Spectrahedral Cone: Theory & Applications

• Spectrahedral cones are convex sets defined by linear matrix inequalities, representing intersections of real vector spaces with the positive semidefinite cone.
• They exhibit rich facial and algebraic structures, making them central to semidefinite programming and applications in systems theory and optimization.
• As a subclass of hyperbolicity cones, spectrahedral cones bridge algebraic geometry with practical optimization, providing frameworks for robust computational methods.

A spectrahedral cone is a convex cone that admits a representation as the solution set of a linear matrix inequality (LMI), equivalently, as the intersection of a real vector space with the cone of positive semidefinite (PSD) matrices. Such cones are central objects in convex algebraic geometry, semidefinite programming, and optimization, generalizing polyhedral cones and providing a key bridge between algebraic and geometric structures.

1. Formal Definition and Representations

A convex cone $K \subseteq \mathbb{R}^d$ is called spectrahedral if there exist real symmetric matrices $A_0, A_1, \dots, A_d \in \mathbb{S}^m$ such that

$$K = \{ x \in \mathbb{R}^d \mid A_0 + x_1 A_1 + \cdots + x_d A_d \succeq 0 \},$$

where “$\succeq 0$” denotes positive semidefiniteness. The integer $m$ is the size of the spectrahedral representation (Saunderson, 2017, Netzer et al., 2012, Tunçel et al., 2022).

Every spectrahedral cone is an affine slice of the convex cone $\mathbb{S}_+^m$ of real $m \times m$ positive semidefinite matrices. The recession cone of a spectrahedron is itself spectrahedral, and the class of spectrahedral cones encompasses many important examples, including the PSD cone itself, the nonnegative orthant, and various cones arising in systems and algebraic geometry (Dörfler et al., 2022, Hildebrand, 2014).

2. Key Structural Properties

Spectrahedral cones are always closed, convex, and exhibit rich facial structures. Given the defining pencil $A(x)$, the faces of $K$ correspond to faces of the PSD cone under the inverse image of $A$. An extreme ray $x$ of $K$ is characterized by the property that $A(x)$ is a rank-one positive semidefinite matrix (Dörfler et al., 2022). Under mild conditions, the extreme rays of full PSD cones are all realized as such rank-one matrices (Hildebrand, 2014, Heaton et al., 2020).

A spectrahedral cone admits an algebraic boundary defined via a “defining polynomial,” frequently arising as the determinant of the pencil restricted to the subspace, which is irreducible in the rank-one generated (ROG) case (Hildebrand, 2014).

Normal form results (cf. Ramana's theorem) reveal that a spectrahedral cone is polyhedral (i.e., defined by finitely many linear inequalities) if and only if its defining pencil can be unitarily diagonalized, and the diagonal entries are linear forms (Bhardwaj et al., 2011).

3. Spectrahedral Cones as Hyperbolicity Cones

Spectrahedral cones are a distinguished subclass of hyperbolicity cones. A homogeneous polynomial $p \in \mathbb{R}[x_1,\dots,x_n]$ is hyperbolic with respect to $e \in \mathbb{R}^n$ if $p(e) \neq 0$ and for every $x$, $t \mapsto p(te + x)$ has only real roots. The associated hyperbolicity cone

$$\Lambda_+(p, e) = \{ x \in \mathbb{R}^n \mid \text{all roots of } p(te + x) \text{ are } \leq 0 \}$$

is a closed, convex, pointed cone (Raghavendra et al., 2017, Netzer et al., 2012).

Spectrahedral cones always arise as hyperbolicity cones of determinantal polynomials: $p(x) = \det(A_0 + x_1A_1 + \cdots + x_dA_d)$ with respect to directions $e$ for which $A(e) \succ 0$. The celebrated Generalized Lax conjecture asserts that every hyperbolicity cone is spectrahedral; while proved in some cases, it remains open in full generality (Saunderson, 2017, Netzer et al., 2012, Brändén, 2012, Raghavendra et al., 2017).

4. Examples and Constructions

The class of spectrahedral cones contains:

• The full PSD cone $\mathbb{S}_+^n$ and its linear sections.
• Cones defined by block-diagonal, banded, or sparse patterns in symmetric matrices, including those determined by underlying chordal graphs. Homogeneous spectrahedral cones with specific sparsity patterns correspond precisely to so-called homogeneous chordal graphs (Chua, 21 Nov 2025, Tunçel et al., 2022).
• Rank-one generated (ROG) cones, where every element is a sum of rank-one matrices also lying in the cone, such as the full PSD cone, diagonal PSD cone, Hankel moment cones, and certain block-structured cones (Hildebrand, 2014).

