Revisiting linearized Bregman iterations under Lipschitz-like convexity condition

Abstract

The linearized Bregman iterations (LBreI) and its variants have received considerable attention in signal/image processing and compressed sensing. Recently, LBreI has been extended to a larger class of nonconvex functions, along with several theoretical issues left for further investigation. In particular, the Lipschitz gradient continuity assumption precludes its use in many practical applications. In this study, we propose a generalized algorithmic framework to unify LBreI-type methods. Our main discovery is that the Lipschitz gradient continuity assumption can be replaced by a Lipschitz-like convexity condition in both convex and nonconvex cases. As a by-product, a class of bilevel optimization problems can be solved in the proposed framework, which extends the main result made by Cai et al. [Math. Comp. 78 (2009), pp. 2127–2136]. At last, provably convergent iterative schemes on modified linear/quadratic inverse problems illustrate our finding.