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On the Regularity of Generalized Conjugate Functions

Konstantinos Oikonomidis, Emanuel Laude, Panagiotis Patrinos

Abstract

We investigate regularity properties of generalized conjugate functions induced by a general coupling function and the associated generalized proximal mapping. Our main results provide verifiable conditions ensuring local single-valuedness, continuity, Lipschitz continuity, and differentiability of the generalized proximal mapping, and transfer these properties to generalized conjugates providing explicit derivative formulas. These results are based on a nonsmooth implicit function theorem for generalized equations, relying on graphical localizations and second-order variational tools. Beyond first-order regularity, we also derive conditions under which generalized conjugates are strictly twice differentiable.

On the Regularity of Generalized Conjugate Functions

Abstract

We investigate regularity properties of generalized conjugate functions induced by a general coupling function and the associated generalized proximal mapping. Our main results provide verifiable conditions ensuring local single-valuedness, continuity, Lipschitz continuity, and differentiability of the generalized proximal mapping, and transfer these properties to generalized conjugates providing explicit derivative formulas. These results are based on a nonsmooth implicit function theorem for generalized equations, relying on graphical localizations and second-order variational tools. Beyond first-order regularity, we also derive conditions under which generalized conjugates are strictly twice differentiable.

Paper Structure

This paper contains 11 sections, 13 theorems, 63 equations, 1 table.

Table of Contents

  1. AMS Subject Classification:
  2. Introduction
  3. Motivation and related work
  4. Outline and summary of contributions
  5. Notation and preliminaries
  6. Generalized convexity
  7. Existence and regularity of Phi-subgradients
  8. Continuity and differentiability properties of Phi-conjugates
  9. Continuous differentiability of Phi-conjugates
  10. Lipschitz Differentiability and C2 properties of Phi-conjugates
  11. Conclusion

Key Result

Proposition 1

Let $X$ and $Y$ be nonempty sets, $\Phi: X \times Y \to \mathbb{R}$ a real-valued coupling and $f:X\to \overline{\mathbb{R}}$. Then we have In addition, $f^{\Phi\Phi}$ is the pointwise largest $\Phi$-convex function below $f$. In particular, this means that $f$ is $\Phi$-convex on $X$ if and only if $f(x) = f^{\Phi\Phi}(x)$ for all $x \in X$. The statements for $g:Y \to \overline{\mathbb{R}}$ are

Theorems & Definitions (43)

  • Definition 1: $\Phi$-convex and $\Phi$-concave functions
  • Definition 2: $\Phi$-conjugate functions
  • Definition 3: $\varepsilon$-$\Phi$-subgradients
  • Proposition 1: $\Phi$-Fenchel--Young inequality
  • Proposition 2: $\Phi$-subgradient equivalences
  • Example 1: $\Phi$-convexity with the quadratic coupling
  • Example 2: $\Phi$-convexity with the left Bregman coupling
  • Example 3: $\Phi$-convexity with the right Bregman coupling
  • Example 4: $\Phi$-convexity with the anisotropic coupling
  • Example 5: $\Phi$-convexity with $\varphi$-divergences
  • ...and 33 more