Hyperbolicity cones of the elementary symmetric polynomials are spectrahedral; explicit constructions use tools from matroid theory and the matrix–tree theorem in combinatorics to build determinantal representations (Brändén, 2012). More generally, the first derivative (Renegar derivative) relaxations of the PSD cone admit explicit spectrahedral representations of size $\binom{n+1}{2}-1$ (Saunderson, 2017, Saunderson et al., 2012).

The table below summarizes key examples:

Example	Matrix Pencil Representation	Special Properties
PSD cone $\mathbb{S}_+^n$	$X \succeq 0$	Symmetric, ROG
Nonnegative orthant	Diagonal matrices $\operatorname{diag}(x_i) \succeq 0$	Polyhedral, ROG
Hankel moment cone	Hankel structure $H \succeq 0$	ROG, moments
Block-arrow sparsity cone	Block-arrow-patterned matrix $X \succeq 0$	Homogeneous, sparse

5. Derivative Cones and Relaxations

Spectrahedral cones are not generally closed under all operations transferring from hyperbolicity cones. The first derivative relaxation of spectrahedral cones, given by the directional derivative of determinantal polynomials, is again spectrahedral. For the full PSD cone, this yields an explicit description in terms of zero-trace subspace pencils (Saunderson, 2017, Sanyal, 2011, Saunderson et al., 2012).

However, higher derivative cones do not, in general, admit simple determinantal representations; beyond the first polar, spectral representability fails for generic polyhedral cones (Sanyal, 2011). This demonstrates subtle boundaries between spectrahedral and hyperbolicity cones.

A hierarchy of derivative relaxations provides increasingly weaker (i.e., larger) outer approximations to the original cone, all semidefinite representable with polynomial-sized lifts (Saunderson et al., 2012).

6. Homogeneous and Symmetric Spectrahedral Cones

All homogeneous cones (those whose automorphism group acts transitively on the interior) are spectrahedral by the Vinberg–Rothaus theory, realized as linear matrix inequality cones derived from simple Euclidean Jordan or $T$-algebras (Tunçel et al., 2022, Chua, 21 Nov 2025). Symmetric cones, the self-dual subclass, correspond to those with matching rank and Carathéodory number.

Block matrix representations (Ishi model) describe homogeneous spectrahedral cones using specified block sizes and subspaces, with compatibilities enforced by algebraic constraints (Chua, 21 Nov 2025). The Carathéodory number and rank coincide in symmetric cones, and in the sparse setting, only homogeneous chordal graphs yield homogeneous spectrahedral cones.

Invariant (symmetry-adapted) spectrahedral cones arise as the intersection of the full PSD cone with subspaces fixed by group actions. Their structure—dimension, extreme rays, and block decomposition—are dictated by the representation theory of the acting group (Heaton et al., 2020).

7. Computational and Optimization Implications

Spectrahedral cones are precisely the feasible sets of semidefinite programs (SDPs). Their structure underpins efficient convex optimization for many classes of problems, including control theory, quadratic programming, combinatorial relaxations, and sums-of-squares certificates (Tunçel et al., 2022, Saunderson et al., 2012, Dörfler et al., 2022, Heaton et al., 2020).

Outer and inner polyhedral approximation algorithms for spectrahedral cones and their projections have been developed, enabling controlled LP relaxations for large-scale computation (Dörfler et al., 2022). The dimension of possible spectrahedral lifts can be enormous: for generic hyperbolicity cones, exponential lower bounds hold for the size of any spectrahedral representation, even allowing small approximations (Raghavendra et al., 2017).

The intersection with sums-of-squares theory is deep: the Gram matrix method for SOS representations works precisely inside a spectrahedral cone corresponding to the function space and symmetries in play (Heaton et al., 2020).

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References:

(Saunderson, 2017, Netzer et al., 2012, Tunçel et al., 2022, Dörfler et al., 2022, Chua, 21 Nov 2025, Hildebrand, 2014, Heaton et al., 2020, Bhardwaj et al., 2011, Brändén, 2012, Raghavendra et al., 2017, Saunderson et al., 2012, Sanyal, 2011)