Polyhedral-SDP Relaxations

• Polyhedral-SDP relaxations are convex optimization techniques that replace difficult positive semidefinite constraints with tractable polyhedral ones, facilitating scalability in SDP and conic programming.
• They unify methodologies such as DNN, DSOS, and SDSOS relaxations, using LP/SOCP approximations, cutting planes, and bundle methods to achieve efficient trade-offs between computational cost and solution quality.
• Applications span combinatorial optimization, polynomial programming, and control, where these relaxations provide tight bounds and improved numerical tractability for large-scale problems.

Polyhedral-SDP relaxations refer to a broad class of convex optimization relaxations in which intractable positive semidefinite (PSD) constraints are replaced or augmented by tractable polyhedral (linear) constraints, or intersections of the PSD cone with polyhedral cones. These relaxations play a central role in making semidefinite programming (SDP) and higher-level conic programming accessible for large-scale applications, especially in combinatorial optimization, polynomial optimization, quadratic programming, and control. Polyhedral-SDP relaxations unify and generalize a variety of classical techniques, including doubly nonnegative (DNN), diagonally dominant (DD), and scaled diagonally dominant (SDD) relaxations, and enable efficient linear programming (LP)-based or second-order cone programming (SOCP)-based algorithmic strategies for approximating hard convex sets.

1. Foundations and Definitions

Polyhedral-SDP relaxations arise in settings where a convex conic constraint such as $X \succeq 0$ (PSD) is replaced, approximated, or augmented by constraints of the form $X \in \mathcal{P}$, with $\mathcal{P}$ a polyhedral cone in the appropriate symmetric matrix space, frequently leading to LP or SOCP tractability. In polynomial optimization, the relaxation replaces the intractable moment or sum-of-squares (SOS) cone $\text{conv}(K)$ by an intersection $\mathbb{S}^\A_+ \cap \mathcal{P}^\A \cap \mathcal{L}^\A$ of the PSD cone, a tractable polyhedral cone, and a consistency cone enforcing symmetries or monomial equalities (Hou et al., 6 Dec 2025). Canonical examples include:

• DNN Relaxations: Imposing $X \ge 0$ and $X \succeq 0$ (Hou et al., 6 Dec 2025).
• DSOS/SDSOS Hierarchies: Enforcing diagonal dominance or scaled diagonal dominance, allowing further reduction to LP or SOCP (Hou et al., 6 Dec 2025, 2220.12374, Wang et al., 2019).
• Expanded SD-Basis Polyhedra: Using sparsely generated collections of rank-1 PSD matrices to form tractable approximations with controlled inclusion properties (Wang et al., 2019).

The core mathematical structure is that polyhedral-SDP relaxations interpolate between full semidefinite relaxations and purely polyhedral (LP) relaxations, enabling flexible trade-offs between computational complexity and solution quality.

2. Core Methodologies and Hierarchical Schemes

Multiple methodological frameworks underlie polyhedral-SDP relaxations:

• LP/SOCP Approximations of SDP: The DD and SDD cones provide inner (for primal) or outer (for dual) polyhedral or SOCP relaxations of the PSD cone, yielding LP or SOCP relaxations (2220.12374, Wang et al., 2019). Algorithms alternate between LP- or SOCP-based “decrease” phases and (optionally) central path “centering” phases to obtain $\epsilon$-optimal solutions (2220.12374).
• Expanded SD Bases and Cutting Planes: The conical hulls of rank-1 PSD matrices, parametrized by expanded bases, form polyhedral inner or outer approximations. Iterative cut-generation—via eigenvalue separation, as in cutting-plane methods—enables refinement (Wang et al., 2019).
• Bundle-Based Polyhedral QP Methods: Approximating the PSD constraint on dual variables by intersecting supporting hyperplanes (bundles) constructed from eigenvectors, leading to QP relaxations with a tunable number of linear constraints chosen according to the optimal solution’s rank (Cui et al., 14 Oct 2025).
• Unified Cone-Approximation Frameworks for POP: By properly selecting the polyhedral cone $\mathcal{P}^\A$—for example, nonnegativity, DD, or SDD—classical relaxations such as DNN, DSOS/SDSOS, and their higher-order variants all emerge as special cases (Hou et al., 6 Dec 2025). The consistency cone structure can be exploited for efficient blockwise projection.
• Lift-and-Project Hierarchies: In combinatorial contexts, polyhedral and SDP-based hierarchies (e.g., Lovász–Schrijver, Lasserre) can be analyzed and compared via their rank, symmetry, and tightness properties (Au et al., 2020).

