Incremental proximal gradient algorithm using penalization strategies and fixed point techniques for solving some convex bilevel optimization problems

The objective of this paper is to propose a novel incremental proximal algorithm that incorporates gradient penalization strategies within a fixed-point framework for solving convex bilevel optimization problems. The outer-level objective is modeled as the sum of two composite convex functions, one of which is nonsmooth. We establish a convergence theorem for the proposed algorithm under suitable assumptions. Additionally, we provide numerical experiments to demonstrate the effectiveness of the method in solving image inpainting problems. Comparative results with existing algorithms in the literature indicate that the proposed approach exhibits superior convergence performance.