Where hyperbolic optimization goes beyond semidefinite representations

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2026-10-14

Math Digest

Family 095 · Convex and metric geometry
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-14

1. The question

Semidefinite programming describes convex constraints by requiring a symmetric matrix, whose entries depend linearly on the variables, to be positive semidefinite. Hyperbolic programming offers another broad source of convex constraints, defined by polynomials with a special real-root property.

Are these two settings ultimately the same after introducing enough extra variables? This family claims that they are not, but its three manuscripts must be distinguished carefully.

M2 constructs a specific cone in twenty-three variables with no direct semidefinite representation. M3 then gives that very cone an exact representation using auxiliary variables. M1 makes a stronger claim about a different, higher-dimensional construction: some hyperbolicity cones have no finite semidefinite lift whatsoever. The positive lift in M3 and the negative theorem in M1 concern different cones and answer different questions.

2. Background and essential definitions

A real symmetric matrix $A$ is positive semidefinite, written $A\succeq0$ , when $z^{\mathsf T}Az\geq0$ for every vector $z$ . A spectrahedron is a set described by one finite linear matrix inequality,

A_0+x_1A_1+\cdots+x_dA_d\succeq0.

The matrices are fixed; the coordinates $x_i$ are the unknowns. Several inequalities can be combined into one block-diagonal matrix.

A spectrahedral shadow, or set with a finite semidefinite lift, allows auxiliary variables:

S=\left\{x:\exists u\in\mathbb R^a, A_0+\sum_i x_iA_i+\sum_j u_jB_j\succeq0\right\}.

Geometrically, this projects a higher-dimensional spectrahedron onto the original coordinates. Equality with the projected set is exact; taking its closure is an additional operation not included in the definition.

A homogeneous polynomial $p$ is hyperbolic in direction $e$ if $p(e)>0$ and, for every $x$ , the one-variable polynomial $t\mapsto p(te-x)$ has only real roots. Its closed hyperbolicity cone consists of those $x$ for which all these roots are nonnegative.

The determinant on symmetric matrices is the basic example: $p(tI-X)=\det(tI-X)$ has the eigenvalues of $X$ as roots, and the cone is exactly the positive semidefinite cone. Gårding’s theory shows that hyperbolicity cones are convex in general, even without a known matrix representation.

3. Development history and previous partial results

The original Lax conjecture concerned homogeneous hyperbolic polynomials in three variables. Helton and Vinnikov’s determinantal theorem, together with the equivalence explained by Lewis, Parrilo and Ramana, gave a positive solution. This success motivated higher-dimensional representation questions.

Two formulations then diverged. A particular polynomial may fail to have a determinantal representation, while its cone could still be described by a different determinant. Brändén gave a real-zero polynomial for which no positive integral power has a definite determinantal representation. Kummer later showed that a related specialized Vámos polynomial nevertheless has a spectrahedral cone. A polynomial obstruction alone therefore does not settle the geometric question.

The generalized Lax conjecture proposes that every hyperbolicity cone is spectrahedral. The weaker Projected Lax Conjecture asks only for a finite semidefinite lift. Netzer and Sanyal’s Smooth hyperbolicity cones are spectrahedral shadows (2015) established liftability under a smoothness condition at every nonzero boundary point of the defining polynomial.

Other positive results cover elementary symmetric cones, particular low-degree cases, and stricter regularity hypotheses. Scheiderer’s smooth-cone result gives second-order-cone lifts under a Nash-smooth boundary condition. Such results leave singular cones outside their assumptions.

For general convex semialgebraic sets, Scheiderer had already developed obstructions to semidefinite lifts using local positivity and sums of squares. M1 adapts that broad strategy to hyperbolicity cones. M2 uses the classical Choi–Lam nonnegative biquadratic form for a direct-representation obstruction, while M3 shows concretely how a lift can bypass that obstruction.

4. What the new manuscripts claim

A cone with no finite lift

M1 states that there exists a real homogeneous hyperbolic polynomial whose closed hyperbolicity cone is not a spectrahedral shadow. It excludes every finite matrix size, every finite number of auxiliary variables and arbitrary real coefficients in an affine pencil.

This is a nonexistence theorem for exact finite representations, not merely a lower bound saying that a representation must be large. If correct, it disproves the Projected Lax Conjecture and consequently the stronger generalized Lax conjecture.

The construction is high-dimensional. It is distinct from the explicit twenty-three-variable example in the other two manuscripts.

A smaller explicit obstruction to direct representation

M2 defines symmetric $4\times4$ matrix variables $X,Z$ and a vector $y\in\mathbb R^3$ , for a total of twenty-three real coordinates. A quadratic matrix map derived from the Choi–Lam form enters a determinant formula producing a degree-twenty hyperbolic polynomial.

The manuscript claims that its cone has no finite homogeneous symmetric linear matrix representation in those original coordinates. It then removes a determinant factor, obtaining a degree-sixteen homogeneous polynomial with the same hyperbolicity cone. Neither the variable count nor the degree is claimed minimal.

For a cone containing zero, a direct affine representation can be homogenized by scaling along rays, so the obstruction also rules out direct affine pencils. It does not, by itself, rule out auxiliary variables.

An exact lift of that same explicit cone

M3 states that the cone from M2 has a homogeneous real symmetric lift with a matrix of size $100$ and $307$ auxiliary variables:

x\in K\quad\Longleftrightarrow\quad \exists v\in\mathbb R^{307}, \sum_{\nu=1}^{23}x_\nu A_\nu+ \sum_{j=1}^{307}v_jB_j\succeq0.

