Moment–SOS Hierarchy Overview

• The Moment–SOS hierarchy is a systematic method that reformulates polynomial optimization as a sequence of semidefinite programs using dual SOS certificates.
• It utilizes moment and localizing matrices to certify nonnegativity, ensuring convergence to the global optimum under conditions like the Archimedean property.
• Extensions of the hierarchy address matrix, robust, sparse, and infinite-dimensional problems, with algorithmic enhancements enabling scalable implementations and finite convergence detection.

The moment–sum-of-squares (SOS) hierarchy, often called the Lasserre hierarchy, is a foundational methodology in polynomial optimization, polynomial matrix optimization, and generalized moment problems. It provides a systematic, convex relaxation framework for approximating infima of polynomials under semialgebraic constraints and supports a wide array of extensions to complex, infinite-dimensional, matrix-valued, robust, and sparse settings. The essential mechanism is the duality between representing a sequence as moments of a measure and certifying nonnegativity via sum-of-squares polynomials, realized via a hierarchy of semidefinite programs (SDPs) whose solutions exhibit strong convergence theorems, finite convergence under structural conditions, and a variety of practical computational relaxations.

1. Foundational Principles and Duality

The classical moment–SOS hierarchy addresses the polynomial optimization problem (POP)

$$f^* = \inf_{x \in K} f(x), \quad K = \{ x \in \mathbb{R}^n : g_j(x) \ge 0, \, j = 1, \dots, m \}$$

by casting it as a generalized moment problem (GMP), seeking a representing measure $\mu$ whose moments $y_\alpha = \int_K x^\alpha d\mu(x)$ satisfy certain positivity constraints (Lasserre, 2018, Tacchi, 2020).

This approach is dual to the sum-of-squares (SOS) certificate of nonnegativity. Under an Archimedean quadratic module (i.e., with $R - \|x\|^2 \in Q(g)$ for some $R$), any polynomial strictly positive on $K$ admits a decomposition

$$f(x) - \lambda = \sigma_0(x) + \sum_j \sigma_j(x) g_j(x), \quad \sigma_j \in \mathrm{SOS}$$

with degree bounds determined by the order of relaxation (Lasserre, 2018). The corresponding moment SDP imposes positive semidefiniteness (PSD) on the (truncated) moment matrix $M_d(y)$ and weighted (localizing) matrices $M_{d-d_j}(g_j y)$, yielding a monotone sequence of relaxations whose optimal values converge to $f^*$ as $\mu$0 under the Putinar-Archimedean condition (Tacchi, 2020, Tran et al., 1 Jul 2025, Schlosser et al., 2024).

The dual–primal relationship extends to a large class of GMPs, such as measure-projected LPs, volume, optimal control, and nonlinear PDEs (Tacchi, 2020, Henrion et al., 2023, Chhatoi et al., 2024).

2. Structure of the Hierarchy: Primal and Dual SDPs

At each order $\mu$1, the primal (moment) SDP uses the truncated moment sequence $\mu$2:

• The moment matrix $\mu$3 collects $\mu$4 for $\mu$5.
• For each constraint $\mu$6, the localizing matrix $\mu$7 imposes PSD constraints reflecting measure support on $\mu$8.

The SDP formulation (primal side) is:

$\mu$9

The dual (SOS) SDP seeks sum-of-squares representations:

$y_\alpha = \int_K x^\alpha d\mu(x)$0

Strong duality holds under interiority and compactness conditions (Tacchi, 2020, Lasserre, 2018).

These constructions extend to generalized moment problems with additional moment constraints and to cases where positivity must be enforced on noncompact sets using regularization and Carleman conditions (Henrion, 5 Dec 2025).

3. Convergence, Flat Extension, and Finite Convergence

If the quadratic module is Archimedean, convergence of the SDP hierarchy is guaranteed:

$y_\alpha = \int_K x^\alpha d\mu(x)$1

where $y_\alpha = \int_K x^\alpha d\mu(x)$2 is the optimal value at relaxation order $y_\alpha = \int_K x^\alpha d\mu(x)$3 (Tacchi, 2020). Under flat extension, i.e., when

$y_\alpha = \int_K x^\alpha d\mu(x)$4

finite convergence is certified and global minimizers can be extracted via linear algebraic procedures (multiplication matrices, joint diagonalization) (Guo et al., 2023, Huang et al., 2024).

Finite convergence also occurs whenever the problem exhibits hidden strong convexity or SOS-convexity, with explicit degree thresholds provided by Putinar-type matrix positivity certificates (Ðurašinović et al., 27 Feb 2026). In matrix polynomial optimization, the nondegeneracy, strict complementarity, and second-order sufficient conditions at all minimizers guarantee finite convergence and eventual flatness of moments (Huang et al., 2024).

