# Projection Algorithms for Finite Sum Constrained Optimization

## TL;DR

This paper introduces parallel and cyclic projection algorithms for optimizing the sum of convex functions over convex sets, proving their convergence under certain conditions in finite-dimensional spaces.

## Contribution

It presents predictor-corrector projection algorithms for finite sum constrained optimization and establishes their convergence in finite-dimensional Hilbert spaces.

## Key findings

- Algorithms converge to an optimal solution under bounded gradients.
- The methods are applicable to convex functions and sets in finite-dimensional spaces.
- Discussion of generalizations and limitations of the algorithms.

## Abstract

Parallel and cyclic projection algorithms are proposed for minimizing the sum of a finite family of convex functions over the intersection of a finite family of closed convex subsets of a Hilbert space. These algorithms are of predictor-corrector type, with each main iteration consisting of an inner cycle of subgradient descent process followed by a projection step. We prove the convergence of these methods to an optimal solution of the composite minimization problem under investigation upon assuming boundedness of the gradients at the iterates of the local functions and the stepsizes being chosen appropriately, in the finite-dimensional setting. We also discuss generalizations and limitations of the proposed algorithms and our techniques.