An Extragradient-Based Alternating Direction Method for Convex Minimization

TL;DR

This paper introduces an extragradient-based alternating direction method for convex minimization problems with linear constraints, effectively handling cases where one function's proximal mapping is difficult, and demonstrates strong empirical performance.

Contribution

The paper proposes a novel extragradient-based ADMM that handles convex problems with challenging proximal mappings, extending applicability beyond classical methods.

Findings

Converges within O(1/ε) iterations for ε-optimal solutions.

Performs well on fused logistic regression problems.

Competitive results on lasso problems compared to existing solvers.

Abstract

In this paper, we consider the problem of minimizing the sum of two convex functions subject to linear linking constraints. The classical alternating direction type methods usually assume that the two convex functions have relatively easy proximal mappings. However, many problems arising from statistics, image processing and other fields have the structure that while one of the two functions has easy proximal mapping, the other function is smoothly convex but does not have an easy proximal mapping. Therefore, the classical alternating direction methods cannot be applied. To deal with the difficulty, we propose in this paper an alternating direction method based on extragradients. Under the assumption that the smooth function has a Lipschitz continuous gradient, we prove that the proposed method returns an $\epsilon$-optimal solution within $O(1/\epsilon)$ iterations. We apply the proposed method to solve a new statistical model called fused logistic regression. Our numerical experiments show that the proposed method performs very well when solving the test problems. We also test the performance of the proposed method through solving the lasso problem arising from statistics and compare the result with several existing efficient solvers for this problem; the results are very encouraging indeed.