4. Extensions: Matrix, Robust, Sparse, Infinite-dimensional, and Regularized Hierarchies

The hierarchy generalizes to:

• Polynomial matrix inequalities (PMIs): Problems with robust or uncertain matrix constraints, including uncertainty sets defined via PMIs, are treated by introducing matrix-valued moment and localizing matrices, with PSD-SOS-convexity yielding strong duality, convergence, and flat extension results (Guo et al., 2023, Huang et al., 2024). Extraction of finitely-atomic matrix-valued measures and global minimizers proceeds analogously to the scalar case.
• Robust and eigenvalue minimization: Applications to robust semidefinite programs and eigenvalue minimization of polynomial matrices are handled with the same architecture (Guo et al., 2023).
• Complex polynomials and Hermitian-SOS: The real moment-HSOS hierarchy matches the lower bounds of the complex version for real-data, achieves speedup via halved matrix sizes, and admits tight rank-detection criteria for global optimality under sphere constraints (Wang et al., 2023).
• Infinite-dimensional problems: The moment–SOS framework extends to evolution PDEs in Hilbert or distribution spaces, reformulated as measure-valued Liouville equations. Discretizing cylindrical polynomials and moments yields SDP relaxations whose solutions converge to the true measure-valued dynamics, with global convergence under dissipativity (Chhatoi et al., 2024, Henrion et al., 2023).
• Unbounded domains and non-Archimedean sets: Regularized versions of the hierarchy employ Carleman-determinacy and L²-density results, providing strong convergence even without compactness or Positivstellensatz, and yield monotonic certified lower bounds after suitable penalization (Henrion, 5 Dec 2025).

5. Algorithmic Enhancements: Exploiting Sparsity and Scalability

Computational demands motivate scalable variants exploiting problem structure:

• Term sparsity and block decompositions: The TSSOS framework and its variants (block, chordal, CS-TSSOS, refined TSSOS) utilize monomial adjacency graphs to decompose large moment/localizing matrices into block-diagonal form (Wang et al., 2019, Wang et al., 2020, Wang et al., 2020, Shaydurova et al., 2024). Support–extension and chordal–extension procedures iteratively build block structures, often drastically reducing computational cost without sacrificing tightness. Refined block-splitting algorithms further optimize block size using combinatorial optimization (Shaydurova et al., 2024).
• Sublevel and interpolatory hierarchies: The sublevel Moment–SOS hierarchy interpolates between consecutive orders by selectively enlarging matrix blocks (“levels” and “depths”). Fine-tuning these hyperparameters balances bound quality and memory footprint, yielding practical improvements in applications such as Max-Cut and neural network verification (Chen et al., 2021).
• Dual sparsity (correlative + term): The CS-TSSOS hierarchy marries correlative (variable) and term (monomial) sparsity to achieve two-level block-diagonal SDPs with strong convergence and demonstrable tractability for problems with thousands of variables (Wang et al., 2020).

6. Quantitative Convergence Rates, Complexity, and Generic Finite Convergence

Recent advances provide explicit convergence rates and complexity bounds:

• Convergence rates: The hierarchy achieves rates $y_\alpha = \int_K x^\alpha d\mu(x)$5 where $y_\alpha = \int_K x^\alpha d\mu(x)$6 is the Lojasiewicz exponent of the set, with $y_\alpha = \int_K x^\alpha d\mu(x)$7 possible in highly regular domains (ball, simplex, sphere), $y_\alpha = \int_K x^\alpha d\mu(x)$8 for polytopes, and $y_\alpha = \int_K x^\alpha d\mu(x)$9 under strong convexity (Tran et al., 1 Jul 2025, Schlosser et al., 2024).
• Bit complexity: Although the SDP at fixed order has polynomial size in $R - \|x\|^2 \in Q(g)$0, the worst-case bit complexity may be exponential (as shown for certain combinatorial and univariate instances). However, under geometric (interior ball) or spectral-gap conditions, polynomial-time solvability at each order is guaranteed (Gribling et al., 2023).
• Finite convergence “in practice”: Generic finite convergence, ensured for generic $R - \|x\|^2 \in Q(g)$1 under the Archimedean property, can be detected by monitoring flatness of the moment matrix (Lasserre, 2020).

7. The Christoffel–Darboux Interpretation and Extraction of Minimizers

Each moment–SOS relaxation may be viewed as constructing a polynomial density in $R - \|x\|^2 \in Q(g)$2 over the feasible set, with the optimal density being the Christoffel–Darboux kernel associated to the global minimizer. This reproducing kernel perspective underlies the extraction of minimizers as solutions to systems involving orthonormal polynomials, and relates the density maximization principle to Dirac-like measure recovery (Lasserre, 2020).

At exactness, the Christoffel function is sharply peaked at the minimizer and inverse Christoffel values are diagnostic of optimality.

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The moment–SOS hierarchy is thus a comprehensive mathematical and algorithmic construction for global polynomial optimization, semidefinite generalized problems of moments, and their matrix-, infinite-dimensional-, and robust extensions. Central issues concerning convergence, extraction of solutions, computational complexity, and practical scalability have led to a rich body of methodologies and theoretical guarantees within this paradigm (Lasserre, 2018, Tran et al., 1 Jul 2025, Tacchi, 2020, Guo et al., 2023, Ðurašinović et al., 27 Feb 2026, Huang et al., 2024).