Proximal primal-dual best approximation algorithm with memory

TL;DR

This paper introduces a novel primal-dual proximal best approximation algorithm that incorporates memory of previous iterates, leading to improved convergence and performance in convex optimization problems, especially in image reconstruction.

Contribution

The paper presents a new primal-dual algorithm with memory that uses projections onto intersections of halfspaces, enhancing convergence properties and practical performance.

Findings

Algorithm with memory shows faster convergence.

Numerical results demonstrate improved image reconstruction.

Memory incorporation enhances stability and accuracy.

Abstract

We propose a new modified primal-dual proximal best approximation method for solving convex not necessarily differentiable optimization problems. The novelty of the method relies on introducing memory by taking into account iterates computed in previous steps in the formulas defining current iterate. To this end we consider projections onto intersections of halfspaces generated on the basis of the current as well as the previous iterates. To calculate these projections we are using recently obtained closed-form expressions for projectors onto polyhedral sets. The resulting algorithm with memory inherits strong convergence properties of the original best approximation proximal primal-dual algorithm. Additionally, we compare our algorithm with the original (non-inertial) one with the help of the so called attraction property defined below. Extensive numerical experimental results on image reconstruction problems illustrate the advantages of including memory into the original algorithm.