Homogeneous Spectrahedral Cone

• Homogeneous spectrahedral cones are convex sets with transitive automorphism groups and LMI descriptions, making them pivotal in optimization and algebraic function theory.
• They are characterized by unique block matrix representations and classifications via Vinberg–Rothaus theory and graph-theoretic conditions such as chordality.
• Their structured properties enable efficient primal-dual interior-point methods and scaling strategies, driving advances in conic optimization.

A homogeneous spectrahedral cone is a closed convex cone in finite-dimensional real vector space that is both homogeneous—meaning its automorphism group acts transitively on its interior—and spectrahedral, i.e., it is describable as the solution set to a linear matrix inequality (LMI). These objects play a central role at the interface of convex geometry, optimization, and algebraic function theory, synthesizing structural notions from the Vinberg–Rothaus theory and combinatorial conditions from matrix sparsity and graph theory. Every homogeneous convex cone admits a spectrahedral representation; conversely, not every spectrahedral cone is homogeneous. This article reviews the defining properties, structural representations, fundamental classification results, carathéodory and facial properties, canonical examples, and optimization-theoretic implications of homogeneous spectrahedral cones.

1. Definitions and Structural Foundations

A convex cone $K \subset \mathbb{R}^n$ is homogeneous if its automorphism group

$\operatorname{Aut}(K) = \{g \in GL(n) : gK = K\}$

acts transitively on $\operatorname{int} K$. That is, for any $x, y \in \operatorname{int} K$, there exists $g \in \operatorname{Aut}(K)$ with $g(x) = y$ (Tunçel et al., 2022, Chua, 21 Nov 2025).

A spectrahedral cone is any convex cone of the form

$$K = \{ x \in \mathbb{R}^n : A_0 + x_1A_1 + \cdots + x_nA_n \succeq 0 \}$$

where $A_0, \ldots, A_n$ are real symmetric $m \times m$ matrices and "$\succeq 0$" denotes positive semidefiniteness. A homogeneous spectrahedral cone is, therefore, a subset of $\operatorname{Aut}(K) = \{g \in GL(n) : gK = K\}$0 that is both homogeneous and admits a spectrahedral representation.

The canonical examples include the positive semidefinite (PSD) cone $\operatorname{Aut}(K) = \{g \in GL(n) : gK = K\}$1, second-order (Lorentz) cones, and more generally, cones arising as the sets of real squares in T-algebras or Euclidean Jordan algebras (Tunçel et al., 2022).

2. Spectrahedral Representations and Vinberg–Rothaus Classification

According to the Vinberg–Rothaus theory, every homogeneous cone admits a spectrahedral (LMI) representation (Tunçel et al., 2022): $\operatorname{Aut}(K) = \{g \in GL(n) : gK = K\}$2 for suitable symmetric matrices $\operatorname{Aut}(K) = \{g \in GL(n) : gK = K\}$3. This is operationally viewed as the Cayley embedding of the underlying algebraic structure into the space of symmetric matrices.

The Ishi block-matrix model provides a constructive approach: every homogeneous cone of rank $\operatorname{Aut}(K) = \{g \in GL(n) : gK = K\}$4 admits a representation as an intersection

$\operatorname{Aut}(K) = \{g \in GL(n) : gK = K\}$5

where $\operatorname{Aut}(K) = \{g \in GL(n) : gK = K\}$6 is the space of symmetric block matrices with specified linear subspaces $\operatorname{Aut}(K) = \{g \in GL(n) : gK = K\}$7 for each block, subject to compatibility axioms (triangular product, mixed transpose-product, self-product) (Chua, 21 Nov 2025). The set of positive-diagonal upper triangular block factors acts simply transitively, confirming homogeneity.

Vinberg’s theorem asserts that homogeneous cones precisely correspond to cones of squares in Jordan/T-algebras, linking the geometric notion of homogeneity to deep algebraic invariants (Tunçel et al., 2022).

