Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Determination of (unbounded) convex functions via Crandall-Pazy directions

Aris Daniilidis, David Salas, Sebastián Tapia-García

TL;DR

The work develops a Neumann-type boundary condition at infinity for convex analysis by introducing the Crandall-Pazy direction $p_f$ and studying subgradient-flow asymptotics. It proves a determination result for convex $\mathcal{C}^{1,1}_{\rm loc}$ functions: if two functions share the same slope field and Crandall-Pazy direction, they differ by an additive constant; initial results are shown for $C^2$, then extended to nonsmooth settings in finite dimensions via Clarke theory and to general Hilbert spaces via separable reduction. The paper also treats the case where the CP direction is attained, establishing a similar determination when $p_f$ lies in the subdifferential range. Together, these results extend function-determination beyond bounded-from-below assumptions and provide a systematic framework across smooth and nonsmooth, finite- and infinite-dimensional settings, with open questions remaining for fully general nonsmooth cases.

Abstract

It has been recently discovered that a convex function can be determined by its slopes and its infimum value, provided this latter is finite. The result was extended to nonconvex functions by replacing the infimum value by the set of all critical and asymptotically critical values. In all these results boundedness from below plays a crucial role and is generally admitted to be a paramount assumption. Nonetheless, this work develops a new technique that allows to also determine a large class of unbounded from below convex functions, by means of a Neumann-type condition related to the Crandall-Pazy direction.

Determination of (unbounded) convex functions via Crandall-Pazy directions

TL;DR

The work develops a Neumann-type boundary condition at infinity for convex analysis by introducing the Crandall-Pazy direction and studying subgradient-flow asymptotics. It proves a determination result for convex functions: if two functions share the same slope field and Crandall-Pazy direction, they differ by an additive constant; initial results are shown for , then extended to nonsmooth settings in finite dimensions via Clarke theory and to general Hilbert spaces via separable reduction. The paper also treats the case where the CP direction is attained, establishing a similar determination when lies in the subdifferential range. Together, these results extend function-determination beyond bounded-from-below assumptions and provide a systematic framework across smooth and nonsmooth, finite- and infinite-dimensional settings, with open questions remaining for fully general nonsmooth cases.

Abstract

It has been recently discovered that a convex function can be determined by its slopes and its infimum value, provided this latter is finite. The result was extended to nonconvex functions by replacing the infimum value by the set of all critical and asymptotically critical values. In all these results boundedness from below plays a crucial role and is generally admitted to be a paramount assumption. Nonetheless, this work develops a new technique that allows to also determine a large class of unbounded from below convex functions, by means of a Neumann-type condition related to the Crandall-Pazy direction.

Paper Structure

This paper contains 11 sections, 18 theorems, 94 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Crandall-Pazy Directions
  4. Relation with slopes and descent sequences
  5. Relation with Thom's conjecture at infinity
  6. Main result: Determination of $\mathcal{C}^{1,1}_{\rm loc}$-convex functions
  7. The case of $\mathcal{C}^2$ functions in a Hilbert space
  8. The case of $\mathcal{C}^{1,1}_{\rm loc}$ functions in finite dimensions
  9. The case of $\mathcal{C}^{1,1}_{\rm loc}$ functions in a separable Hilbert space
  10. The case of $\mathcal{C}^{1,1}_{\rm loc}$ functions in a general Hilbert space
  11. The case where Crandall-Pazy direction is attained

Key Result

Proposition 2.2

Let $x\in \overline{\textup{dom}}\,f$ and let $\gamma:[0,+\infty)\to \mathcal{H}$ be the subgradient curve emanating from $x$. Then the map $t\mapsto \|\partial^\circ f(\gamma(t))\|$ is nonincreasing and $\|\partial^\circ f(\gamma(t))\|\xrightarrow{t\to+\infty} \inf s_f$.

Figures (1)

  • Figure 1: Construction of $\phi$ and $\varphi$.

Theorems & Definitions (39)

  • Definition 2.1: Crandall-Pazy direction
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Remark 2.5
  • Corollary 2.6
  • Theorem 2.7
  • Proposition 2.8
  • ...and 29 more