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Local geometry of feasible regions via smooth paths

Adrian S. Lewis, Adriana Nicolae, Tonghua Tian

TL;DR

The paper introduces smooth approximate convexity (SAC) as a local geometric regularity for feasible regions shaped as $C=F^{-1}(D)$ with $F$ being ${ m C}^{1}$-smooth and $D$ convex, showing SAC is weaker than both amenability and prox-regularity yet stronger than Clarke regularity in a way that guarantees near-straight smooth paths within the set. It proves amenability and prox-regularity imply SAC, and that SAC, in turn, yields normal embedding-like properties and Uniform Approximation by Geodesics (UAG). The authors extend SAC to Riemannian manifolds (embedded submanifolds and general manifolds via Nash embedding) and formulate a weaker manifold notion, smooth weak convexity, to handle tangent-space variations, showing amenability/prox-regularity preserve SAC or its weak form. They connect SAC to optimization by establishing first-order descent implications: if smooth weak convexity holds and a first-order condition fails, there exists a ${C}^{1}$-smooth descent path in the feasible set with linear decrease, suggesting a path-based alternative to classical constraint qualifications. Overall, SAC provides a potentially practical substitute for stronger regularity assumptions in variational analysis and optimization on both Euclidean and Riemannian settings, with a focus on constructive, path-based geometric regularity.

Abstract

Variational analysis presents a unified theory encompassing in particular both smoothness and convexity. In a Euclidean space, convex sets and smooth manifolds both have straightforward local geometry. However, in the most basic hybrid case of feasible regions consisting of pre-images of convex sets under maps that are once (but not necessarily twice) continuously differentiable, the geometry is less transparent. We define a new approximate convexity property, that holds both for such feasible regions and also for all prox-regular sets. This new property requires that nearby points can always be joined by smooth feasible paths that are almost straight. In particular, in the terminology of real algebraic geometry, such feasible regions are locally normally embedded in the Euclidean space.

Local geometry of feasible regions via smooth paths

TL;DR

The paper introduces smooth approximate convexity (SAC) as a local geometric regularity for feasible regions shaped as with being -smooth and convex, showing SAC is weaker than both amenability and prox-regularity yet stronger than Clarke regularity in a way that guarantees near-straight smooth paths within the set. It proves amenability and prox-regularity imply SAC, and that SAC, in turn, yields normal embedding-like properties and Uniform Approximation by Geodesics (UAG). The authors extend SAC to Riemannian manifolds (embedded submanifolds and general manifolds via Nash embedding) and formulate a weaker manifold notion, smooth weak convexity, to handle tangent-space variations, showing amenability/prox-regularity preserve SAC or its weak form. They connect SAC to optimization by establishing first-order descent implications: if smooth weak convexity holds and a first-order condition fails, there exists a -smooth descent path in the feasible set with linear decrease, suggesting a path-based alternative to classical constraint qualifications. Overall, SAC provides a potentially practical substitute for stronger regularity assumptions in variational analysis and optimization on both Euclidean and Riemannian settings, with a focus on constructive, path-based geometric regularity.

Abstract

Variational analysis presents a unified theory encompassing in particular both smoothness and convexity. In a Euclidean space, convex sets and smooth manifolds both have straightforward local geometry. However, in the most basic hybrid case of feasible regions consisting of pre-images of convex sets under maps that are once (but not necessarily twice) continuously differentiable, the geometry is less transparent. We define a new approximate convexity property, that holds both for such feasible regions and also for all prox-regular sets. This new property requires that nearby points can always be joined by smooth feasible paths that are almost straight. In particular, in the terminology of real algebraic geometry, such feasible regions are locally normally embedded in the Euclidean space.
Paper Structure (14 sections, 24 theorems, 164 equations)

This paper contains 14 sections, 24 theorems, 164 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Euclidean spaces
  4. Metric spaces
  5. Smooth approximate convexity in Euclidean spaces
  6. Other regularity conditions
  7. Amenability vs. smooth approximate convexity
  8. Prox-regularity vs. smooth approximate convexity
  9. Smooth approximate convexity on Riemannian manifolds
  10. Embedded submanifolds
  11. General manifolds
  12. First-order optimality condition on Riemannian manifolds
  13. Appendix: amenability on Riemannian manifolds
  14. Appendix: prox-regularity on Riemannian manifolds

Key Result

Proposition 3.4

If a function $f: {\mathbf R}^n \rightarrow {\mathbf R}$ is locally Lipschitz continuous at $\bar{x}$ with local Lipschitz modulus $0$, then $\mathrm{epi}\, f$ is smoothly approximately convex at $(\bar{x}, \bar{s})$ for all $\bar{s} \geq f(\bar{x})$.

Theorems & Definitions (52)

  • Definition 3.1
  • Remark 3.2
  • Remark 3.3
  • Proposition 3.4: Smooth approximate convexity of epigraphs
  • Lemma 3.5: Smooth embeddings preserve smooth approximate convexity
  • Proposition 3.6: Implications of smooth approximate convexity
  • Example 3.7: Clarke regularity does not imply smooth approximate convexity
  • Example 3.8: UAG does not imply smooth approximate convexity
  • Remark 3.9
  • Definition 3.10
  • ...and 42 more