Moment-SOS Hierarchies Overview

• Moment-SOS hierarchies are a sequence of semidefinite programming relaxations that use polynomial and convex analysis to provide convergent bounds for nonconvex optimization problems.
• They leverage dual certificates, moment and localizing matrices, and the flat extension criterion to recover global minimizers under the Archimedean condition.
• The framework scales to large and sparse problems, with applications in control, PDEs, stochastic processes, and combinatorial optimization.

A Moment-SOS (Sum-of-Squares) hierarchy is a sequence of increasingly tight semidefinite programming (SDP) relaxations for polynomial optimization problems, grounded in the theory of the K-moment problem and dual algebraic certificates for positivity on semialgebraic sets. It delivers convergent upper and lower bounds for a broad class of problems, including static optimization, pathwise properties of stochastic processes, generalized moment problems, and polynomial optimal control. Moment-SOS hierarchies exploit both polynomial algebra and convex analysis, and serve as the mathematical backbone for measure-LP duality, occupation measures in dynamical systems, and many scalable algorithms in systems control, combinatorial optimization, signal processing, and PDEs.

1. Mathematical Foundations and Basic Construction

The archetypal problem addressed by the Moment-SOS hierarchy is global polynomial optimization over a compact basic semialgebraic set,

$$K = \{ x \in \mathbb{R}^n : g_j(x) \geq 0,\ j=1,\dots,m \}$$

with $f \in \mathbb{R}[x]$. The minimum is equivalently cast as

$$f^* = \inf_{x \in K} f(x) = \inf_{\mu \in P(K)} \int f(x)\, d\mu(x)$$

where $\mu$ is a Borel probability measure on $K$ ($P(K)$), or dually as

$$f^* = \sup\{ \lambda : f(x) - \lambda \geq 0 \quad \forall x \in K \}.$$

Moment sequences $y = (y_\alpha)_{|\alpha| \leq 2d}$ with $y_\alpha = \int x^\alpha\, d\mu(x)$ are introduced. The moment matrix $M_d(y)$ and, for each constraint, the localizing matrix $f \in \mathbb{R}[x]$0 encode the Gram–type positivity constraints associated to measures supported on $f \in \mathbb{R}[x]$1: $f \in \mathbb{R}[x]$2 The $f \in \mathbb{R}[x]$3-th order relaxation is the SDP: $f \in \mathbb{R}[x]$4 This is the primal moment relaxation. The dual (SOS) relaxation seeks certificates

$f \in \mathbb{R}[x]$5

where $f \in \mathbb{R}[x]$6 denotes the cone of sum-of-squares polynomials.

The standard setting for convergence is the Archimedean property,

$f \in \mathbb{R}[x]$7

ensuring $f \in \mathbb{R}[x]$8 is compact and Putinar’s Positivstellensatz applies (Lasserre, 2018).

2. Duality, Certificates, and Flat Extension

Primal-dual strong duality holds under the Archimedean condition: the primal moment relaxation and the dual SOS relaxation attain the same value for large enough $f \in \mathbb{R}[x]$9, and the sequence of relaxations is non-decreasing and convergent to $$f^* = \inf_{x \in K} f(x) = \inf_{\mu \in P(K)} \int f(x)\, d\mu(x)$$0 (Lasserre, 2018). Practical extraction of minimizers leverages the "flat extension" criterion: if the optimal $$f^* = \inf_{x \in K} f(x) = \inf_{\mu \in P(K)} \int f(x)\, d\mu(x)$$1 at level $$f^* = \inf_{x \in K} f(x) = \inf_{\mu \in P(K)} \int f(x)\, d\mu(x)$$2 satisfies

$$f^* = \inf_{x \in K} f(x) = \inf_{\mu \in P(K)} \int f(x)\, d\mu(x)$$3

then $$f^* = \inf_{x \in K} f(x) = \inf_{\mu \in P(K)} \int f(x)\, d\mu(x)$$4 arises from a finitely supported measure on $$f^* = \inf_{x \in K} f(x) = \inf_{\mu \in P(K)} \int f(x)\, d\mu(x)$$5, i.e., the global optimizers are recovered (Lasserre, 2018). This algebraic-geometric finite convergence is generically achieved under nondegeneracy conditions such as strict complementarity and second-order sufficiency (Huang et al., 2024).

3. Structure, Variants, and Extensions

Several significant extensions and variants exist:

• Generalized Problems of Moments (GPM): Application to optimal control and PDEs interprets the dynamic or infinite-dimensional constraints as linear constraints on occupation measures, yielding measure-LP formulations and associated hierarchies (Lasserre, 2018, Chhatoi et al., 2024, Sehnalová et al., 17 Mar 2025).
• Polynomial Matrix and Matrix Inequality Problems: The matrix moment-SOS hierarchy generalizes the scalar case to polynomial matrix constraints, introducing matrix-valued moment and localizing matrices; finite convergence under matrix analogues of KKT and second-order conditions has been established (Guo et al., 2023, Huang et al., 2024, Handa et al., 2 Sep 2025).
• Complex and Sparse Hierarchies: The theory supports efficient real moment-HSOS hierarchies for complex polynomial problems and highly scalable sparse variants exploiting term and/or correlative sparsity (TSSOS, chordal-TSSOS, CS-TSSOS), often yielding orders-of-magnitude reductions in SDP block sizes (Wang et al., 2023, Wang et al., 2019, Wang et al., 2020, Shaydurova et al., 2024, Wang et al., 2020).
• Regularized Hierarchies for Non-compact Sets: Using the Carleman condition, density theorems guarantee convergence of a regularized moment-SOS hierarchy on non-Archimedean domains or unbounded sets (Henrion, 5 Dec 2025).

