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The Aubin Property for Generalized Equations over $C^2$-cone Reducible Sets

Jiaming Ma, Defeng Sun

TL;DR

The paper addresses stability questions for canonically perturbed generalized equations over $C^2$-cone reducible sets by proving the Aubin property and strong regularity are equivalent for both $y\in\varphi(x)+N_S(x)$ and $y\in\varphi(x)+N_S^{-1}(x)$. It introduces a degree-theoretic, topological approach that leverages a new local structure lemma for the normal cone map and shows the associated index is $\pm1$, enabling a homological inverse mapping theorem to conclude a local homeomorphism and hence strong regularity. The results generalize the classical polyhedral case to a broad non-polyhedral class and provide a unified framework that applies to NLP, SOCP, and SDP stability under a single condition. The work hinges on $C^2$-cone reducibility; it also outlines an open problem for extending the equivalence to arbitrary closed convex sets and discusses implications for cone-constrained optimization across several domains.

Abstract

This paper establishes the equivalence of the Aubin property and the strong regularity for generalized equations over $C^2$-cone reducible sets. This result resolves a long-standing question in variational analysis and extends the well-known equivalence theorem for polyhedral sets to a significantly broader class of non-polyhedral cases. Our proof strategy departs from traditional variational techniques, integrating insights from convex geometry with powerful tools from algebraic topology. A cornerstone of our analysis is a new fundamental lemma concerning the local structure of the normal cone map for arbitrary closed convex sets, which reveals how the dimension of normal cones varies in the neighborhood of a boundary point. This geometric insight is the key to applying degree theory, allowing us to prove that a crucial function associated with the problem has a topological index of $\pm1$. This, via a homological version of the inverse mapping theorem, implies that the function is a local homeomorphism, which in turn yields the strong regularity of the original solution map. This result unifies and extends several existing stability results for problems such as conventional nonlinear programming, nonlinear second-order cone programming, and nonlinear semidefinite programming under a single general framework.

The Aubin Property for Generalized Equations over $C^2$-cone Reducible Sets

TL;DR

The paper addresses stability questions for canonically perturbed generalized equations over -cone reducible sets by proving the Aubin property and strong regularity are equivalent for both and . It introduces a degree-theoretic, topological approach that leverages a new local structure lemma for the normal cone map and shows the associated index is , enabling a homological inverse mapping theorem to conclude a local homeomorphism and hence strong regularity. The results generalize the classical polyhedral case to a broad non-polyhedral class and provide a unified framework that applies to NLP, SOCP, and SDP stability under a single condition. The work hinges on -cone reducibility; it also outlines an open problem for extending the equivalence to arbitrary closed convex sets and discusses implications for cone-constrained optimization across several domains.

Abstract

This paper establishes the equivalence of the Aubin property and the strong regularity for generalized equations over -cone reducible sets. This result resolves a long-standing question in variational analysis and extends the well-known equivalence theorem for polyhedral sets to a significantly broader class of non-polyhedral cases. Our proof strategy departs from traditional variational techniques, integrating insights from convex geometry with powerful tools from algebraic topology. A cornerstone of our analysis is a new fundamental lemma concerning the local structure of the normal cone map for arbitrary closed convex sets, which reveals how the dimension of normal cones varies in the neighborhood of a boundary point. This geometric insight is the key to applying degree theory, allowing us to prove that a crucial function associated with the problem has a topological index of . This, via a homological version of the inverse mapping theorem, implies that the function is a local homeomorphism, which in turn yields the strong regularity of the original solution map. This result unifies and extends several existing stability results for problems such as conventional nonlinear programming, nonlinear second-order cone programming, and nonlinear semidefinite programming under a single general framework.

Paper Structure

This paper contains 9 sections, 16 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Classification of normal vectors
  4. Degree theory and inverse mapping theorem
  5. Strictly stationary property
  6. C2-cone reduction
  7. Index of N
  8. Equivalence of the Aubin property and the strong regularity
  9. Conclusions

Key Result

Lemma 1

Let $f:{\mathbb R}^n\to{\mathbb R}^m$ be a continuous function and $x_0\in{\mathbb R}^n$. If $f^{-1}$ is inner semicontinuous around $(f(x_0),x_0)$, then $f$ is open around $x_0$.

Theorems & Definitions (26)

  • Lemma 1
  • proof
  • Lemma 2: schneider2013convex
  • Proposition 1
  • Lemma 3: barreto2016inverse
  • Lemma 4: dontchev1994inverse
  • Lemma 5
  • Lemma 6
  • proof
  • Lemma 7
  • ...and 16 more