pierskalla quasi-subgradient of the geodesic quasi-convex function in the setting of Riemannian manifolds are established. The convergence result of the proposed algorithm with the dynamic step size is presented in the case when the optimal solution is unknown. To demonstrate practical applicability, we show that the proposed method can be used to find a solution of the (geodesic) quasi-convex feasibility problems and the sum of ratio problems in the setting of Riemannian manifolds.

Keywords:

• Quasi-convex optimization problems
• sum-minimization problems
• subgradient method
• Greenberg-Pierskalla quasi-subdifferential
• Riemannian manifolds

Acknowledgements

Authors are grateful to the reviewers and handling editor for their constructive suggestions which improved the previous draft of this paper. In this research, the first and last author were support by a research grant from DST-SERB No. EMR/2016/00124.