Title: First Order Methods beyond Convexity and Lipschitz Gradient Continuity with Applications to Quadratic Inverse Problems,
Abstract: We focus on nonconvex and nonsmooth minimization problems with a composite
objective, where the differentiable part of the objective is freed from the
usual and restrictive global Lipschitz gradient continuity assumption. This
longstanding smoothness restriction is pervasive in first order methods (FOM),
and was recently circumvent for convex composite optimization by Bauschke,
Bolte and Teboulle, through a simple and elegant framework which captures, all
at once, the geometry of the function and of the feasible set. Building on this
work, we tackle genuine nonconvex problems. We first complement and extend
their approach to derive a full extended descent lemma by introducing the
notion of smooth adaptable functions. We then consider a Bregman-based proximal
gradient methods for the nonconvex composite model with smooth adaptable
functions, which is proven to globally converge to a critical point under
natural assumptions on the problem's data. To illustrate the power and
potential of our general framework and results, we consider a broad class of
quadratic inverse problems with sparsity constraints which arises in many
fundamental applications, and we apply our approach to derive new globally
convergent schemes for this class.