The equivalence holds on the entire closed cone, including points where $X$ is singular. The stated sizes are upper bounds, not optimality claims. M3 uses M2’s nonrepresentation theorem to highlight the distinction, but presents the lift construction itself as self-contained.

5. Why the advance matters

Importance: 5/5. The strongest claim would place a fundamental limit on representing hyperbolic optimization exactly by finite semidefinite programs. Auxiliary variables can greatly expand the expressive power of matrix inequalities, so excluding every finite lift is a substantial structural statement.

The explicit pair M2–M3 is also instructive. It shows why one must ask whether an obstruction concerns a particular polynomial, a direct representation of its cone, or every possible lifted representation. These are successively stronger demands, and failure at one level need not imply failure at the next.

The result does not say hyperbolic optimization is computationally hopeless. Hyperbolicity cones have their own barrier theory and optimization methods. Nor does absence of an exact finite lift prevent semidefinite approximations, converging hierarchies or useful representations of particular slices.

The score reflects the broad conjectural separation and the concrete illustration of representation power. It does not claim a minimal-dimensional counterexample or an algorithmic lower bound for every optimization task over the constructed cones.

6. The main idea

Start with positivity that resists a square decomposition

M2 begins with a nonnegative biquadratic form $b(z,y)=z^{\mathsf T}Q(y)z$ . It is quadratic in each group of variables. The selected Choi–Lam form has a rigid property: if the square of a bilinear form is everywhere bounded above by $b$ , that bilinear form must vanish.

A determinant construction turns $Q(y)\succeq0$ into a hyperbolic polynomial. For each fixed $y$ , a Gram factorization gives a symmetric matrix whose eigenvalues describe the roots along the hyperbolicity direction. The factorization is chosen separately at each $y$ ; it is not already a linear pencil in all variables jointly.

Assuming an arbitrary direct pencil existed, rescaling and rank-one limits would produce an operator-norm identity for its kernel projections. Differentiating that identity yields bilinear squares dominated by $b$ . The special positivity property forces those terms to vanish, contradicting the identity. The argument targets any hypothetical pencil rather than comparing one determinant with the defining polynomial.

A lift can recover the explicit cone through angular slices

M3 slices the $y$ -space by planes parameterized by an angle. On each plane, the quadratic matrix has a factorization linear in the two plane coordinates and trigonometric in the angle. This gives a size-twenty pencil for each slice.

The continuously varying angle is replaced by a finite linear functional on trigonometric polynomials. Requiring nonnegativity on a bounded-degree family of quadratic tests becomes one finite matrix inequality. Evaluation at a single angle immediately supplies feasible lift data for points of the cone.

The converse is harder: a positive functional need not come from evaluation at an angle or from an actual measure. A degree-two matching identity provides enough tests to recover the slice inequalities for every feasible functional. At singular $X$ , the same tests enforce the necessary range conditions, which is why the result represents the closed boundary exactly rather than only the interior or a closure of a projection.

Exclude all lifts with a different local obstruction

M1 builds a different positive quadratic matrix map using finite-dimensional spaces of polynomials and Hankel moment matrices. Such a matrix records a linear functional applied to products of lower-degree polynomials. Positivity on squares need not mean that the functional is integration against a measure.

The construction produces an affine section and projection of a hyperbolicity cone with a carefully designed positivity obstruction. A finite positive moment functional applied to a small polynomial patch creates a violation of order four in a scale parameter.

A hypothetical finite lift would supply local analytic lift coordinates and Gram factors. Applying the same functional to their Taylor expansions nearly preserves feasibility. The manuscript then repairs the much smaller residual error on the common matrix support, without assuming strict feasibility. The repaired feasible point would remain too close to the violating patch to eliminate the order-four discrepancy.

This contradiction is the claimed obstruction to every finite lift. The finite-field and moment calculations described in the source are components of its construction; they have not been rerun as proof checks for this digest.

7. Scope, limitations, and open questions

The three conclusions should be read together: the explicit twenty-three-variable cone has no direct pencil but does have a lift; the no-lift theorem concerns another construction. M3 therefore refines the understanding of M2 rather than overturning its direct-representation claim.

All representations are finite and exact over the reals. Arbitrarily many but finitely many auxiliary variables are allowed in M1’s negative theorem. Approximation by semidefinite sets and infinite limiting constructions are outside that conclusion.

Smooth or strictly hyperbolic positive results have hypotheses that the counterexamples need not satisfy. Singular boundary behavior is central to the obstruction mechanisms. The sources do not claim the smallest possible degree, dimension or lift size.

M2 discusses a conflicting general positive claim in an appendix. The digest does not adjudicate that argument; it reports the present manuscripts’ stated results and keeps the logical distinction between direct and projected representations explicit.

8. Sources and further reading

M1 is the source for the no-finite-lift theorem. M2 gives the explicit polynomial, degree reduction and direct-pencil obstruction. M3 gives the size-100 lift and its exact treatment of singular boundary points.

Helton–Vinnikov’s Linear matrix inequality representation of sets (2007) and Lewis–Parrilo–Ramana’s The Lax conjecture is true (2005) provide the three-variable background. Choi–Lam’s Extremal positive semidefinite forms (1977), Netzer–Sanyal’s smooth-cone theorem and Scheiderer’s work supply the main positivity and representation precedents cited by the family.

Source manuscripts

M1. OpenAI, Hyperbolicity Cones Without Semidefinite Lifts (October 5, 2026).
M2. OpenAI, A nonspectrahedral hyperbolicity cone (September 24, 2026).
M3. OpenAI, An Exact Semidefinite Lift of a Nonspectrahedral Hyperbolicity Cone (October 5, 2026).

Sources consulted on 14 October 2026 at repository revision 1581d4c. Download the manuscript citations.