Spectrahedral Relaxation

• Spectrahedral relaxation is a technique that approximates nonconvex sets with spectrahedra defined by LMIs, enabling tractable semidefinite programming formulations.
• It extends semidefinite programming and convex algebraic geometry to relax combinatorial, polynomial, and spectral optimization problems with tight outer approximations.
• Key constructions include derivative relaxations, spectral polyhedra, and MSLMP formulations, which enhance scalability and convergence in complex optimization scenarios.

A spectrahedral relaxation is a technique for approximating a convex semialgebraic set by a spectrahedron—the feasible region of a linear matrix inequality (LMI), i.e., an affine slice of the positive semidefinite (PSD) cone. This framework unifies and extends methods from semidefinite programming (SDP), convex algebraic geometry, and combinatorial optimization, and serves as a core tool in the relaxation of intractable polynomial, combinatorial, invariant, and spectral optimization problems. The spectrahedral relaxation paradigm subsumes classical polyhedral and Schur-Horn-type relaxations, recognizing spectrahedra as a natural next step in the moment/SOS (sum-of-squares) hierarchy and in the study of hyperbolicity cones and their approximations.

1. Definitions and Fundamental Objects

A spectrahedron is the solution set $\mathcal{S}(L)=\{x\in\mathbb{R}^n:L(x)\succeq0\}$, where $L(x)=A_0+\sum_{i=1}^n x_i A_i$ is a symmetric matrix pencil, and "$\succeq0$" denotes positive semidefiniteness. The intersection of a linear subspace $L\subset S^n$ with the PSD cone $S^n_+$ is called a spectrahedral cone. The spectrahedral shadow is the image of a spectrahedron under a linear projection, often needed to express feasible regions in original variable coordinates after lifting.

Spectrahedral relaxation refers to the replacement of a possibly nonconvex or intractable set with a spectrahedron or its shadow, producing a tractable semidefinite program approximation. This arises naturally as the convex relaxation of nonconvex QCQPs (quadratically constrained quadratic programs), polynomial optimization, combinatorial problems with convex hulls of rank-one matrices, spectral polyhedra, and hyperbolicity cones—where the true feasible set is captured as a special section or lift of the spectrahedral relaxation (Hildebrand, 2014, Sanyal et al., 2020, Schweighofer, 2019).

2. Principle of Spectrahedral Relaxation

Given a nonconvex problem, such as a QCQP requiring a matrix variable $X$ to be rank-one and positive semidefinite, relaxing the rank constraint admits the convex feasible set $K=L\cap S^n_+$ (a spectrahedral cone). The SDP relaxation is exact precisely if the optimal extreme point is rank-one. This principle generalizes as follows:

• Every nonconvex feasible set admitting a "lift" as a section/projection of $S^n_+$ can be relaxed to a spectrahedron.
• For real zero polynomials $p(x)$ (e.g., determinants or elementary symmetric polynomials), the "rigidly convex set" $C(p)$ is approximated from outside by a spectrahedron $S(p) =\{x: M_p(x)\succeq0\}$, where $M_p(x)$ is a cubic matrix pencil constructed from low-degree Taylor data of $p$ (Schweighofer, 2019, Nevado, 4 Jul 2025, Kummer, 2020).
• For optimization, the original problem $\min\{\phi(x): x\in C(p)\}$ is relaxed to $\min\{\phi(x): M_p(x)\succeq 0\}$.

Spectrahedral relaxations are also canonical in the context of convex hulls of matrix-valued functions of eigenvalues, e.g., spectral polyhedra associated to symmetric polyhedral sets in $\mathbb{R}^d$, via Schur-Horn-type orbitope lifts (Sanyal et al., 2020).

3. Key Constructions and Spectrahedral Representations

Derivative Relaxations: Given a hyperbolic polynomial $p(x)$, the $k$th Renegar derivative $D_e^{(k)}p$ produces a sequence of outer approximating hyperbolicity cones, each strictly containing its predecessor and, for many cases, admitting an explicit spectrahedral description (Saunderson et al., 2012, Saunderson, 2017, Sanyal, 2011, Brändén, 2012, Kummer, 2020).

Spectral Polyhedra and Convex Invariant Lifts: Any symmetric convex body $K\subset \mathbb{R}^d$ gives rise to a spectral convex set $\hat{K}=\{A\in S^d: \lambda(A)\in K\}$, which is convex and, if $K$ is a polyhedron, is a spectrahedron by an explicit system of LMIs using Schur block maps (Sanyal et al., 2020, Kummer, 2020).

Monic Symmetric Linear Matrix Polynomial (MSLMP) for Multivariate Rigidly Convex Sets: Outer relaxations for rigidly convex sets defined by real zero polynomials are constructed via cubic pencils that depend on low-degree power series data. For multivariate Eulerian polynomials and general combinatorial RZ polynomials, the resulting spectrahedral relaxations achieve extremely fine approximations, particularly on diagonal slices (Nevado, 4 Jul 2025).

Spectrahedral Shadows: Small-lift projected representations are provided (e.g., Ben-Tal–Nemirovski majorization LMIs for spectral polyhedra), reducing the size of semidefinite constraints and improving scalability (Sanyal et al., 2020).

Spectral Relaxations for Polynomial Optimization: The "constant trace property" (CTP) moment/SOS hierarchy can be reduced to spectral minimization, i.e., minimizing $\lambda_{\max}$ of an affine pencil under linear side constraints, providing robustness and scalability for high-dimensional polynomial optimization (Mai et al., 2020).

