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Tight error bounds for log-determinant cones without constraint qualifications

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

TL;DR

This work provides tight error bounds for the log-determinant cone $\mathcal{K}_{\mathrm{logdet}}$ in conic feasibility problems without requiring constraint qualifications. By leveraging the one-step facial residual function framework and a precise classification of the cone's facial structure, the authors derive explicit Hölderian, entropic, and log-type bounds associated with distinct faces ($\mathcal{F}_{\mathrm{d}}$, $\mathcal{F}_{\#}$, $\mathcal{F}_{\infty}$, $\mathcal{F}_{\mathrm{r}}$), and prove their sharpness. The results extend the error-bound paradigm to a high-dimensional, non-polyhedral setting, revealing non-Hölder behavior in certain regimes and providing robust tools for analyzing interior-point and related algorithms on log-determinant-constrained problems. Overall, the paper advances theoretical understanding of feasibility stability for logdet-cone problems and informs algorithm design by clarifying when strong error bounds can be expected under minimal assumptions.

Abstract

In this paper, without requiring any constraint qualifications, we establish tight error bounds for the log-determinant cone, which is the closure of the hypograph of the perspective function of the log-determinant function. This error bound is obtained using the recently developed framework based on one-step facial residual functions.

Tight error bounds for log-determinant cones without constraint qualifications

TL;DR

This work provides tight error bounds for the log-determinant cone in conic feasibility problems without requiring constraint qualifications. By leveraging the one-step facial residual function framework and a precise classification of the cone's facial structure, the authors derive explicit Hölderian, entropic, and log-type bounds associated with distinct faces (, , , ), and prove their sharpness. The results extend the error-bound paradigm to a high-dimensional, non-polyhedral setting, revealing non-Hölder behavior in certain regimes and providing robust tools for analyzing interior-point and related algorithms on log-determinant-constrained problems. Overall, the paper advances theoretical understanding of feasibility stability for logdet-cone problems and informs algorithm design by clarifying when strong error bounds can be expected under minimal assumptions.

Abstract

In this paper, without requiring any constraint qualifications, we establish tight error bounds for the log-determinant cone, which is the closure of the hypograph of the perspective function of the log-determinant function. This error bound is obtained using the recently developed framework based on one-step facial residual functions.
Paper Structure (12 sections, 17 theorems, 148 equations)

This paper contains 12 sections, 17 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Matrices
  4. Error bounds for conic feasibility problems
  5. Error bounds for the log-determinant cones
  6. Facial structure
  7. One-step facial residual functions
  8. $\mathcal{F}_{{\rm d}}$: the unique $({\rm sd}(d)+1)$-dimensional faces
  9. $\mathcal{F}_{\#}$: the family of $d-rank(n_Z)+1$-dimensional faces
  10. $\mathcal{F}_{\infty}$: the exceptional 1-dimensional face
  11. $\mathcal{F}_{{\rm r}}$: the family of 1-dimensional faces
  12. Error bounds

Key Result

Lemma 2.1

Let $Z \in \mathcal{S}_{+}^d \setminus \{\bm{0}\}$. Then for any $\eta > 0$, there exists $C>0$ so that

Theorems & Definitions (37)

  • Lemma 2.1
  • proof
  • Definition 2.2: Error bounds LL22Pang97
  • Definition 2.3: One-step facial residual function ($\mathds{1}$-FRF)
  • Lemma 2.4: Formula of $\| \bm{w} - \bm{u} \|$
  • proof
  • Proposition 2.5: Error bound for positive semidefinite cones
  • proof
  • Proposition 3.1: Facial structure of $\mathcal{K}_{{\rm logdet}}$
  • proof
  • ...and 27 more