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Normalizations of factorizations over convex cones and their effects on extension complexity

Adam Brown, Kanstantsin Pashkovich, Levent Tunçel

TL;DR

The paper investigates when factorizations over a regular convex cone $\mathcal{C}$ (and its dual $\mathcal{C}^\star$) can be normalized, i.e., when factorization vectors can be rescaled with a cone-dependent constant $f_{\mathcal{C}}$ without increasing the objective inner products. It shows that, if normalization is possible, it can be realized via automorphisms of the cone, and it establishes precise bounds and constructions for symmetric and homogeneous cones using self-scaled barriers and Carathéodory-type arguments; it also provides counterexamples in certain hyperbolic cones. The paper then connects normalization to extension complexity, using nets and rounding techniques to argue that a cone with polynomial $f_{\mathcal{C}}$ and dimension cannot express all $0/1$ polytopes, yielding Rothvoss-style lower bounds for polytopes relative to $\mathcal{C}$ and extending to cyclic polytopes. Overall, it broadens the normalization toolkit beyond PSD and nonnegative orthant cones, clarifying when normalization helps or fails and its impact on polyhedral representations. $f_{\mathcal{C}}$ and Carathéodory numbers play central roles in quantifying these effects, with the Nesterov–Todd scaling providing a concrete realization in symmetric cones.

Abstract

Factorizations over cones and their duals play central roles for many areas of mathematics and computer science. One of the reasons behind this is the ability to find a representation for various objects using a well-structured family of cones, where the representation is captured by the factorizations over these cones. Several major questions about factorizations over cones remain open even for such well-structured families of cones as non-negative orthants and positive semidefinite cones. Having said that, we possess a far better understanding of factorizations over non-negative orthants and positive semidefinite cones than over other families of cones. One of the key properties that led to this better understanding is the ability to normalize factorizations, i.e., to guarantee that the norms of the vectors involved in the factorizations are bounded in terms of an input and in terms of a constant dependent on the given cone. Our work aims at understanding which cones guarantee that factorizations over them can be normalized, and how this effects extension complexity of polytopes over such cones.

Normalizations of factorizations over convex cones and their effects on extension complexity

TL;DR

The paper investigates when factorizations over a regular convex cone (and its dual ) can be normalized, i.e., when factorization vectors can be rescaled with a cone-dependent constant without increasing the objective inner products. It shows that, if normalization is possible, it can be realized via automorphisms of the cone, and it establishes precise bounds and constructions for symmetric and homogeneous cones using self-scaled barriers and Carathéodory-type arguments; it also provides counterexamples in certain hyperbolic cones. The paper then connects normalization to extension complexity, using nets and rounding techniques to argue that a cone with polynomial and dimension cannot express all polytopes, yielding Rothvoss-style lower bounds for polytopes relative to and extending to cyclic polytopes. Overall, it broadens the normalization toolkit beyond PSD and nonnegative orthant cones, clarifying when normalization helps or fails and its impact on polyhedral representations. and Carathéodory numbers play central roles in quantifying these effects, with the Nesterov–Todd scaling providing a concrete realization in symmetric cones.

Abstract

Factorizations over cones and their duals play central roles for many areas of mathematics and computer science. One of the reasons behind this is the ability to find a representation for various objects using a well-structured family of cones, where the representation is captured by the factorizations over these cones. Several major questions about factorizations over cones remain open even for such well-structured families of cones as non-negative orthants and positive semidefinite cones. Having said that, we possess a far better understanding of factorizations over non-negative orthants and positive semidefinite cones than over other families of cones. One of the key properties that led to this better understanding is the ability to normalize factorizations, i.e., to guarantee that the norms of the vectors involved in the factorizations are bounded in terms of an input and in terms of a constant dependent on the given cone. Our work aims at understanding which cones guarantee that factorizations over them can be normalized, and how this effects extension complexity of polytopes over such cones.
Paper Structure (10 sections, 12 theorems, 46 equations)

This paper contains 10 sections, 12 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Problem of Normalizing Factorizations
  3. Normalization and Automorphisms
  4. Normalization of Cones
  5. Symmetric Cones
  6. Homogeneous Cones
  7. Rescaling of Cones and Extension Complexity
  8. Relationship to Polytopes
  9. Compact Encoding
  10. Relationship to Extension Complexity

Key Result

Lemma 1

Let $\mathcal{C}$ be a regular convex cone and let $\mathcal{C}^*$ be its dual. Let $\mathcal{A}$ and $\mathcal{B}$ be subsets of $\mathcal{C}$ and $\mathcal{C}^*$ respectively. If $\mathcal{A}$ and $\mathcal{B}$ both span $\mathbb{R}^n$ and there exist two sets $\widetilde{\mathcal{A}}\subseteq \ma then $g$ and $q$ are invertible linear maps. Moreover, $q$ is the inverse-adjoint of $g$.

Theorems & Definitions (23)

  • Lemma 1
  • proof
  • Theorem 2
  • proof : Proof of Theorem \ref{['thm:rescaling_automorphism']}
  • Remark 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Theorem 6
  • ...and 13 more