Incremental proximal gradient scheme with penalization for constrained composite convex optimization problems

TL;DR

This paper introduces an incremental proximal gradient algorithm with penalization for constrained convex optimization, proving convergence under certain conditions and demonstrating its effectiveness through image inpainting and geometric problems.

Contribution

It proposes a novel algorithm combining incremental proximal gradient with smooth penalization for constrained convex problems, with proven convergence guarantees.

Findings

Algorithm converges to an optimal solution under specific conditions.

Numerical experiments show effectiveness in image inpainting.

Method applies to generalized Heron problems with least squares constraints.

Abstract

We consider the problem of minimizing a finite sum of convex functions subject to the set of minimizers of a convex differentiable function. In order to solve the problem, an algorithm combining the incremental proximal gradient method with smooth penalization technique is proposed. We show the convergence of the generated sequence of iterates to an optimal solution of the optimization problems, provided that a condition expressed via the Fenchel conjugate of the constraint function is fulfilled. Finally, the functionality of the method is illustrated by some numerical experiments addressing image inpainting problems and generalized Heron problems with least squares constraints.