3. Sparse-PSD Slices and Graph-Theoretic Criteria

A central class of spectrahedral cones is those arising as sparse slices of the PSD cone: $\operatorname{Aut}(K) = \{g \in GL(n) : gK = K\}$8 where $\operatorname{Aut}(K) = \{g \in GL(n) : gK = K\}$9 is a symmetric subset of pairs (dictating zero constraints, i.e., sparsity). The key result is:

• $\operatorname{int} K$0 is homogeneous if and only if the underlying graph on $\operatorname{int} K$1 with edge set $\operatorname{int} K$2 is a nested block-arrow graph—a combinatorial subclass of chordal graphs permitting a linearly ordered clique tree with an “arrow” terminal clique. In this case, the automorphism group includes all conjugations by Cholesky and inverse-Cholesky factors respecting the block structure (Tunçel et al., 2022).

More generally, Chua demonstrates that the only sparse spectrahedral cones that are homogeneous are those arising from homogeneous chordal graphs, i.e., chordal graphs admitting a perfect-elimination ordering that forbids induced 4-paths ($\operatorname{int} K$3) (Chua, 21 Nov 2025). This is formalized in the table below:

Feature	Condition for Homogeneity
PSD zero-pattern cone	Nested block-arrow / homogeneous chordal graph
Arbitrary sparsity	Typically not homogeneous unless above holds

For such cones, the automorphism group acts transitively via block-wise operations, crucial for algorithmic decomposability in optimization contexts.

4. Rank, Carathéodory Number, Self-Duality

For homogeneous spectrahedral cones, the rank $\operatorname{int} K$4 is the length of the longest chain of proper faces. The Carathéodory number $\operatorname{int} K$5 is the minimal $\operatorname{int} K$6 such that every point in the cone is a nonnegative combination of at most $\operatorname{int} K$7 extreme rays.

Chua proves that for the Ishi-type cones,

$\operatorname{int} K$8

and analogously for the dual cone (Chua, 21 Nov 2025). This block-dimension alignment is equivalent to the cone being symmetric (i.e., self-dual). The equivalence tightly links geometric symmetry, facial structure, and the combinatorics of the spectrahedral model.

The classical families of homogeneous spectrahedral cones and their invariants:

Cone	Rank $\operatorname{int} K$9	Carathéodory $x, y \in \operatorname{int} K$0
Real PSD $x, y \in \operatorname{int} K$1	$x, y \in \operatorname{int} K$2	$x, y \in \operatorname{int} K$3
Second-order cone $x, y \in \operatorname{int} K$4	$x, y \in \operatorname{int} K$5	$x, y \in \operatorname{int} K$6
Complex Hermitian PSD	$x, y \in \operatorname{int} K$7	$x, y \in \operatorname{int} K$8
Quaternionic Hermitian PSD	$x, y \in \operatorname{int} K$9	$g \in \operatorname{Aut}(K)$0
Exceptional 27-dim ($g \in \operatorname{Aut}(K)$1)	$g \in \operatorname{Aut}(K)$2	$g \in \operatorname{Aut}(K)$3

For general homogeneous cones not satisfying the self-duality criterion, $g \in \operatorname{Aut}(K)$4, reflecting the richer extreme-ray structure and deviation from maximal symmetry (Chua, 21 Nov 2025).

5. Illustrative Examples

• Symmetric cones:
  • The PSD cone $g \in \operatorname{Aut}(K)$5 ($g \in \operatorname{Aut}(K)$6) is homogeneous and self-dual, with full automorphism group (conjugations and inversion) (Tunçel et al., 2022).
  • The second-order (Lorentz) cone $g \in \operatorname{Aut}(K)$7 is realized by the $g \in \operatorname{Aut}(K)$8 block LMI:

  $g \in \operatorname{Aut}(K)$9

  and is also homogeneous and self-dual (Tunçel et al., 2022).

• Non-self-dual homogeneous cones: For instance, consider a sparse-PSD cone on $g(x) = y$0 with maximal cliques: $g(x) = y$1, $g(x) = y$2, $g(x) = y$3. This pattern, being nested block-arrow, yields a non-self-dual homogeneous cone. Its automorphism group is a product of block-wise Cholesky and inverse-Cholesky actions, but unlike symmetric cases, the dual is not equivalent (Tunçel et al., 2022, Chua, 21 Nov 2025).
• Hyperbolicity cones of elementary symmetric polynomials: The cones $g(x) = y$4 for the elementary symmetric polynomial $g(x) = y$5 in $g(x) = y$6 variables are homogeneous spectrahedral cones, constructed explicitly via the matrix-tree theorem and series-parallel graph expansion; these LMI lifts can be made explicit for all $g(x) = y$7 (Brändén, 2012).