4. Convergence: Rates, Finite Convergence, Exactness

Convergence behavior exhibits fine analytic structure:

• Asymptotic Convergence: Under Putinar’s condition, the moment-SOS hierarchy yields a monotonic sequence converging to the exact minimum (or solution to the associated generalized problem) (Lasserre, 2018, Henrion et al., 2021, Chhatoi et al., 2024).
• Finite Convergence via Hidden Convexity: Strong convexity or SOS-convexity of the objective (sometimes only locally) or of constraint data can ensure finite convergence at a relaxation order dictated by the degree of convexity certificates, even in globally non-convex settings (Ðurašinović et al., 27 Feb 2026).
• Rate Analysis: The convergence rate is governed by the geometry of the feasible set, described by its Łojasiewicz exponent $$f^* = \inf_{x \in K} f(x) = \inf_{\mu \in P(K)} \int f(x)\, d\mu(x)$$6. The error at order $$f^* = \inf_{x \in K} f(x) = \inf_{\mu \in P(K)} \int f(x)\, d\mu(x)$$7 is $$f^* = \inf_{x \in K} f(x) = \inf_{\mu \in P(K)} \int f(x)\, d\mu(x)$$8: for the ball, $$f^* = \inf_{x \in K} f(x) = \inf_{\mu \in P(K)} \int f(x)\, d\mu(x)$$9; for general polytopes, $\mu$0; for locally strongly convex sets, $\mu$1; for general sets, the rate is dictated by the exponent $\mu$2 (Tran et al., 1 Jul 2025, Schlosser et al., 2024). Effective bounds are obtained using explicit degree estimates for Putinar’s Positivstellensatz and polynomial approximation theory.
• Exactness Cones: The set of objective polynomials for which the relaxation is exact at finite $\mu$3 forms a union of spectrahedral cones, directly tied to the geometry of the moment spectrahedron and the faces exposed by Dirac measures at global minimizers (Henrion, 2023).

5. Computational Complexity and Scalability

While the dimension of the corresponding SDP at order $\mu$4 is polynomial in the number of variables ($\mu$5 for most problems), the bit complexity of certificates may be exponential in $\mu$6 and/or $\mu$7 in pathological cases, as demonstrated for Boolean problems (Gribling et al., 2023). Polynomial-time solvability is guaranteed under explicit algebraic and geometric strict feasibility and spectral-gap conditions, providing inner-ball and outer-ball certificates for the feasible set of the SDP (Gribling et al., 2023). In practical settings, these conditions are often achieved via explicit bounding constraints or verified numerically.

Advanced implementations scale to hundreds or thousands of variables leveraging modern sparsity-exploiting frameworks (TSSOS, chordal-TSSOS, CS-TSSOS, refined TSSOS), which block-diagonalize the moment and localizing matrices according to a hierarchy of monomial and variable interaction graphs. Benchmarks illustrate dramatic speedup and scalability for large sparse combinatorial, power system, and control instances (Wang et al., 2019, Wang et al., 2020, Wang et al., 2020, Shaydurova et al., 2024).

6. Applications Across Domains

Moment-SOS hierarchies are foundational in a wide range of applications:

• Control and PDEs: Computation of reachable sets, region of attraction, value functions, occupation measures for optimal control, and Liouville-driven measure relaxations for nonlinear and stochastic PDEs, always with rigorously convergent monotonic lower/upper bounds (Chhatoi et al., 2024, Sehnalová et al., 17 Mar 2025, Henrion et al., 2021).
• Stochastic Processes: Quantitative bounding of exit time and location distributions for polynomial SDEs, via the measure-LP and sum-of-squares duality with direct convergence guarantee (Henrion et al., 2021).
• Combinatorial and Large-Scale Optimization: Tight convex relaxations for Max-Cut, mixed-integer quadratic programs, AC-OPF, robust design, and code design; partial or fully sparse hierarchies target scalable bounds in memory and time (Wang et al., 2019, Wang et al., 2020, Shaydurova et al., 2024).
• Signal and System Recovery: Sparse superresolution, system identification, optimal experimental design, sparse polynomial interpolation, and matrix-valued extensions for robust optimization (Lasserre, 2018, Guo et al., 2023).
• Theoretical Insights: Analysis of exactness geometry, structure of spectrahedral cones of objective functionals, and links to Christoffel-Darboux kernels for the interpretation of finite convergence (Lasserre, 2020, Henrion, 2023).

7. Interpretations, Limitations, and Future Directions

Current research elucidates the algebraic and geometric underpinnings of finite/infinite convergence, the tight connections to real algebraic geometry via Putinar’s and Scheiderer’s theorems, and the analytic rate proxy via the Łojasiewicz exponent. Challenges remain in quantifying effective rates for higher-dimensional and non-smooth domains, extending finite convergence certificates, and optimizing solver architectures for the largest instances. Regularized hierarchies using Carleman determinacy and penalization extend the reach to non-compact or non-Archimedean feasible sets (Henrion, 5 Dec 2025). Applications to infinite-dimensional problems (PDEs, control, measure-driven evolution) are rapidly expanding in both theory and computation.

The Moment-SOS hierarchy constitutes a rigorous, versatile analytic-computational framework for polynomial optimization and measure-driven analysis with rapidly maturing scalability and depth across mathematical disciplines (Lasserre, 2018, Henrion et al., 2021, Gribling et al., 2023, Wang et al., 2019, Chhatoi et al., 2024, Tran et al., 1 Jul 2025, Ðurašinović et al., 27 Feb 2026).