Semidefinite Programming Relaxations

• Semidefinite Programming Relaxations are convex formulations that lift nonconvex quadratic or polynomial problems into positive semidefinite constraints, enabling tractable, polynomial-time optimization.
• Hierarchical approaches, like Lasserre’s and sum-of-squares hierarchies, incrementally tighten approximations to converge to global optima under specific regularity conditions.
• These methods extend to diverse applications such as combinatorial optimization, quantum information, and signal processing, while also highlighting theoretical limits with exponential SDP sizes for exact bounds.

Semidefinite programming (SDP) relaxations are a cornerstone methodology in convex optimization and polynomial programming, providing tractable convex approximations for a vast class of otherwise intractable nonconvex problems. The approach hinges on reformulating discrete, polynomial, or quadratic constraints as positive semidefinite matrix constraints and subsequently relaxing nonconvex rank or integrality restrictions to obtain a convex program solvable in polynomial time. SDP relaxations have become fundamental not only for classical polynomial optimization and combinatorial optimization but also for quantum information, signal processing, and global optimization, enabling both approximation guarantees and, under certain conditions, exact recovery.

1. Basic Methodology and Hierarchies of SDP Relaxations

At its core, an SDP relaxation converts a nonconvex problem—typically quadratic or polynomial in nature—into a convex semidefinite program by 'lifting' the original variable into a positive semidefinite matrix. For polynomial optimization, this commonly arises through the sum-of-squares (SOS) paradigm, where the condition that a polynomial $p(x) \geq 0$ over a domain is relaxed to require $p(x)$ to admit a sum-of-squares decomposition. This can equivalently be encoded as the existence of a positive semidefinite 'Gram' matrix or as an SOS expression of certain fixed degree. Explicitly, the classical Lasserre's hierarchy for polynomial optimization constructs a sequence of SDPs indexed by the SOS degree $2k$: $$f_k := \max \{ \gamma : f(x) - \gamma \in Q_k(\mathcal{A}) \},$$ where $Q_k(\mathcal{A})$ is the truncated quadratic module generated by the problem's equality and inequality constraints, with the inclusion of SOS polynomials of bounded degree (Wang et al., 2013, Jeyakumar et al., 2015, Guo et al., 2015).

Hierarchical approaches are prevalent: in each step, additional degree or moment constraints refine the relaxation, yielding an increasing sequence of lower bounds that typically converge to the global infimum under mild conditions. In the noncommutative (e.g., quantum) setting, analogous hierarchies employ moment matrices indexed by words in noncommuting variables, as in the Navascués–Pironio–Acín (NPA) or DPS hierarchies for quantum correlations (Tavakoli et al., 2023, Wittek, 2013).

2. SDP Relaxations for Discrete and Combinatorial Optimization

SDP relaxations are especially effective for discrete optimization problems such as binary integer programs (BIP) and combinatorial structures (MAXCUT, TSP, stable set, etc.). The Lovász–Schrijver (L–S) SDP relaxation for BIP illustrates a canonical approach: the original variables $x \in \{0,1\}^n$ are lifted to a matrix $X = [1; x][1, x]^T$; convex relaxations are obtained by dropping the rank and integrality restrictions and enforcing a suite of linear matrix inequalities encoding both the original constraints and additional 'cuts' specific to the combinatorial structure (Paparella, 2012). Such methods produce feasible regions that tightly approximate the convex hull of integer solutions and are provably stronger than standard LP relaxations.

For max–CSPs such as MAXCUT, the Goemans–Williamson SDP relaxation leverages a matrix decomposition to encode the quadratic objective in terms of vector inner products, which is then relaxed to a PSD constraint. Notably, efficient approximation algorithms with provable performance guarantees derive from such relaxations, either via direct solution rounding or probabilistic methods (Xu et al., 2014).

3. Hierarchical and Hierarchy-Collapsing Effects

An essential phenomenon in SDP relaxations is the behavior of hierarchical refinement. In the commutative case, as with Lasserre’s and SOS hierarchies, increased degrees systematically tighten the lower bound, converging to the true optimum—often with finite convergence under suitable regularity conditions and, in moment-based duals, sometimes enabling exact extraction of minimizers (Wang et al., 2013, Jeyakumar et al., 2015, Guo et al., 2015). For noncommutative polynomial optimization relevant in quantum physics (e.g., reduced density matrix or bosonic Hamiltonian minimization), there can be a theoretical collapse: the hierarchy stabilizes after the first step, providing the best possible bound classically. However, numerical SDP implementations may yield strictly improving lower bounds due to floating-point rounding errors artificially perturbing the data, as rigorously analyzed in the context of Weyl algebra positivity (Navascues et al., 2012).