6. Hyperbolicity Cones, Spectrahedrality, and the Generalized Lax Conjecture

A hyperbolicity cone associated to a hyperbolic polynomial $g(x) = y$8 in direction $g(x) = y$9 is

$$K = \{ x \in \mathbb{R}^n : A_0 + x_1A_1 + \cdots + x_nA_n \succeq 0 \}$$0

Hyperbolicity cones are always homogeneous, but the question of whether they are always spectrahedral—the generalized Lax conjecture—remains open in general (Schweighofer, 2019, Netzer et al., 2012). For cases where the polynomial is quadratic, a determinant of a matrix pencil, or an elementary symmetric polynomial, spectrahedrality is affirmed.

Several relaxation and representation techniques for spectrahedrality of hyperbolicity cones have been developed:

• Moment-matrix LMI relaxations: For any real-zero polynomial $$K = \{ x \in \mathbb{R}^n : A_0 + x_1A_1 + \cdots + x_nA_n \succeq 0 \}$$1, the rigidly convex set $$K = \{ x \in \mathbb{R}^n : A_0 + x_1A_1 + \cdots + x_nA_n \succeq 0 \}$$2 can be approximated or represented by a spectrahedron $$K = \{ x \in \mathbb{R}^n : A_0 + x_1A_1 + \cdots + x_nA_n \succeq 0 \}$$3, via construction of specific moment matrices and leveraging the Helton–Vinnikov theorem in low dimensions (Schweighofer, 2019). The inclusion $$K = \{ x \in \mathbb{R}^n : A_0 + x_1A_1 + \cdots + x_nA_n \succeq 0 \}$$4 is universal, with exactness in special cases (e.g., for degree-2 polynomials, or when $$K = \{ x \in \mathbb{R}^n : A_0 + x_1A_1 + \cdots + x_nA_n \succeq 0 \}$$5 is determinantal with a perfect subspace spanned by the $$K = \{ x \in \mathbb{R}^n : A_0 + x_1A_1 + \cdots + x_nA_n \succeq 0 \}$$6).
• Clifford algebra criterion: For a homogeneous polynomial $$K = \{ x \in \mathbb{R}^n : A_0 + x_1A_1 + \cdots + x_nA_n \succeq 0 \}$$7, the associated Clifford algebra allows the translation of spectrahedrality of $$K = \{ x \in \mathbb{R}^n : A_0 + x_1A_1 + \cdots + x_nA_n \succeq 0 \}$$8 to the question of whether $$K = \{ x \in \mathbb{R}^n : A_0 + x_1A_1 + \cdots + x_nA_n \succeq 0 \}$$9 is a sum of Hermitian squares. If not, $A_0, \ldots, A_n$0 is spectrahedral (Netzer et al., 2012). This criterion can be checked computationally via a single semidefinite program; its feasibility certifies the spectrahedrality.

7. Implications for Conic Optimization and Interior-Point Methods

Homogeneous spectrahedral cones directly inform algorithm design for conic optimization, particularly with respect to primal–dual symmetry and scaling strategies (Tunçel et al., 2022):

• For symmetric cones, self-scaled barriers enable polynomial-time primal–dual interior-point methods with full symmetry.
• For general homogeneous cones, even when not self-dual, the existence of a transitive automorphism subgroup supports the design of primal–dual algorithms via group-conjugated barrier functions, restoring much of the symmetry.
• In the case of sparse-PSD homogeneous slices, this translates into efficient, clique-wise Cholesky-based steps, benefiting from decomposability while maintaining essential symmetries.
• The spectrahedral representation assures that interior-point methods based on log-det or general homogeneous barriers can be implemented efficiently.

These features bridge the gap between symmetric-cone programming and the broader class of homogeneous (non-self-dual) conic problems, ensuring that key algorithmic tools from the symmetric case extend to the full homogeneous spectrahedral setting (Tunçel et al., 2022).

---

References:

(Tunçel et al., 2022, Chua, 21 Nov 2025, Schweighofer, 2019, Netzer et al., 2012, Brändén, 2012)