# A New Homotopy Proximal Variable-Metric Framework for Composite Convex Minimization

## TL;DR

This paper introduces a novel homotopy proximal variable-metric framework for composite convex minimization, achieving finite iteration bounds and computational efficiency, especially in covariance estimation problems without matrix inversion.

## Contribution

It develops a new parameterization for optimality conditions enabling homotopy proximal variable-metric methods and introduces a primal-dual-primal framework for proximal-Newton methods.

## Key findings

- Achieves finite global iteration-complexity bounds under certain conditions.
- Develops a primal-dual-primal framework that improves computational features.
- Applies the method to covariance estimation without matrix inversion.

## Abstract

This paper suggests two novel ideas to develop new proximal variable-metric methods for solving a class of composite convex optimization problems. The first idea is a new parameterization of the optimality condition which allows us to develop a class of homotopy proximal variable-metric methods. We show that under appropriate assumptions such as strong convexity-type and smoothness, or self-concordance, our new schemes can achieve finite global iteration-complexity bounds. Our second idea is a primal-dual-primal framework for proximal-Newton methods which can lead to some useful computational features for a subclass of nonsmooth composite convex optimization problems. Starting from the primal problem, we formulate its dual problem, and use our homotopy proximal Newton method to solve this dual problem. Instead of solving the subproblem directly in the dual space, we suggest to dualize this subproblem to go back to the primal space. The resulting subproblem shares some similarity promoted by the regularizer of the original problem and leads to some computational advantages. As a byproduct, we specialize the proposed algorithm to solve covariance estimation problems. Surprisingly, our new algorithm does not require any matrix inversion or Cholesky factorization, and function evaluation, while it works in the primal space with sparsity structures that are promoted by the regularizer. Numerical examples on several applications are given to illustrate our theoretical development and to compare with state-of-the-arts.