Global convergence of splitting methods for nonconvex composite optimization

TL;DR

This paper proves convergence properties of splitting methods like ADMM and proximal gradient for nonconvex composite optimization problems common in engineering and machine learning, under certain conditions.

Contribution

It establishes convergence and stationarity results for ADMM and proximal gradient methods applied to nonconvex problems with composite structure, including semi-algebraic functions.

Findings

ADMM converges to stationary points with large penalty parameters.

Whole sequence convergence under semi-algebraic assumptions.

Conditions for boundedness of generated sequences.

Abstract

We consider the problem of minimizing the sum of a smooth function $h$ with a bounded Hessian, and a nonsmooth function. We assume that the latter function is a composition of a proper closed function $P$ and a surjective linear map $\cal M$, with the proximal mappings of $\tau P$, $\tau > 0$, simple to compute. This problem is nonconvex in general and encompasses many important applications in engineering and machine learning. In this paper, we examined two types of splitting methods for solving this nonconvex optimization problem: alternating direction method of multipliers and proximal gradient algorithm. For the direct adaptation of the alternating direction method of multipliers, we show that, if the penalty parameter is chosen sufficiently large and the sequence generated has a cluster point, then it gives a stationary point of the nonconvex problem. We also establish convergence of the whole sequence under an additional assumption that the functions $h$ and $P$ are semi-algebraic. Furthermore, we give simple sufficient conditions to guarantee boundedness of the sequence generated. These conditions can be satisfied for a wide range of applications including the least squares problem with the $\ell_{1/2}$ regularization. Finally, when $\cal M$ is the identity so that the proximal gradient algorithm can be efficiently applied, we show that any cluster point is stationary under a slightly more flexible constant step-size rule than what is known in the literature for a nonconvex $h$.