3. Containment Properties and Theoretical Guarantees

Precise containment hierarchies shape the theoretical understanding of polyhedral-SDP relaxations:

• The cone of diagonally dominant matrices $\mathcal{DD}_n$ coincides with the conical hull generated by canonical SD bases, i.e., $$\mathrm{cone}(\mathcal{B}_+ \cup \mathcal{B}_-) = \mathcal{DD}_n$$ (Wang et al., 2019).
• Expanded SD bases $\overline{\mathcal{B}}(\alpha)$ (for finite $H \subset \mathbb{R}\setminus\{0, -1\}$) satisfy

$$\mathcal{DD}_n \subseteq \mathcal{SDB}_n \subseteq \mathcal{SDD}_n$$

and, in the infinite basis case, generate the full SDD cone (Wang et al., 2019).

• Polyhedral relaxations provide either inner or outer approximations to the PSD cone, ensuring that feasible solutions to the true SDP remain feasible, with improved efficiency and numerical tractability.
• In polynomial optimization, polyhedral-SDP relaxations can provide tight lower bounds, but the number of variables and constraints grows as $\Omega(n^{2\tau})$ with the relaxation order $\tau$ (Hou et al., 6 Dec 2025).

4. Algorithms and Computational Schemes

A variety of efficient computational schemes exploit the structures above:

• Globally Convergent Decrease-and-Center Algorithms: Alternating between LP or SOCP relaxations (with DD/SDD constraints) and log-barrier-based centering steps to track the central path of the true SDP, with polynomial iteration bounds to reach $\epsilon$-optimality (2220.12374).
• Polyhedral Bundle Methods: Constructing lower-cut polyhedral approximations of the PSD cone in the dual space, with a bundle size empirically chosen as $l_{\max} = \frac{1}{2} r^*(r^* + 1) + r^*$ (for optimal rank $r^*$), and QP subproblems of bounded dimension. Bundle elements are pruned or aggregated to maintain efficiency (Cui et al., 14 Oct 2025).
• Low-Rank Augmented Lagrangian Methods: Applying factorized ALM techniques to handle the facial structure and large constraint sets in polyhedral-SDP relaxations for polynomial optimization. A key innovation is a tailored blockwise projection for monomial consistency and nonnegativity constraints, scaling as $O(n^{2\tau})$ (Hou et al., 6 Dec 2025).
• Instance-Specific LP Relaxations: By exploiting commutativity of the data matrices, a single tight LP or SOC relaxation can, for some SDPs, be provably exact or quantifiably tight, greatly accelerating solution (Roux et al., 2023).
• Cutting Plane and Kelley-Type Methods: Iteratively generating eigenvector-based cuts to enforce ray constraints $v^\top X v \ge 0$ for selected $v$. This allows practical tightness with very small LPs, quickly converging to SDP quality (Roux et al., 2023).

