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Math-2306 · mathematics · Analysis

Polynomial-time lifted LMI representations of convex cones

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Characterize full-dimensional pointed closed convex cones that admit lifted linear-matrix-inequality representations with polynomial-time complexity. Is this efficiently representable class strictly larger than the class of hyperbolicity cones?

Formulation references: [S1], [S2].

Short English statement

Characterize full-dimensional pointed closed convex cones that admit lifted linear-matrix-inequality representations with polynomial-time complexity. Is this efficiently representable class strictly larger than the class of hyperbolicity cones?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2306-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2306

The exact lift gives one explicit positive example, and the nonliftable construction defeats a universal hyperbolicity inclusion. Neither result classifies all cones or gives the requested general polynomial-time representation criterion.

Polynomial-time lifted LMI representations of convex cones

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 095; 2026-10-05), the asserted result is that the explicit twenty-three-variable nonspectrahedral cone nonetheless has an exact lift with a size-$100$ pencil and $307$ auxiliary variables, including singular boundary points; In manuscript 2 (family 095; 2026-10-05), the asserted result is that some hyperbolicity cones remain outside the class of projections of finite real semidefinite systems, whatever real coefficients and auxiliary variables are used.

Mathematical statement of the claimed result

Manuscript 1.
For cyclic $i\in\{1,2,3\}$ set $Q(y)_{ii}=y_i^2+y_{i+1}^2$, $Q(y)_{ij}=-y_i y_j$ otherwise, and \[ \Phi_y\begin{pmatrix}a&r^{\mathsf T}\\r&T\end{pmatrix} =\begin{pmatrix}\operatorname{tr}(Q(y)T)&-r^{\mathsf T}Q(y)\\-Q(y)r&aQ(y)\end{pmatrix}, \quad p(X,Z,y)=\det((\det X)Z-\Phi_y(\operatorname{adj}X)). \] Let $K$ be the closed hyperbolicity cone of $p$ in direction $(I_4,I_4,0)$ on the real $23$-dimensional space of symmetric $X,Z$ and $y\in\mathbb R^3$. The claim gives fixed real symmetric $100\times100$ matrices $A_1,\ldots,A_{23},B_1,\ldots,B_{307}$ with \[ x\in K\ \Longleftrightarrow\ \exists v\in\mathbb R^{307}:\quad \sum_{\nu=1}^{23}x_{\nu} A_{\nu}+\sum_{j=1}^{307}v_jB_j\succeq0. \] Coordinates list upper-triangular entries of $X$ then $Z$, followed by $y$. Singular boundary points are included. Auxiliary variables make this a lift; it does not assert direct spectrahedrality.
Manuscript 2.
The existence claim is a real homogeneous polynomial $p$ and a direction $e$ with $p(e)>0$ such that every univariate $t\mapsto p(te-x)$ is real-rooted, but no finite semidefinite lift represents the resulting closed hyperbolicity cone $K$. Precisely, \[ \nexists N,r<\infty,\ A_i,B_j\in\operatorname{Sym}_N(\mathbb R):\quad K=\{x:\exists y\in\mathbb R^r,\ A_0+\sum_ix_iA_i+\sum_jy_jB_j\succeq0\}. \] This excludes representations even with auxiliary variables, and is stronger than excluding a direct linear matrix inequality.

Outline of the argument

For manuscript 1 (family 095; 2026-10-05), the three-variable parameter is covered by rotating two-dimensional planes. Each angular slice has a trigonometric Gram factorization and an exact pencil representation; a finite matching identity combines these slices into the full homogeneous lift.

For manuscript 2 (family 095; 2026-10-05), a positive semidefinite Hankel seed with controlled multiplication kernel produces finitely many inverse-limit directions. Separation from Hankel images yields a polynomial patch whose moment deformation contradicts analytic data that any semidefinite lift would require.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 095; 2026-10-05), manuscript 2 (family 095; 2026-10-05).

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

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