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On the natural domain of Bregman operators

Andreas Themelis, Ziyuan Wang

TL;DR

This work rethinks Bregman proximity theory by replacing the traditional full-space domain with a domain-aware framework that restricts functions to $\mathcal{X}=\mathrm{dom}\,\phi$ and $\mathcal{Y}=\mathrm{int}\,\mathrm{dom}\,\phi$. This shift clarifies that Bregman proximal mappings and Moreau envelopes depend only on the interior and domain values of the distance-generating function, enabling a unified treatment of nonconvex objects like relatively weakly convex functions and yielding simplified proofs. The authors develop both left and right Bregman operators, rectify prior claims, and connect these objects to a Phi-conjugacy perspective, yielding new characterizations of relative smoothness and cocoercivity in the domain-restricted setting. The framework broadens applicability, reduces technical overhead, and offers a coherent duality-based view that aligns with recent advances in $\Phi$-convexity and anisotropic inequalities, with potential implications for domain-aware optimization methods.

Abstract

The Bregman proximal mapping and Bregman-Moreau envelope are traditionally studied for functions defined on the entire space $\mathbb{R}^n$, even though these constructions depend only on the values of the function within (the interior of) the domain of the distance-generating function (dgf). While this convention is largely harmless in the convex setting, it leads to substantial limitations in the nonconvex case, as it fails to embrace important classes of functions such as relatively weakly convex ones. In this work, we revisit foundational aspects of Bregman analysis by adopting a domain-aware perspective: we define functions on the natural domain induced by the dgf and impose properties only relative to this set. This framework not only generalizes existing results but also rectifies and simplifies their statements and proofs. Several examples illustrate both the necessity of our assumptions and the advantages of this refined approach.

On the natural domain of Bregman operators

TL;DR

This work rethinks Bregman proximity theory by replacing the traditional full-space domain with a domain-aware framework that restricts functions to and . This shift clarifies that Bregman proximal mappings and Moreau envelopes depend only on the interior and domain values of the distance-generating function, enabling a unified treatment of nonconvex objects like relatively weakly convex functions and yielding simplified proofs. The authors develop both left and right Bregman operators, rectify prior claims, and connect these objects to a Phi-conjugacy perspective, yielding new characterizations of relative smoothness and cocoercivity in the domain-restricted setting. The framework broadens applicability, reduces technical overhead, and offers a coherent duality-based view that aligns with recent advances in -convexity and anisotropic inequalities, with potential implications for domain-aware optimization methods.

Abstract

The Bregman proximal mapping and Bregman-Moreau envelope are traditionally studied for functions defined on the entire space , even though these constructions depend only on the values of the function within (the interior of) the domain of the distance-generating function (dgf). While this convention is largely harmless in the convex setting, it leads to substantial limitations in the nonconvex case, as it fails to embrace important classes of functions such as relatively weakly convex ones. In this work, we revisit foundational aspects of Bregman analysis by adopting a domain-aware perspective: we define functions on the natural domain induced by the dgf and impose properties only relative to this set. This framework not only generalizes existing results but also rectifies and simplifies their statements and proofs. Several examples illustrate both the necessity of our assumptions and the advantages of this refined approach.

Paper Structure

This paper contains 24 sections, 20 theorems, 59 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries on maps defined on subsets of Rn
  3. Relative topology
  4. Set-valued operators
  5. Extended-real-valued functions
  6. Convex analysis
  7. Canonical extensions to the whole space Rn
  8. Bregman proximal map and Moreau envelope
  9. Definitions
  10. (Left) prox-boundedness and consequences
  11. Rectified claims
  12. A more general statement
  13. The role of 1-coercivity
  14. Alternative expressions
  15. Expressions involving convex conjugates
  16. ...and 9 more sections

Key Result

Lemma 2.1

Let $C\subseteq X\subseteq\m@thbbch@rR^n$, and equip $X$ with the subspace topology inherited from $\m@thbbch@rR^n$. Then $C$ is compact in $X$ if and only if $C$ is compact in $\m@thbbch@rR^n$.

Figures (1)

  • Figure 1: Interrelations among different properties for a set-valued operator $T:X\rightrightarrows Y$ between nonempty subsets $X,Y\subseteq\m@thbbch@rR^n$. Dashed red arrows represent implications that are valid provided that $Y$ is closed in $\m@thbbch@rR^n$. References are omitted when the corresponding implications are either obvious or directly follow from other implications.

Theorems & Definitions (66)

  • Lemma 2.1: Compactness is an "intrinsic" property
  • proof
  • Definition 2.2: local boundedness
  • Definition 2.3: outer semicontinuity
  • Example 2.4: osc: role of the target space $Y$
  • Definition 2.6: upper semicontinuity
  • Lemma 2.7
  • proof
  • Lemma 2.8
  • proof
  • ...and 56 more