5. Applications and Performance Comparisons

Polyhedral-SDP relaxations have been applied effectively to a broad range of paradigmatic problems:

• Combinatorial Optimization: Maximum stable set, max-cut, clique number, and hypergraph matching problems. Polyhedral hierarchies or expanded SD-basis relaxations consistently deliver bounds approaching those of full SDPs at significantly reduced computational cost (Au et al., 2020, Wang et al., 2019).
• Polynomial Optimization: General polynomial optimization, quadratic programming, moment-SOS relaxations, tensor copositivity, and discrete nonconvex problems (Hou et al., 6 Dec 2025).
• Sparse Pseudoinverse Construction: Trade-offs between sparsity and Moore–Penrose property exactness via LP/SDP relaxations (Fuentes et al., 2016).
• Empirical Performance: In the context of large-scale SDPs—random graphs, SDPLib instances, and Max-Cut relaxations—polyhedral LPrelaxations such as CPSDB (expanded SD-basis LP) were found to be the fastest to reduce optimality gaps and matched bound quality of SOCP and SDP methods, while being significantly more efficient at scale (Wang et al., 2019, 2220.12374, Cui et al., 14 Oct 2025, Hou et al., 6 Dec 2025).

Relaxation/Algorithm	Complexity per Iteration	Bound Quality	Applicability
DD (LP) / SDD (SOCP) relax.	$O(M^2 N^2)$ — $O(MN^3)$	Closely tracks SDP	General SDP, combinatorics
Expanded SD-basis LP (CPSDB)	Sparse, efficient LP	Matches SOCP & SDP	Stable set, max-cut, DNN-relax.
Polyhedral bundle QP	Bounded by rank-optimum	Approximates SDP, tunable	Random SDPs, Max-Cut, ill-conditioned SDPs
Low-rank ALM (RiNNAL-POP)	$O(n^{2\tau})$ per step	Outperforms SDPNAL+,	POPs, large-scale lifts

6. Analysis of Hierarchies and Symmetry

The analytic power of polyhedral-SDP relaxations is often enhanced by group and algebraic symmetries:

• Association Schemes and Symmetry Reduction: Many key SDP relaxations and their polyhedral analogues exploit association schemes (Johnson, Hamming, or hypermatching schemes) for systematic reduction, allowing semidefinite feasibility to be checked via a small set of eigenvalue conditions (Au et al., 2020).
• Lift-and-Project Hierarchy Rank: The Lovász–Schrijver ($\text{LS}_+$) SDP lift typically collapses to the integer hull in few steps for symmetric graphs, while the LP-based ($\text{LS}$) hierarchy may require many more steps (Au et al., 2020).
• Exactness via Commutativity: For SDPs on simultaneously diagonalizable data, polyhedral relaxations become exact; strong instance-specific LP bounds are available for max-cut and Lovász theta-type problems (Roux et al., 2023).

7. Limitations, Extensions, and Ongoing Research

Although polyhedral-SDP relaxations dramatically improve scalability and applicability, they have limitations:

• The rank and tightness of polyhedral relaxations are inherently problem-dependent. For complete $q$-uniform hypergraph $K_p^q$, the integrality gap in the matching relaxation survives nearly all hierarchy steps (Au et al., 2020).
• The $O(n^{2\tau})$ scaling for higher-order relaxations is a bottleneck in polynomial optimization (Hou et al., 6 Dec 2025).
• Polyhedral relaxations always lie (for the primal problem) inside the feasible set of the true SDP; they may not always match optimality up to desired tolerances unless further refined by higher-order or instance-specific cuts.

Current and future research directions include:

• Theoretical frameworks to link relaxation fidelity (e.g., bundle cap $l_{\max}$ or symmetry structure) to condition number and approximation error (Cui et al., 14 Oct 2025).
• Dynamic or adaptive cut-generation, hybrid LP/SOCP/SDP-algorithmics, and leveraging problem structure for improved scalability in emerging large-scale applications.
• Extensions of the polyhedral-SDP relaxation paradigm to mixed-integer settings, tensor relaxations, and classes of nonconvex optimization.

Polyhedral-SDP relaxations thus constitute an essential set of methodologies that interpolate between LP tractability and full semidefinite expressiveness, unifying and extending a broad landscape of optimization models and algorithms (Hou et al., 6 Dec 2025, Wang et al., 2019, Au et al., 2020, Cui et al., 14 Oct 2025, Roux et al., 2023, 2220.12374).