Proximal Gradient Algorithms under Local Lipschitz Gradient Continuity: A Convergence and Robustness Analysis of PANOC

Abstract

Composite optimization offers a powerful modeling tool for a variety of applications and is often numerically solved by means of proximal gradient methods. In this paper, we consider fully nonconvex composite problems under only local Lipschitz gradient continuity for the smooth part of the objective function. We investigate an adaptive scheme for PANOC-type methods (Stella et al. in Proceedings of the IEEE 56th CDC, 1939-1944, namely accelerated linesearch algorithms requiring only the simple oracle of proximal gradient. While including the classical proximal gradient method, our theoretical results cover a broader class of algorithms and provide convergence guarantees for accelerated methods with possibly inexact computation of the proximal mapping. These findings have also significant practical impact, as they widen scope and performance of existing, and possibly future, general purpose optimization software that invoke PANOC as inner solver. Keywords. Nonsmooth nonconvex optimization • locally Lipschitz gradient • forward-backward splitting • linesearch methods AMS subject classifications. 49J52 • 65K05 • 90C30 * A. Themelis acknowledges the support of the Japan Society for the Promotion of Science (JSPS) KAKENHI grant JP21K17710.

SPIRAL: A Superlinearly Convergent Incremental Proximal Algorithm for Nonconvex Finite Sum Minimization

“…Also, note that the third block (steps 4,5) involves two linesearches, performing the linesearches in this intertwined fashion is observed to result in acceptance of good directions and reduction in the overall computational complexity [38]. We refer the reader to [13] for the theoretical justification for the effectiveness of this procedure. In Algorithm 2 in the Euclidean case, the same backtrackings can be used with dgfs h i = 1 2 • 2 .…”