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$K-$Lorentzian Polynomials, Semipositive Cones, and Cone-Stable EVI Systems

Papri Dey

Abstract

Lorentzian and completely log-concave polynomials have recently emerged as a unifying framework for negative dependence, log-concavity, and convexity in combinatorics and probability. We extend this theory to variational analysis and cone-constrained dynamics by studying $K$-Lorentzian and $K$-completely log-concave polynomials over a proper convex cone $K\subset\mathbb{R}^n$. For a $K$-Lorentzian form $f$ and $v\in\operatorname{int}K$, we define an open cone $K^\circ(f,v)$ and a closed cone $K(f,v)$ via directional derivatives along $v$, recovering the usual hyperbolicity cone when $f$ is hyperbolic. We prove that $K^\circ(f,v)$ is a proper cone and equals $\operatorname{int}K(f,v)$. If $f$ is $K(f,v)$-Lorentzian, then $K(f,v)$ is convex and maximal among convex cones on which $f$ is Lorentzian. Using the Rayleigh matrix $M_f(x)=\nabla f(x)\nabla f(x)^T - f(x)\nabla^2 f(x)$, we obtain cone-restricted Rayleigh inequalities and show that two-direction Rayleigh inequalities on $K$ are equivalent to an acuteness condition for the bilinear form $v^T M_f(x) w$. This yields a cone-restricted negative-dependence interpretation linking the curvature of $\log f$ to covariance properties of associated Gibbs measures. For determinantal generating polynomials, we identify the intersection of the hyperbolicity cone with the nonnegative orthant as the classical semipositive cone, and we extend this construction to general proper cones via $K$-semipositive cones. Finally, for linear evolution variational inequality (LEVI) systems, we show that if $q(x)=x^T A x$ is (strictly) $K$-Lorentzian, then $A$ is (strictly) $K$-copositive and yields Lyapunov (semi-)stability on $K$, giving new Lyapunov criteria for cone-constrained dynamics.

$K-$Lorentzian Polynomials, Semipositive Cones, and Cone-Stable EVI Systems

Abstract

Lorentzian and completely log-concave polynomials have recently emerged as a unifying framework for negative dependence, log-concavity, and convexity in combinatorics and probability. We extend this theory to variational analysis and cone-constrained dynamics by studying -Lorentzian and -completely log-concave polynomials over a proper convex cone . For a -Lorentzian form and , we define an open cone and a closed cone via directional derivatives along , recovering the usual hyperbolicity cone when is hyperbolic. We prove that is a proper cone and equals . If is -Lorentzian, then is convex and maximal among convex cones on which is Lorentzian. Using the Rayleigh matrix , we obtain cone-restricted Rayleigh inequalities and show that two-direction Rayleigh inequalities on are equivalent to an acuteness condition for the bilinear form . This yields a cone-restricted negative-dependence interpretation linking the curvature of to covariance properties of associated Gibbs measures. For determinantal generating polynomials, we identify the intersection of the hyperbolicity cone with the nonnegative orthant as the classical semipositive cone, and we extend this construction to general proper cones via -semipositive cones. Finally, for linear evolution variational inequality (LEVI) systems, we show that if is (strictly) -Lorentzian, then is (strictly) -copositive and yields Lyapunov (semi-)stability on , giving new Lyapunov criteria for cone-constrained dynamics.
Paper Structure (6 sections, 26 theorems, 130 equations, 4 figures)

This paper contains 6 sections, 26 theorems, 130 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Cone associated with -Lorentzian polynomials
  3. Rayleigh Differences and ${\mathcal{K}}$-Semipositive Cones
  4. Semipositive cones and hyperbolic barriers in conic optimization
  5. Hyperbolic generating polynomials
  6. Stability Analysis via -Lorentzian polynomials

Key Result

Proposition 2.3

Let $f(x)$ be a nonzero ${\mathcal{K}}$-CLC over a proper convex cone ${\mathcal{K}}$. Then for any $x,v \in \mathop{\mathrm{int}}\nolimits {\mathcal{K}}$, the coefficients of $f(x+tv)$ are positive.

Figures (4)

  • Figure 1: ${\mathcal{K}}(f,v)\cap\{x+y+z=1\}$. White indicates feasibility.
  • Figure 2: Compact base of ${\mathcal{K}}_A=\Lambda_{+}(f_A,1)\cap{\mathbb R}^4_{\ge0}$ at $x_4=1$.
  • Figure 3: ${\mathcal{K}}(f,v)\cap{\mathbb R}^3_{\ge0}$.
  • Figure 4: Stability of the origin w.r.t. ${\mathcal{K}}$.

Theorems & Definitions (66)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Remark 2.6
  • Lemma 2.7
  • ...and 56 more