Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Isolated Calmness in Regularized Convex Optimization

Tran T. A. Nghia, Huy N. Pham

TL;DR

This work addresses the stability of the optimal solution map in regularized convex composite optimization by introducing a geometric characterization of isolated calmness in terms of kernels and tangent cones tied to the second-order subgradient structure. It develops a zero-product property for the subgradient graphical derivative and leverages quadratic growth conditions to connect isolated calmness with strong second-order optimality. A central result provides necessary and sufficient conditions for isolated calmness in the general composite problem min f(x,p)+g(Kx), with RCQ- and SRCQ-based implications, including concrete corollaries for least-squares with regularizers such as the nuclear norm and group Lasso. The findings yield verifiable criteria for stability of KKT/Lagrange systems and establish equivalences to strong local minimality and, in key cases, to solution uniqueness, with potential applications in inference and inverse problems.

Abstract

This paper studies the isolated calmness of the optimal solution mapping and the associated Lagrange system for regularized convex composite optimization problems. Several necessary and sufficient conditions for this property are established. These conditions are geometric in nature and relatively simple to verify. To support the analysis, we also develop a so-called zero-product property for second-order structures, namely the graphical derivative of the subgradient mapping of convex functions.

Isolated Calmness in Regularized Convex Optimization

TL;DR

This work addresses the stability of the optimal solution map in regularized convex composite optimization by introducing a geometric characterization of isolated calmness in terms of kernels and tangent cones tied to the second-order subgradient structure. It develops a zero-product property for the subgradient graphical derivative and leverages quadratic growth conditions to connect isolated calmness with strong second-order optimality. A central result provides necessary and sufficient conditions for isolated calmness in the general composite problem min f(x,p)+g(Kx), with RCQ- and SRCQ-based implications, including concrete corollaries for least-squares with regularizers such as the nuclear norm and group Lasso. The findings yield verifiable criteria for stability of KKT/Lagrange systems and establish equivalences to strong local minimality and, in key cases, to solution uniqueness, with potential applications in inference and inverse problems.

Abstract

This paper studies the isolated calmness of the optimal solution mapping and the associated Lagrange system for regularized convex composite optimization problems. Several necessary and sufficient conditions for this property are established. These conditions are geometric in nature and relatively simple to verify. To support the analysis, we also develop a so-called zero-product property for second-order structures, namely the graphical derivative of the subgradient mapping of convex functions.
Paper Structure (5 sections, 12 theorems, 143 equations)

This paper contains 5 sections, 12 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Kernel and zero-product property of the subgradient graphical derivative of convex functions
  4. Isolated Calmness for Variational Convex Systems
  5. Conclusion

Key Result

Lemma 3.1

Let $F:\mathbb{X}\rightrightarrows\mathbb{Y}$ be a set-valued mapping whose graph is locally closed around $(\bar{x},\bar{y})\in \hbox{\rm gph}\, F$. Then we have Moreover, if $F$ is metrically subregular at $\bar{x}$ for $\bar{y}$ with some modulus $\kappa>0$, then Consequently, if the proper l.s.c. convex function $\varphi:\mathbb{X}\to \overline{\mathbb{R}}$ satisfies the quadratic growth con

Theorems & Definitions (26)

  • Lemma 3.1: Kernel of graphical derivative
  • proof
  • Lemma 3.2
  • Theorem 3.3: Kernels of the subgradient graphical derivative and the second subderivative
  • proof
  • Definition 3.4: Primal-dual quadratic growth condition
  • Remark 3.5
  • Theorem 3.6: Zero-product property of subgradient graphical derivative
  • proof
  • Theorem 4.1: Sufficient conditions for the isolated calmness of solution mapping
  • ...and 16 more