# General parameterized proximal point algorithm with applications in statistical learning

## TL;DR

This paper introduces a flexible, parameterized proximal point algorithm with a relaxation step for multi-block convex optimization, demonstrating improved efficiency and convergence properties in statistical learning applications.

## Contribution

It develops a general parameterized PPA with relaxation for multi-block convex problems, establishing convergence and efficiency improvements over existing methods.

## Key findings

- Proposed algorithm has a global convergence guarantee.
- Achieves an $ ext{O}(1/t)$ convergence rate.
- Numerical experiments show superior efficiency in sparse matrix minimization.

## Abstract

In the literature, there are a few researches to design some parameters in the Proximal Point Algorithm (PPA), especially for the multi-objective convex optimizations. Introducing some parameters to PPA can make it more flexible and attractive. Mainly motivated by our recent work (Bai et al., A parameterized proximal point algorithm for separable convex optimization, Optim. Lett. (2017) doi: 10.1007/s11590-017-1195-9), in this paper we develop a general parameterized PPA with a relaxation step for solving the multi-block separable structured convex programming. By making use of the variational inequality and some mathematical identities, the global convergence and the worst-case $\mathcal{O}(1/t)$ convergence rate of the proposed algorithm are established. Preliminary numerical experiments on solving a sparse matrix minimization problem from statistical learning validate that our algorithm is more efficient than several state-of-the-art algorithms.