Linear Regression Problem Relaxations Solved by Nonconvex ADMM With Convergence Analysis

In this work, we focus on studying the differentiable relaxations of several linear regression problems, where the original formulations are usually both nonsmooth with one nonconvex term. Unfortunately, in most cases, the standard alternating direction method of multipliers (ADMM) cannot guarantee global convergence when addressing these kinds of problems. To address this issue, by smoothing the convex term and applying a linearization technique before designing the iteration procedures, we employ nonconvex ADMM to optimize challenging nonconvex-convex composite problems. In our theoretical analysis, we prove the boundedness of the generated variable sequence and then guarantee that it converges to a stationary point. Meanwhile, a potential function is derived from the augmented Lagrange function, and we further verify that the objective function is monotonically nonincreasing. Under the Kurdyka-Łojasiewicz (KŁ) property, the global convergence is analyzed step by step. Finally, experiments on face reconstruction, image classification, and subspace clustering tasks are conducted to show the superiority of our algorithms over several state-of-the-art ones.