4. Representative Examples

Problem/Class	Relaxation Object	Key Representation/Structure
Quadratic Eq. QCQP	SDP (spectrahedron or shadow)	$K=L\cap S^n_+$
Rig. convex set of RZ polynomial $p$	$S(p)=\{x:M_p(x)\succeq0\}$	$M_p$ cubic pencil from $-\log p(-x)$
Spectral polyhedron (permutahedron, etc.)	Spectrahedron	Block-diagonal LMI with Schur functors
Derivative relax. (e.g., Renegar’s cone)	Spectrahedron (or shadow)	LMI in elementary symmetric eigenvalues
Rigidly convex set from multiv. Eulerian $A_n$	Spectrahedral outer bound	LMP with explicit linearization/eigenvector bounds

Spectrahedral relaxations can also be understood as the core of facial reduction for degenerate SDPs (Im et al., 2024), or as the base level in SDP hierarchies for set containment (Kellner et al., 2013).

5. Quality, Diagonal Accuracy, and Tightness

Spectrahedral relaxations provide tight and, in some cases, best-possible outer approximations:

• For symmetric polynomials (e.g., elementary symmetric), spectrahedral relaxations give exact representations of hyperbolicity cones (Brändén, 2012, Kummer, 2020).
• For multivariate Eulerian polynomials, diagonal restriction of the cubic LMP yields bounds for $|q_1^{(n)}|$ (extremal roots) with first-order sharp asymptotics, improved further by optimized eigenvector choices. Explicit linearization sequences yield gaps that grow exponentially with $n$, highlighting the diagonal measure of accuracy of the spectrahedral method compared to previous linear or SDP bounds (Nevado, 4 Jul 2025).
• For optimization over such relaxations, one can extract minimizers with approximate tightness matching or exceeding previous literature, with theoretical justification for convergence rates and optimality (Nevado, 4 Jul 2025, Mai et al., 2020).

6. Extensions: Polar Bodies, Duals, and Generalized Lax Conjecture

Spectrahedrality is preserved under polarity for spectral polyhedra, making their polars again spectrahedra and ensuring spectral polyhedra are doubly-spectrahedral (Sanyal et al., 2020). Derivative relaxations and spectral lifts are deeply connected to the generalized Lax conjecture, which posits that every hyperbolicity cone is spectrahedral (Saunderson, 2017, Kummer, 2020). Explicit relaxations and partial "wrapping" results provide evidence and technical machinery for this conjecture, especially in low dimensions or special cases (plane hyperbolic curves, hyperbolicity cones of elementary symmetric polynomials), and enhance the understanding of when spectrahedral relaxations are exact (Kummer et al., 2018, Brändén, 2012, Schweighofer, 2019).

7. Applications and Algorithmic Aspects

Spectrahedral relaxation is fundamental in:

• SDPs for QCQP and polynomial optimization (including reduction to spectral eigenvalue minimization for CTP problems) (Mai et al., 2020).
• Convexification of combinatorially structured optimization problems (e.g., assignment, clustering, spectral norm minimization) (Sanyal et al., 2020, Hildebrand, 2014).
• Certification of convex set containment and hierarchy interpolation between positivity and complete positivity for matrix maps (Kellner et al., 2013).
• Construction of small semidefinite outer approximations for hyperbolic programs with explicit accuracy guarantees on extremal statistics (Schweighofer, 2019, Nevado, 4 Jul 2025).
• Structural analysis in optimization of hyperbolicity cones, facial structure of spectrahedra, and exactness of relaxation for rank-one generated cones (Hildebrand, 2014).

Algorithmic strategies exploit spectrahedral structure for efficient eigenvalue minimization, scalable SDP extraction, and robust facial reduction, handling high-dimensional instances unattainable by general-purpose SDP solvers (Im et al., 2024, Mai et al., 2020). The spectrahedral paradigm allows for modular composition, isomorphic classification, and systematic geometric understanding of relaxation tightness and facial structure.

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References:

• (Sanyal et al., 2020) Sanyal & Saunderson, "Spectral Polyhedra"
• (Schweighofer, 2019) Schweighofer, "Spectrahedral relaxations of hyperbolicity cones"
• (Nevado, 4 Jul 2025) González Nevado, "Spectrahedral relaxations of Eulerian rigidly convex sets"
• (Nevado, 24 Jul 2025) González Nevado, "Guessing sequences of eigenvectors for LMPs defining spectrahedral relaxations of Eulerian rigidly convex sets"
• (Hildebrand, 2014) Sanyal, "Spectrahedral cones generated by rank 1 matrices"
• (Saunderson et al., 2012) Saunderson, "Polynomial-sized Semidefinite Representations of Derivative Relaxations of Spectrahedral Cones"
• (Sanyal, 2011) Sanyal, "On the derivative cones of polyhedral cones"
• (Saunderson, 2017) Saunderson, "A spectrahedral representation of the first derivative relaxation of the positive semidefinite cone"
• (Kummer, 2020) Kummer, "Spectral linear matrix inequalities"
• (Mai et al., 2020) Mai–Lasserre–Magron, "A hierarchy of spectral relaxations for polynomial optimization"
• (Im et al., 2024) Saunderson et al., "Projection, Degeneracy, and Singularity Degree for Spectrahedra"
• (Kummer et al., 2018) Kummer–Naldi–Plaumann, "Spectrahedral representations of plane hyperbolic curves"
• (Kellner et al., 2013) Kellner–Theobald–Trabandt, "A Semidefinite Hierarchy for Containment of Spectrahedra"
• (Brändén, 2012) Brändén, "Hyperbolicity cones of elementary symmetric polynomials are spectrahedral"
• (Cifuentes et al., 2018) Nie, "The Geometry of SDP-Exactness in Quadratic Optimization"