bibkey: nishijimanakata2024generalizations
authors: M. Nishijima and K. Nakata
year: 2024
title: Generalizations of doubly nonnegative cones and their comparison
doi: 10.15807/jorsj.67.84
url: https://arxiv.org/abs/2204.12119
claim: "Conjecture 5.1 states that the NN-type generalized doubly nonnegative cone is contained in the BD-type cone for a direct product of a nonnegative orthant and second-order cones."
strata_touched:
  - D5/S3/QuadraticForms/ConicGDNN/Result
license: citation-only
triage: anchor

Generalized doubly nonnegative cone inclusion

Verified locator

DOI: 10.15807/jorsj.67.84

Source: https://arxiv.org/abs/2204.12119

Pinned version: https://arxiv.org/abs/2204.12119v4, Section 5.1, Conjecture 5.1. Journal publication: Journal of the Operations Research Society of Japan 67(3) (2024), 84–109.

The source definitions used here are the zeroth homogeneous sum of squares approximation in Theorem 3.5 with relaxation order zero, its dual in Definition 3.6, and the BD-type cone in Definition 3.7. Section 4.2 also considers products involving positive semidefinite cone factors; those factors are outside the delivered Lean claim.

Mathematical correspondence

For the product cone $K={\mathbb{R}}_{+}^p\times {\prod }_{j=1}^r{\mathcal{L}}_{k_j+1}$, the polynomial map is the Jordan square: orthant coordinates are squared, and a Lorentz block $(t,z)$ maps to $(t^2+\Vert z{\Vert }^2,2tz)$. The quartic sum of squares cone is the real convex hull of squares of homogeneous quadratic polynomials. The symmetric trace dual of its matrix pullback is the NN-type cone. The BD-type cone consists of positive semidefinite matrices sending every extreme ray generator of $K^{\ast }$ into $K$.

D5/S3/QuadraticForms/ConicGDNN/Result.result proves the inclusion. D5/S3/QuadraticForms/ConicGDNN/Duality.mulVec_mem_cone gives the stronger preservation of every cone vector, including zero spatial dimensions.

The statement and definitions were read in arXiv:2204.12119v4. Exhaustive literature completeness is ASSUMED-UNVERIFIED.