Rank-One Generated (ROG) Cone

• The ROG cone is a convex conic subset of positive semidefinite matrices in which every extreme ray is generated by a rank-one matrix, establishing a clear geometric structure.
• ROG cones ensure the exactness of semidefinite programming relaxations for nonconvex quadratically constrained quadratic programs, linking optimization with combinatorics and algebraic geometry.
• They exhibit rich structural, algebraic, and combinatorial characterizations, with applications in spectrahedral analysis, hyperbolicity cones, and moduli space geometry.

A rank-one generated (ROG) cone is a closed convex conic subset whose extreme rays all correspond to rank-one elements. ROG cones provide a precise geometric framework underpinning the exactness of semidefinite programming (SDP) relaxations for a broad class of nonconvex quadratically constrained quadratic programs (QCQPs), and they occur prominently in convex geometry, optimization, combinatorics, and algebraic geometry. ROG cones also admit a rich set of structural, algebraic, and combinatorial characterizations, connecting matrix theory, spectrahedra, hyperbolic polynomials, and moduli spaces.

1. Formal Definition and Fundamental Properties

Let $$S^n_+ = \{ X \in \mathbb{R}^{n \times n}: X^\top = X,\, X \succeq 0 \}$$ denote the positive semidefinite cone of symmetric $n \times n$ matrices. A closed convex cone $K \subset S^n_+$ is called rank-one generated (ROG) if

$$K = \mathrm{conv}\{ xx^\top : x \in \mathbb{R}^n \} \cap K$$

or, equivalently, every extreme ray of $K$ is generated by a rank-one matrix; that is, for all nonzero $X \in K$ that are extreme, $\operatorname{rank}(X) = 1$ (2007.07433, Hildebrand, 2014). Equivalently, for spectrahedral cones $K = L \cap S^n_+$ for some linear subspace $L \subset S^n$, $K$ is ROG if and only if every $n \times n$0 admits a decomposition

$n \times n$1

with $n \times n$2 and the $n \times n$3 linearly independent (Hildebrand, 2014). The Carathéodory number of $n \times n$4 is then precisely $n \times n$5. The facial structure is stable: every face of a ROG cone is again ROG, and in a ROG cone, higher-rank matrices are never extreme (Hildebrand, 2014).

2. ROG Cones in Spectrahedral, Hyperbolicity, and Algebro-geometric Settings

The ROG property extends beyond classical spectrahedral cones into the broader class of hyperbolicity cones. For a hyperbolic polynomial $n \times n$6 of degree $n \times n$7 and direction $n \times n$8, the associated hyperbolicity cone is

$n \times n$9

Defining $K \subset S^n_+$0 as the number of strictly positive Gårding eigenvalues, a pointed hyperbolicity cone is ROG if all its nonzero extreme rays are generated by elements of hyperbolic rank one. The PSD cone $K \subset S^n_+$1 and nonnegative orthant $K \subset S^n_+$2 are both ROG in this sense. Under Renegar derivatives, the automorphism groups of ROG hyperbolicity cones contract to subgroups fixing the distinguished direction, with full characterizations for derivative relaxations (e.g., $K \subset S^n_+$3, $K \subset S^n_+$4) (Ito et al., 2022).

In algebraic geometry, cones generated by first Chern classes of rank-one vector bundles of conformal blocks (with combinatorial classifications via quantum Kostka numbers or $K \subset S^n_+$5-invariance) furnish ROG cones within nef cones of moduli spaces such as $K \subset S^n_+$6, with polyhedral and combinatorial structure governed by representation theory (Hobson, 2015, Kazanova, 2014).

3. Structural and Characterization Theorems

Several formulations provide both necessary and sufficient criteria for a spectrahedral cone to be ROG:

• General Sufficient Conditions: For $K \subset S^n_+$7, if for every pair $K \subset S^n_+$8, there exist $K \subset S^n_+$9 with $$K = \mathrm{conv}\{ xx^\top : x \in \mathbb{R}^n \} \cap K$$0, then $$K = \mathrm{conv}\{ xx^\top : x \in \mathbb{R}^n \} \cap K$$1 is ROG (pairwise-PSD-aggregation) (2007.07433).
• Envelope Lemma: $$K = \mathrm{conv}\{ xx^\top : x \in \mathbb{R}^n \} \cap K$$2 is ROG if and only if, for every nonzero $$K = \mathrm{conv}\{ xx^\top : x \in \mathbb{R}^n \} \cap K$$3, there exists $$K = \mathrm{conv}\{ xx^\top : x \in \mathbb{R}^n \} \cap K$$4 in $$K = \mathrm{conv}\{ xx^\top : x \in \mathbb{R}^n \} \cap K$$5 such that $$K = \mathrm{conv}\{ xx^\top : x \in \mathbb{R}^n \} \cap K$$6 for all $$K = \mathrm{conv}\{ xx^\top : x \in \mathbb{R}^n \} \cap K$$7 (Lemma 2.11) (2007.07433).
• Two-LMI Case: $$K = \mathrm{conv}\{ xx^\top : x \in \mathbb{R}^n \} \cap K$$8 is ROG if and only if either there exists a nontrivial PSD aggregation or a special rank-two structure: $$K = \mathrm{conv}\{ xx^\top : x \in \mathbb{R}^n \} \cap K$$9, $K$0 for some $K$1 (2007.07433).

