Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the growth of nonconvex functionals at strict local minimizers

Alberto Domínguez Corella, Trí Minh Lê

TL;DR

The paper addresses how nonconvex functionals grow near a strict local minimizer by proving three equivalent conditions: a growth condition $f(x)\ge f(\bar{x})+\gamma\|x-\bar{x}\|^p$, tilt stability $\ orm{x-\bar{x}\}\le\kappa\|\xi\|^{q/p}$ for linearly perturbed local minimizers, and a Polyak–Łojasiewicz–type inequality $f(x)-f(\bar{x})\le\mu\|\xi\|^q$ with $p^{-1}+q^{-1}=1$. A tilting principle shows that stability under linear perturbations implies stability under nonlinear perturbations, and the results extend to Banach spaces with the Radon–Nikodým property. The authors apply these insights to prove convergence of a proximal point algorithm with explicit rates and to analyze a tracking problem governed by an elliptic PDE, linking second-order optimality to data-sensitivity of solutions. Collectively, the work provides a comprehensive framework for error estimates and robustness in nonconvex optimization, including PDE-constrained settings.

Abstract

We give new characterizations of growth conditions at strict local minimizers. The main characterizations are a variant of the so-called tilt stability property and an analog of the classical Polyak--Łojasiewicz condition, where the gradient is replaced by linear perturbations. As a consequence, we derive a tilting principle that relates the stability of minimizers under linear perturbations to their stability under nonlinear ones.

On the growth of nonconvex functionals at strict local minimizers

TL;DR

The paper addresses how nonconvex functionals grow near a strict local minimizer by proving three equivalent conditions: a growth condition , tilt stability for linearly perturbed local minimizers, and a Polyak–Łojasiewicz–type inequality with . A tilting principle shows that stability under linear perturbations implies stability under nonlinear perturbations, and the results extend to Banach spaces with the Radon–Nikodým property. The authors apply these insights to prove convergence of a proximal point algorithm with explicit rates and to analyze a tracking problem governed by an elliptic PDE, linking second-order optimality to data-sensitivity of solutions. Collectively, the work provides a comprehensive framework for error estimates and robustness in nonconvex optimization, including PDE-constrained settings.

Abstract

We give new characterizations of growth conditions at strict local minimizers. The main characterizations are a variant of the so-called tilt stability property and an analog of the classical Polyak--Łojasiewicz condition, where the gradient is replaced by linear perturbations. As a consequence, we derive a tilting principle that relates the stability of minimizers under linear perturbations to their stability under nonlinear ones.
Paper Structure (18 sections, 16 theorems, 71 equations, 1 algorithm)

This paper contains 18 sections, 16 theorems, 71 equations, 1 algorithm.

Table of Contents

  1. Overview of results and literature review
  2. Overview of results
  3. Literature review
  4. Preliminaries
  5. Notation
  6. Set--valued mappings
  7. Radon-Nikodým property
  8. Technical lemmas
  9. Proofs
  10. Proof of Theorem \ref{['thm.growth']}
  11. Proof of Corollary \ref{['thm.L_vs_NL']}
  12. Proof of Theorem \ref{['thm.proximal']}
  13. Further discussion
  14. Subregularity of the Fenchel-Moreau subdifferential
  15. On the Polyak--Łojasiewicz--type inequality
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $p, q \in (1, + \infty)$ be such that $p^{-1} + q^{-1}=1$. The following statements are equivalent.

Theorems & Definitions (40)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Corollary 1.5
  • Remark 1.6
  • Remark 1.7
  • Corollary 1.8
  • Definition 2.1
  • Definition 2.2
  • ...and 30 more