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Monotonicity in Quadratically Regularized Linear Programs

Alberto González-Sanz, Marcel Nutz, Andrés Riveros Valdevenito

TL;DR

The paper studies monotonicity of the solution curve to quadratically regularized linear programs over polytopes, revealing that monotonicity is exactly characterized by two geometric conditions on the polytope: the global minimum-norm point must lie in the relative interior, and the minimum-norm point of every affine face must lie in that face. This leads to a sharp dichotomy: the unit simplex is monotone for all costs, while the Birkhoff polytope is non-monotone in higher dimensions (specifically when the dimension $d\ge25$, i.e., $N\ge5$ for transport problems). The authors provide a rigorous abstract result (linking invariance to face-wise projection properties), prove a detailed monotonicity breakdown for the Birkhoff polytope, and give a constructive Erdős-type application showing that certain doubly stochastic matrices arise as minimum-norm projections onto centered faces, with all such matrices necessarily having rational entries. Together, these results explain why empirical monotonicity of optimal transport supports may fail in high-dimensional settings and offer a precise geometric lens for analyzing regularized LPs and related combinatorial problems.

Abstract

In optimal transport, quadratic regularization is a sparse alternative to entropic regularization: the solution measure tends to have small support. Computational experience suggests that the support decreases monotonically to the unregularized counterpart as the regularization parameter is relaxed. We find it useful to investigate this monotonicity more abstractly for linear programs over polytopes, regularized with the squared norm. Here, monotonicity can be stated as an invariance property of the curve mapping the regularization parameter to the solution: once the curve enters a face of the polytope, does it remain in that face forever? We show that this invariance is equivalent to a geometric property of the polytope, namely that each face contains the minimum norm point of its affine hull. Returning to the optimal transport problem and its associated Birkhoff polytope, we verify this property for low dimensions, but show that it fails for marginals with five or more point masses. As a consequence, the conjectured monotonicity of the support fails in general, even if experiments suggest that monotonicity holds for many cost matrices. Separately, we apply our geometric point of view to a problem of Erdős, namely to characterize the doubly stochastic matrices whose maximal trace equals their squared norm.

Monotonicity in Quadratically Regularized Linear Programs

TL;DR

The paper studies monotonicity of the solution curve to quadratically regularized linear programs over polytopes, revealing that monotonicity is exactly characterized by two geometric conditions on the polytope: the global minimum-norm point must lie in the relative interior, and the minimum-norm point of every affine face must lie in that face. This leads to a sharp dichotomy: the unit simplex is monotone for all costs, while the Birkhoff polytope is non-monotone in higher dimensions (specifically when the dimension , i.e., for transport problems). The authors provide a rigorous abstract result (linking invariance to face-wise projection properties), prove a detailed monotonicity breakdown for the Birkhoff polytope, and give a constructive Erdős-type application showing that certain doubly stochastic matrices arise as minimum-norm projections onto centered faces, with all such matrices necessarily having rational entries. Together, these results explain why empirical monotonicity of optimal transport supports may fail in high-dimensional settings and offer a precise geometric lens for analyzing regularized LPs and related combinatorial problems.

Abstract

In optimal transport, quadratic regularization is a sparse alternative to entropic regularization: the solution measure tends to have small support. Computational experience suggests that the support decreases monotonically to the unregularized counterpart as the regularization parameter is relaxed. We find it useful to investigate this monotonicity more abstractly for linear programs over polytopes, regularized with the squared norm. Here, monotonicity can be stated as an invariance property of the curve mapping the regularization parameter to the solution: once the curve enters a face of the polytope, does it remain in that face forever? We show that this invariance is equivalent to a geometric property of the polytope, namely that each face contains the minimum norm point of its affine hull. Returning to the optimal transport problem and its associated Birkhoff polytope, we verify this property for low dimensions, but show that it fails for marginals with five or more point masses. As a consequence, the conjectured monotonicity of the support fails in general, even if experiments suggest that monotonicity holds for many cost matrices. Separately, we apply our geometric point of view to a problem of Erdős, namely to characterize the doubly stochastic matrices whose maximal trace equals their squared norm.
Paper Structure (11 sections, 19 theorems, 63 equations, 2 figures)

This paper contains 11 sections, 19 theorems, 63 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Summary of Results
  4. Preliminaries
  5. Abstract Result
  6. Proof of \ref{['th:main']}
  7. Applications
  8. Soft-Min
  9. Optimal Transport
  10. Proof of \ref{['le:BirkhoffSptMonEquiv']} and \ref{['th:BirkhoffMon']}
  11. On a Problem of Erdős

Key Result

Lemma 2.1

Denote $\mathcal{P}(\delta):=\{x\in\mathcal{P}:\, \|x\|\leq\delta\}$ for $\delta\geq0$ and Then For each $\delta\in \overline{D }=[\delta_{\min},\delta_{\max}]$, the problem Moreover, $[\delta_{\min},\delta_{\max}]\ni \delta\mapsto x^{\delta}$ is continuous and $\|x^{\delta}\|=\delta$. In particular, $x^{\delta_{\min}}$ is the singleton $\mathcal{P}(\delta_{\min})$, and $x^{\delta_{\max}}$ is t

Figures (2)

  • Figure 1: A non-monotone (left) and a monotone (right) polytope with cost $c$ (red). The curve $[\delta_{\min},\delta_{\max}]\ni \delta\mapsto x^{\delta}$ follows the blue arrows and has the invariance property only in the right example.
  • Figure 2: Probability mass $\gamma_{\eta}(X_{1},Y_{1})$ plotted against $1/(2\eta)$, showing that $(X_{1},Y_{1})$ is in the support for small and large values of $\eta$ but outside for an intermediate interval.

Theorems & Definitions (39)

  • Lemma 2.1: Eulerian Formulation
  • proof
  • Lemma 2.2: Projection
  • Lemma 2.3: Lagrangian Formulation
  • proof
  • Lemma 2.4: Euler $\leftrightarrow$ Lagrange
  • proof
  • Definition 3.1
  • Theorem 3.2
  • Lemma 3.3
  • ...and 29 more