For spectrahedral cones, the minimal defining polynomial is always the restriction of the determinant, and the degree of $K$2 is the maximal rank of elements in $K$3—and in the ROG case, the facial and algebraic structures are uniquely determined (up to congruence) by the geometric data (Hildebrand, 2014).

4. Construction Methods and Classes of ROG Cones

Multiple techniques for constructing large classes of ROG cones exist through operations preserving the ROG property:

• Direct Sums: Block-diagonal sums of ROG cones remain ROG; the degree is additive (Hildebrand, 2014).
• Full Extensions (Projections): Full extensions (lifting by adding zero blocks) preserve ROG structure (Hildebrand, 2014).
• Intertwinings: Gluing along isomorphic faces of positive rank—intertwining—produces new ROG cones (Hildebrand, 2014).

Key examples include:

• Chordal-graph cones: $K$4 is ROG if and only if the graph $K$5 is chordal, with $K$6 encoding the zero-pattern (Hildebrand, 2014).
• Hankel and block-Hankel cones: arising from moment and SOS representations, always ROG under appropriate function space conditions (Hildebrand, 2014).
• Simplicial cones of divisors inside $K$7 generated by conformal blocks Chern classes at level one (Kazanova, 2014).

Classification results hold for small matrix sizes, where all simple ROG cones can be explicitly listed (Hildebrand, 2014).

5. Connections to Optimization and Convexification

A central motivation for studying ROG cones lies in conic optimization. In QCQPs, replacing the rank-one constraint $K$8 by a PSD constraint and working over a convex cone $K$9 leads to an SDP relaxation: $X \in K$0 If $X \in K$1 is ROG, the SDP relaxation is always exact; that is, the minimum is achieved at a rank-one solution unless infeasible or unbounded (2007.07433, Hildebrand, 2014). The ROG condition translates to exactness—no duality gap—of homogeneous QCQPs over $X \in K$2 and enables explicit convex-hull characterizations for feasibility regions and epigraphs of QCQPs under mild compactness and definiteness conditions (2007.07433).

Further, inhomogeneous QCQPs with a single quadratic equality (as in trust-region or mixed-integer quadratic problems) inherit tightness of the SDP relaxation from the ROG property, and the maximal possible matrix rank in optimal solutions increases from one to at most two (2007.07433).

Applications include the perspective reformulation of mixed-binary feasible sets, explicit convex hulls for sets defined by quadratic constraints, and connections to the moment-SOS hierarchy in function spaces (2007.07433, Hildebrand, 2014).

6. Automorphisms and Uniqueness

For hyperbolicity cones, the automorphism group $X \in K$3 is the group of invertible linear maps preserving the cone. For ROG hyperbolicity cones of degree $X \in K$4, derivative relaxations shrink the automorphism group to those elements fixing a reference ray $X \in K$5 (Ito et al., 2022). For spectral cones, automorphisms are congruences up to scale, and for block-diagonal cones, automorphisms correspond to simultaneous block permutations and congruence transformations (Ito et al., 2022).

In the real symmetric case, if two ROG cones are linearly isomorphic as convex cones, they are isomorphic as spectrahedral cones (i.e., the linear subspace and the matrix size are determined up to congruence), which is not true for general spectrahedral cones (Hildebrand, 2014).

7. Combinatorial and Algebraic Geometry Contexts

The ROG property appears in the structure of nef cones in the moduli of curves. For example, the nef cone of $X \in K$6 admits a simplicial ROG subcone generated by the first Chern classes of rank-one $X \in K$7-invariant conformal block bundles for $X \in K$8, described as nonnegative combinations of level-one boundary divisors $X \in K$9 (Kazanova, 2014). In $\operatorname{rank}(X) = 1$0 with rectangular weights, the ROG cone generated by the first Chern classes of all rank-one bundles is finitely generated, and explicit decomposition formulas exist. Open problems include classifying the full conformal blocks divisorial cone and extending these frameworks to other Lie types and more general weight data (Hobson, 2015, Kazanova, 2014).

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For an in-depth development of foundations, structural theory, and applications, see (2007.07433, Hildebrand, 2014), and (Ito et al., 2022). For algebro-geometric and representation-theoretic implementations, see (Hobson, 2015) and (Kazanova, 2014).