Bregman Proximal Mappings and Bregman–Moreau Envelopes Under Relative Prox-Regularity

Abstract

We systematically study the local single-valuedness of the Bregman proximal mapping and local smoothness of the Bregman-Moreau envelope under relative prox-regularity, an extension of prox-regularity for nonconvex functions which has been originally introduced by Poliquin and Rockafellar. Although, we focus on the left Bregman proximal mapping, a translation result yields analogue (and partially sharp) results for the right Bregman proximal mapping. The class of relatively prox-regular functions significantly extends the recently considered class of relatively hypoconvex functions. In particular, relative prox-regularity allows for functions with possibly nonconvex domain. Moreover, as a main source of examples, in analogy to the classical setting, we introduce relatively amenable functions by invoking the recently proposed notion of smooth adaptability or relative smoothness. Exemplarily we apply our theory to interpret joint alternating Bregman minimization with proximal regularization, locally, as a Bregman proximal gradient algorithm.Keywords: Bregman-Moreau envelope · Bregman proximal mapping · prox-regularity · amenable functions 2000 Mathematics Subject Classification: 49J52 · 65K05 · 65K10 · 90C26

A Bregman Forward-Backward Linesearch Algorithm for Nonconvex Composite Optimization: Superlinear Convergence to Nonisolated Local Minima

“…What we provide next is instead a local result that requires local properties of ϕ around nondegenerate fixed points of the proximal map. We remark that after the first submission of our paper a similar result appeared in [45] in a more general setting. We, however, offer our alternative proof as a means of emphasizing the favorable theoretical implications of the Euclidean equivalence stated in Theorem 3.8, which, ultimately, will lead us to the second-order result of Theorem 3.11 which is instead novel.…”

“…Nonetheless, in the nonconvex setting, there is a big discrepancy between the well-studied Euclidean setting and the less mature Bregman generalization. In an attempt to partially fill this gap, this section complements the analysis of [42,45] for the proximal mapping and the Moreau envelope in the Bregman setting. The extension-or better, the "translation"-of the results for the proximal gradient will then be derived as simple byproducts in the following section.…”

“…Thanks to the local equivalence shown in Theorem 3.8, these requirements are the same as those ensuring similar properties in the Euclidean case. These amount to prox-regularity, a condition which was first proposed in [71] and which has been recently extended to itsĥ-relative version in [45]. Definition 3.9 (prox-regularity).…”

“…(i) Bregman-Moreau envelope analysis. We provide new insights on the Bregman-Moreau envelope complementing the ones in [42,15,45]. Among these, we highlight properties of fixed points, a local equivalence with the forwardbackward envelope (FBE) [68], and ultimately a second-order differentiability result with classical generalized differentiability tools [69,73].…”

Dualities for non-Euclidean smoothness and strong convexity under the light of generalized conjugacy

“…In this section we recover the well-known notions of (Bregman) strong and weak convexity relative to φ as in Assumption I as instantiations of Φ-convexity. Thereby we will also clarify the correspondences between (left and right) Φ-conjugates and Bregman-Moreau and Bregman-Klee (left and right) envelopes [14,9,22,17,24]. Another correspondence is observed between Φ-biconjugates and Bregman proximal hulls, a recently introduced extension [34, Section 2.3] of Euclidean proximal hulls [32,Example 1.44].…”