Convergence of Nonmonotone Proximal Gradient Methods under the Kurdyka-Lojasiewicz Property without a Global Lipschitz Assumption

TL;DR

This paper proves that nonmonotone proximal gradient methods converge globally with favorable rates for composite optimization problems under the Kurdyka-Lojasiewicz property, without requiring a global Lipschitz condition or prior boundedness of iterates.

Contribution

It establishes that nonmonotone proximal gradient methods share convergence properties with monotone methods under the Kurdyka-Lojasiewicz property, without strong assumptions.

Findings

Global convergence of nonmonotone methods proven

Convergence rates comparable to monotone methods shown

No need for global Lipschitz or boundedness assumptions

Abstract

We consider the composite minimization problem with the objective function being the sum of a continuously differentiable and a merely lower semicontinuous and extended-valued function. The proximal gradient method is probably the most popular solver for this class of problems. Its convergence theory typically requires that either the gradient of the smooth part of the objective function is globally Lipschitz continuous or the (implicit or explicit) a priori assumption that the iterates generated by this method are bounded. Some recent results show that, without these assumptions, the proximal gradient method, combined with a monotone stepsize strategy, is still globally convergent with a suitable rate-of-convergence under the Kurdyka-Lojasiewicz property. For a nonmonotone stepsize strategy, there exist some attempts to verify similar convergence results, but, so far, they need stronger assumptions. This paper is the first which shows that nonmonotone proximal gradient methods for composite optimization problems share essentially the same nice global and rate-of-convergence properties as its monotone counterparts, still without assuming a global Lipschitz assumption and without an a priori knowledge of the boundedness of the iterates.