An Adaptive Proximal ADMM for Nonconvex Linearly Constrained Composite Programs

TL;DR

This paper introduces an adaptive proximal ADMM method for nonconvex linearly constrained problems, which is parameter-free, flexible, and achieves state-of-the-art convergence with demonstrated computational advantages.

Contribution

It develops a novel adaptive proximal ADMM that handles weakly convex smooth components and convex non-smooth parts without rank assumptions, with inexact subproblem solutions.

Findings

Achieves approximate stationary points with optimal iteration complexity.

Does not require rank assumptions on constraint matrices.

Numerical experiments show significant computational benefits.

Abstract

This paper develops an adaptive proximal alternating direction method of multipliers (ADMM) for solving linearly constrained, composite optimization problems under the assumption that the smooth component of the objective is weakly convex, while the non-smooth component is convex and block-separable. The proposed method is adaptive to all problem parameters, including smoothness and weak convexity constants, and allows each of its block proximal subproblems to be inexactly solved. Each iteration of our adaptive proximal ADMM consists of two steps: the sequential solution of each block proximal subproblem; and adaptive tests to decide whether to perform a full Lagrange multiplier and/or penalty parameter update(s). Without any rank assumptions on the constraint matrices, it is shown that the adaptive proximal ADMM obtains an approximate first-order stationary point of the constrained problem in a number of iterations that matches the state-of-the-art complexity for the class of proximal ADMM's. The three proof-of-concept numerical experiments that conclude the paper suggest our adaptive proximal ADMM enjoys significant computational benefits.