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Optimal error bounds in the absence of constraint qualifications with applications to the $p$-cones and beyond

Scott B. Lindstrom, Bruno F. Lourenço, Ting Kei Pong

TL;DR

The Holderian error bounds are obtained under the framework of facial residual functions, and the framework is expanded by establishing for general cones an optimality criterion under which the resulting error bound must be tight.

Abstract

We prove tight Hölderian error bounds for all $p$-cones. Surprisingly, the exponents differ in several ways from those that have been previously conjectured; moreover, they illuminate $p$-cones as a curious example of a class of objects that possess properties in 3 dimensions that they do not in 4 or more. Using our error bounds, we analyse least squares problems with $p$-norm regularization, where our results enable us to compute the corresponding KL exponents for previously inaccessible values of $p$. Another application is a (relatively) simple proof that most $p$-cones are neither self-dual nor homogeneous. Our error bounds are obtained under the framework of facial residual functions, and we expand it by establishing for general cones an optimality criterion under which the resulting error bound must be tight.

Optimal error bounds in the absence of constraint qualifications with applications to the $p$-cones and beyond

TL;DR

The Holderian error bounds are obtained under the framework of facial residual functions, and the framework is expanded by establishing for general cones an optimality criterion under which the resulting error bound must be tight.

Abstract

We prove tight Hölderian error bounds for all -cones. Surprisingly, the exponents differ in several ways from those that have been previously conjectured; moreover, they illuminate -cones as a curious example of a class of objects that possess properties in 3 dimensions that they do not in 4 or more. Using our error bounds, we analyse least squares problems with -norm regularization, where our results enable us to compute the corresponding KL exponents for previously inaccessible values of . Another application is a (relatively) simple proof that most -cones are neither self-dual nor homogeneous. Our error bounds are obtained under the framework of facial residual functions, and we expand it by establishing for general cones an optimality criterion under which the resulting error bound must be tight.

Paper Structure

This paper contains 13 sections, 25 theorems, 133 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Facial residual functions and error bounds
  4. Computing $\mathds{1}$-FRFs
  5. Best error bounds
  6. Error bounds for $p$-cones
  7. Facial structure of ${\mathcal{K}_{p}^{n+1}}$ for $n\ge 2$ and $p \in (1,\infty)$
  8. One-step facial residual functions for the faces of ${\mathcal{K}_{p}^{n+1}}$ when $n\ge 2$ and $p \in (1,\infty)$
  9. How $p$-cones of dimension 3 differ from those of dimension 4
  10. Error bounds
  11. Applications
  12. Least squares with $p$-norm regularization
  13. Self-duality and homogeneity of $p$-cones

Key Result

Proposition 2.1

If eq:feas satisfies the PPS condition, then a Lipschitzian error bound holds.

Figures (3)

  • Figure 1: The framework of Lemma \ref{['lem:facialresidualsbeta']}, Theorem \ref{['thm:1dfacesmain']}, and Lemma \ref{['lem:infratio']} is illustrated. The right image shows a 2D slice of the left image, where the slice is in the plane given by ${\rm span}\{v,w,u\}$.
  • Figure 2: Example \ref{['ex:3d4d']} is illustrated.
  • Figure 3: Example \ref{['ex:3d4d_3halves']} is illustrated.

Theorems & Definitions (52)

  • Definition 2.1: Lipschitzian and Hölderian error bounds
  • Proposition 2.1: Error bound under the PPS condition
  • Proposition 2.2: LiLoPo20
  • Definition 2.2: One-step facial residual function ($\mathds{1}$-FRF)
  • Theorem 2.1: LiLoPo20
  • Lemma 2.1
  • proof
  • Lemma 2.2: $\mathds{1}$-FRF for the zero face
  • proof
  • Remark 2.1: Geometric intuition for finding suitable $\mathds{1}$-FRF
  • ...and 42 more