Spectrahedral Relaxations in Rigidly Convex Sets

• Spectrahedral relaxations are outer approximations of convex sets defined by real-zero polynomials using LMIs to capture key combinatorial structures.
• They are constructed from linear matrix polynomials whose structured eigenvector sequences yield explicit quantitative bounds on extreme roots of Eulerian polynomials.
• Numerical experiments demonstrate that optimized eigenvector choices produce exponentially tighter spectral bounds compared to classical univariate relaxations.

A spectrahedral relaxation is an outer approximation of a convex set, typically defined by polynomial inequalities, using a spectrahedron—i.e., the solution set of a Linear Matrix Inequality (LMI). In the context of Eulerian rigidly convex sets (RCSs), which are convex sets defined by real-zero multivariate Eulerian polynomials, spectrahedral relaxations provide a tractable approach for approximating these sets and deriving explicit quantitative bounds on associated quantities, such as extreme roots of univariate Eulerian polynomials occurring on the diagonal specialization of their multivariate counterparts.

1. Definitions and Theoretical Background

Spectrahedral relaxations in this context are constructed as follows. Given a real-zero (RZ) polynomial $p(x_1, \ldots, x_n)$, its rigidly convex set is defined as the closure of the connected component containing the origin in $\mathbb{R}^n \setminus V(p)$, where $V(p)$ is the zero set of $p$. When $p$ is a multivariate lifting of a univariate Eulerian polynomial, these rigidly convex sets exhibit rich combinatorial structure.

A spectrahedral relaxation to such a set is derived from a linear matrix polynomial (LMP) of the form

$M(x) = A_0 + x_1A_1 + \cdots + x_nA_n,$

where the $A_i$ are real symmetric matrices, and the associated spectrahedron is

$S(p) = \{ x \in \mathbb{R}^n : M(x) \succeq 0 \}.$

This is an outer approximation in the sense that the rigidly convex set defined by $p$ is contained in $S(p)$, but not generally equal to it.

Multivariate Eulerian polynomials used here are real-stable (hence real-zero upon suitable dehomogenization), and are constructed using stability-preserving operators and combinatorial refinements of descents in permutations. Their specialized diagonal evaluations recover the classical univariate Eulerian polynomials.

2. Methodology: Vector Linearization and Eigenvector Sequences

Earlier approaches to bounding the extreme roots of univariate Eulerian polynomials via spectrahedral relaxations relied on "linearizing" the determinant of the LMP along the diagonal (i.e., the line $x_1 = x_2 = \cdots = x_n = x$). Specifically, for given $x$, one would consider a quadratic form

$v^\top (A_0 + x A_1 + \cdots + x A_n) v \geq 0$

for some vector $v \in \mathbb{R}^{n+1}$. Solving for $x$ when the left-hand side equals zero yields a bound for the extreme (leftmost) root.

Earlier constructions ([ale1]) selected simple candidate vectors (e.g., all-ones or sign-alternating patterns), which resulted in improvement over classical bounds that shrank to zero for large $n$. The new approach exploits richer structure in the eigenvectors of the LMP to improve the linearization.

Through extensive numerical experimentation, the eigenvectors of the relevant LMPs (restricted to the diagonal) were analyzed. This led to the observation that, for even $n = 2m$, the approximate eigenvector entries have the form: $$\Bigl( y,\, \underbrace{-2^{m-3},\, -2^{m-4},\, \ldots,\, -2^0}_{m-2\ \text{terms}},\, 0,\, \tfrac{1}{2},\, \underbrace{1, 1, \ldots, 1}_m \Bigr)$$ as $n$ increases. This observation guided the construction of a "structured" vector for linearization: $$v^{(n)} = \bigl( y,\, (-2^{m-i})_{i=3}^{m},\, 0,\, \tfrac{1}{2},\, (1)_{i=1}^m \bigr),\quad n=2m,$$ where $y$ is a free parameter (often optimized for tightness).

3. Numerical Experiments and Quantitative Results

Having specified the sequence of approximating vectors, the impact on extreme root bounds was examined in detail. The prior linearization approach in [ale1] yielded improvement over best-known univariate bounds, but the difference between the new bound and the univariate bound decayed to zero as $n \to \infty$.

In contrast, using the numerically suggested and now analytically formalized vector sequence, the linearized bound for the diagonal spectral relaxation provides a gap that grows exponentially with $n$: $$\Delta_n \sim \frac{3}{8} \left(\frac{9}{8}\right)^m \quad \text{for } n=2m$$ Thus, as $n$ grows, the difference between the newly established spectral relaxation bound and prior bounds increases rapidly—unlike previous approaches where it vanished. The paper demonstrates this effect explicitly by computing, for increasing $n$, the leftmost roots certified by the new relaxation and showing the quantitative divergence from the univariate bound.

The essential mechanism behind this improvement is that the structured eigenvector sequence harnesses "multivariate information" that is lost when relying on simple, constant or sign-alternating patterns. The structure reflects the intrinsic combinatorics of the multivariate Eulerian lifting and captures additional moment information encoded by the higher-degree terms in the LMP.

4. Mathematical Formulation of the Approximating Eigenvector

The key formal mathematical result is the proposal and implementation of the approximating sequence for the eigenvector used in the linearization: $$v^{(n)} = \left( y,\, (-2^{m-i})_{i=3}^{m},\, 0,\, \frac{1}{2},\, (1)_{i=1}^m \right) \in \mathbb{R}^{n+1},\quad n = 2m$$ This vector is used to linearize the LMP $M(x)$ along the diagonal. The associated bound for the diagonal root (i.e., a lower bound for the leftmost root of the univariate Eulerian polynomial given by setting all variables equal) is then given by the solution $x^*$ to

$$v^{(n)\top} \left( M_0 + x^* M_1 \right) v^{(n)} = 0,$$

i.e.,

$$x^* = - \frac{v^{(n)\top} M_0 v^{(n)}}{v^{(n)\top} M_1 v^{(n)}}.$$

Here $M_0$ and $M_1$ are the constant and diagonal-coefficient matrices, respectively, from the LMP specialized to the diagonal case ($x_1 = x_2 = \cdots = x_n = x$).

The explicit computation of these quadratic forms is intricate but yields the exponential estimate for the improvement of the bound.

5. Implications and Directions for Further Research

The exponentially growing separation between the new multivariate spectrahedral relaxation bound and the best univariate relaxation suggests that substantial multivariate combinatorial information can be captured and leveraged via suitable eigenvector selection. This points toward the principle that, for families of structured real-zero polynomials (especially those with combinatorial or stability-theoretic origin), the choice of linearizing vector in a spectrahedral relaxation is critical for quantitative tightness.

Potential future directions include:

• Optimizing the free parameter $y$ in the eigenvector sequence for each $n$ to further tighten bounds.
• Investigating additional sequence structures reflecting more detailed combinatorial features of the multivariate polynomials.
• Extending this methodology to spectrahedral relaxations of other classes of rigidly convex sets and polynomials arising from combinatorial or algebraic constructions.
• Exploring dual or alternative relaxations capturing further non-diagonal multivariate information.

These results reinforce the notion that incorporating refined multivariate structure—via both the initial lifting of univariate polynomials and the numerical-structural analysis of LMI eigenvectors—enables spectrahedral relaxations to achieve much stronger approximations than previously thought possible, at least when measured by diagonal behavior. This has implications for optimization, convex algebraic geometry, and the combinatorial theory of real-rooted and real-zero polynomials.