Convergence for nonconvex ADMM, with applications to CT imaging

TL;DR

This paper establishes new convergence guarantees for the ADMM algorithm in nonconvex, nonsmooth optimization problems, including practical CT image reconstruction, expanding theoretical understanding beyond existing smoothness assumptions.

Contribution

The work provides the first convergence analysis for nonconvex ADMM without requiring smoothness, under a restricted strong convexity condition.

Findings

Convergence guaranteed for nonconvex, nonsmooth problems

Validated theoretical results with simulated CT reconstruction

Demonstrated effectiveness on nondifferentiable functions

Abstract

The alternating direction method of multipliers (ADMM) algorithm is a powerful and flexible tool for complex optimization problems of the form $\min\{f(x)+g(y) : Ax+By=c\}$. ADMM exhibits robust empirical performance across a range of challenging settings including nonsmoothness and nonconvexity of the objective functions $f$ and $g$, and provides a simple and natural approach to the inverse problem of image reconstruction for computed tomography (CT) imaging. From the theoretical point of view, existing results for convergence in the nonconvex setting generally assume smoothness in at least one of the component functions in the objective. In this work, our new theoretical results provide convergence guarantees under a restricted strong convexity assumption without requiring smoothness or differentiability, while still allowing differentiable terms to be treated approximately if needed. We validate these theoretical results empirically, with a simulated example where both $f$ and $g$ are nondifferentiable -- and thus outside the scope of existing theory -- as well as a simulated CT image reconstruction problem.