Hierarchical Behavior	Example	Phenomenon
Strictly Tightening	Polynomial optimization (commutative)	Sequence of lower bounds converges under Archimedean/Putinar conditions
Theoretical Collapse	Noncommutative bosonic energy (Weyl algebra)	No improvement after first level, unless perturbed by numerical error
Finite Convergence	Jacobian SOS relaxation	Under genericity, hierarchy terminates finitely with global solution

4. Limitations and Lower Bounds on SDP Relaxation Power

Despite their power, SDP relaxations have fundamental theoretical limits. The semidefinite extension complexity of a polytope—the minimal size of a spectrahedron whose linear image is the polytope—can be superpolynomial or exponential for key combinatorial objects (cut polytope, TSP, stable set). Lower bound techniques exploit connections between the positive semidefinite (psd) rank of certain nonnegative matrices and the sos degree needed to certify nonnegativity. For example, (Lee et al., 2014) proves that no family of polynomial-size SDP relaxations can express the convex hulls of these polytopes or beat 7/8-approximations for MAX–3–SAT, due to high sos degree requirements and corresponding high psd rank. This demonstrates the necessity of exponential-sized SDPs for further gains and underscores a precise limitation of the method.

5. Applications Beyond Classical Optimization

SDP relaxations have become fundamental in quantum information, where they are used to characterize quantum correlations, entanglement, network nonlocality, and quantum key distribution. The NPA hierarchy and its variants exploit moment matrices of noncommuting operators to yield outer approximations of achievable correlation sets, facilitating rigorous device-independent certification and entropic security bounds through SDP feasibility and optimization (Tavakoli et al., 2023). In prepare–and–measure, network, and contextuality scenarios, SDP hierarchies can encode and/or bound operational quantities such as Tsirelson bounds, entropies, and guessing probabilities, enabling powerful numerical and analytical techniques in scenarios previously out of reach.

In control, signal processing, and polynomial matrix inequality (PMI) optimization, SDP hierarchies, including exchange and homogenization approaches for semi-infinite polynomial programming (Wang et al., 2013, Guo et al., 2015), allow global solution schemes for problems involving infinitely many constraints, as well as efficient certificate extraction techniques.

6. Recent Advances: Scalability, Approximate Methods, and Exactness

Recent developments address the computational scalability of SDP relaxations and their exactness properties:

• Dimensionality Reduction: Sparse sub-gaussian random projections have been used to 'sketch' the PSD matrix variable, creating a lower-dimensional SDP; bounds on preservation of feasibility and optimality now depend on both projection dimension and sparsity parameter (Guedes-Ayala et al., 20 Jun 2024). These approximate methods yield high-quality solutions with substantially reduced computational effort, especially when the number of constraints is moderate.
• Biconvex and Nonconvex Reparameterizations: Factorizing the lifted variable as $Y = XX^T$ and solving the resulting biconvex formulation via alternating minimization with careful initialization yields substantial computational gains in large SDP instances (notably, computer vision applications), while maintaining solution quality (Shah et al., 2016).
• Cutting Plane and Projection-based Algorithms: Projective Cutting-Planes algorithms efficiently update both inner and outer approximations by maximizing feasible step-lengths along projection directions, leveraging fast matrix factorization routines for scalability on matrix sizes beyond $2000 \times 2000$ (Porumbel, 2023).
• Exactness and Stability: The conditions under which SDP relaxations (and the broader sum-of-squares hierarchy) are exact—i.e., recover the nonconvex global minimum—have been characterized in terms of strict complementarity, rank-1 generation of the feasible cone, and geometric properties of the constraint set (Cifuentes et al., 2017, Kojima et al., 21 Feb 2025). There exist broad classes of QCQP and polynomial problems for which the SDP relaxation is provably tight, sometimes regardless of the objective function, provided the quadratic constraint system satisfies certain disjointness and summability properties (rank-one generation) or compatibility conditions (e.g., for block-structured constraints).
• Paradoxes and Numerical Phenomena: Practical computation may exhibit paradoxical behavior where hierarchical improvements appear in theory-collapsed cases due to finite precision, as in bosonic energy SDP relaxations, which are then fully explained by the paper's detailed analysis of rounding-induced SOS approximants (Navascues et al., 2012).

7. Impact and Prospective Directions

SDP relaxations now underpin many of the best-known approximation algorithms for NP-hard problems, yield certifiably tight bounds in mixed-integer and noncommutative polynomial optimization, and have profoundly influenced fields such as quantum information and computational algebraic geometry. However, intrinsic complexity lower bounds delimit their reach, steering current research toward hybrid relaxations (combining linear and semidefinite constraints), improved preconditioning and dimensionality reduction, exploitation of symmetry and sparsity, and rigorous understanding of when exactness and efficient solution recovery are achievable.

Active directions include the development of enhanced SDP formulations capturing more structural problem information (such as explicit phase and amplitude constraints in complex quadratic programs (Xu et al., 2023)), construction of scalable and approximate algorithms for large-scale convex relaxations (Guedes-Ayala et al., 20 Jun 2024, Porumbel, 2023), and complete characterizations of exactness and tightness in parametric nonconvex quadratic settings (Cifuentes et al., 2017, Kojima et al., 21 Feb 2025). In quantum and network science, the lens of SDP relaxation continues to expand the capacity for nonlocality certification, device independence, and network inference.

The evolution of both theory and implementation of SDP relaxations thus exemplifies the synergy between convex geometry, algebraic methods, computational optimization, and applications across the mathematical sciences.