Lasserre Hierarchy for Polynomial Optimization

• Lasserre Hierarchy is a systematic framework that globally solves polynomial optimization by approximating nonconvex problems with convex SDP relaxations.
• It leverages sum-of-squares certificates and moment theory to guarantee monotone convergence under specific algebraic and geometric conditions.
• Extensions to sparse, noncommutative, and SOCP variants enhance its applicability for large-scale and structured optimization problems.

The Lasserre hierarchy is a systematic framework for globally solving polynomial optimization problems (POPs) via a sequence of semidefinite programming (SDP) relaxations of increasing size. Each level in the hierarchy approximates the generally nonconvex problem by a convex SDP, yielding a monotone sequence of lower bounds converging to the global optimum. The approach is grounded in real algebraic geometry, specifically in sum-of-squares (SOS) certificates of nonnegativity and moment theory. The hierarchy's versatility allows extensions to structured problems (e.g., sparse, complex, noncommutative), and its convergence and duality properties are tightly linked to the underlying geometric and algebraic assumptions.

1. Problem Setup and Canonical Formulation

A generic polynomial optimization problem is defined as follows: minimize a real polynomial $f(x)$ over a semialgebraic set $K \subseteq \mathbb{R}^n$: $$\min_{x \in K} f(x), \quad K = \{ x \in \mathbb{R}^n : g_i(x) \geq 0,\, i=1,\dots,m \},$$ where $f,g_i \in \mathbb{R}[x]$. The nonconvexity and potential multi-extremality render such problems computationally hard in the general case (Josz et al., 2014).

To approximate $f^* = \inf_{x \in K} f(x)$, the Lasserre hierarchy constructs, for each relaxation order $d \geq d_{\min}$ (where $d_{\min}$ depends on the degrees of the $g_i$), a primal moment relaxation and a dual sum-of-squares (SOS) SDP.

2. Moment and SOS Relaxations

At each level $d$, the primal (moment) SDP seeks a truncated moment sequence $y = (y_\alpha)_{|\alpha| \leq 2d}$:

• Moment matrix: $M_d(y)$ is the symmetric matrix indexed by monomials up to degree $d$, with entries $[M_d(y)]_{\alpha, \beta} = y_{\alpha+\beta}$.
• Localizing matrices: For each constraint $g_i$, $M_{d-d_i}(g_i y)$ encodes the moment conditions weighted by $g_i$.

The primal SDP is: $$\min_{y} \sum_{|\alpha| \leq \deg(f)} f_\alpha y_\alpha \quad \text{subject to} \quad y_0 = 1,\quad M_d(y) \succeq 0,\quad M_{d-d_i}(g_i y) \succeq 0, \ i = 1, \dots, m$$ where $d_i = \lceil \deg(g_i)/2 \rceil$ (Josz et al., 2014).

The dual relaxation seeks the largest $z \in \mathbb{R}$ such that: $$f(x) - z = \sigma_0(x) + \sum_{i=1}^m \sigma_i(x) g_i(x)$$ with each $\sigma_i$ an SOS polynomial, $\deg(\sigma_i g_i) \leq 2d$.

3. Convergence, Duality, and Finite Termination

Under an Archimedean condition (e.g., $R^2 - \|x\|^2$ included as a redundant constraint to guarantee compactness of $K$), the sequence of lower bounds from the hierarchy converges monotonically to $f^*$: $\rho_0 \leq \rho_1 \leq \cdots \uparrow f^*$ Strong duality holds at each level (no duality gap) provided the redundant ball constraint ensures the compactness of the feasible set, even if $K$ has empty interior (Josz et al., 2014).

Finite convergence, i.e., $\rho_{d_0} = f^*$ for some $d_0<\infty$, is guaranteed under additional regularity conditions at all global minimizers: constraint qualification (CQ), strict complementarity (SC), and second order sufficiency (SOSC) (Nie, 2012). For finite varieties or finite feasible sets, such as when $K$ is finite or the real variety of equalities is finite, finite convergence holds generically and is independent of the specific defining polynomials (Nie, 2012).

A generalized "flat truncation" or flat extension criterion—that is, rank stability of the moment matrix, $$\operatorname{rank} M_d(\bar y) = \operatorname{rank} M_{d-1}(\bar y)$$—enables constructive extraction of the global minimizers through spectral decomposition (Josz et al., 2014, Quijorna, 2017).

4. Quantitative Convergence Rates

The convergence rate of the hierarchy can be made explicit on specific domains. For the unit hypercube $[-1,1]^n$, the tightest general result is $O(1/r^2)$ for the error after $r$ levels, established both through kernel approaches and analysis of extremal zeros of orthogonal polynomials (Klerk et al., 2018, Slot, 2021, Klerk et al., 2019). More precisely, on the cube or the hypersphere, the error in the lower bound scales as: $f^* - f_r = O\left(\frac{1}{r^2}\right)$ where $f_r$ is the $r$-th Lasserre relaxation value (Slot, 2021, Klerk et al., 2019). Lower bounds matching this rate are also constructed, for instance, using linear polynomials in coordinate directions and explicit analysis of zeros of Jacobi and Chebyshev polynomials.

For general compact semialgebraic sets or those only satisfying a "fatness" condition, a convergence rate of $O(\log^2 r / r^2)$ is achievable via measure-based univariate kernel constructions (Slot et al., 2020, Gribling et al., 1 May 2025), and $O(1/r)$ in settings with only Putinar-type Positivstellensatz (Gribling et al., 1 May 2025). For compact convex sets, these rates hold for the standard hierarchy, and extensions to Schmüdgen-type and multivariate settings exist via Christoffel-Darboux kernel techniques (Slot, 2021).

5. Advancements, Variants, and Computational Strategies

Tightened and Sparse Hierarchies

By incorporating explicit polynomial representations for Lagrange multipliers (KKT multipliers), one can derive tightened hierarchies that achieve finite convergence under milder conditions and often require fewer relaxation levels in practice (Nie, 2017). The multi-ordered Lasserre hierarchy adapts the relaxation order per constraint and exploits sparsity patterns, which is crucial for large-scale problems, such as AC optimal power flow with thousands of variables and constraints (Josz et al., 2017).

Bounded-Degree and SOCP Variants

To control the computational complexity, bounded-degree hierarchies and mixed SDP/SOCP relaxations have been formulated, leveraging Krivine-Stengle certificates and enabling convergence with fixed-size matrix blocks, particularly suitable for SOCP-convex polynomials or polynomials with essentially nonpositive coefficients (Chuong et al., 2017).

Extensions to Noncommutative and Complex Variables

The Lasserre hierarchy admits powerful generalizations to noncommutative polynomial optimization, yielding the NPA hierarchy in quantum information, and to optimization in the complex domain. In the complex case, convergence and solution extraction rely on hyponormality conditions and require additional operator-theoretic structure (Josz et al., 2017, Klep et al., 2024, Wang, 2024).

Numerical Implementation and First-Order Methods

Practical variants of the hierarchy, including fine-grained relaxations that incrementally add PSD blocks, allow for efficient warm-start and scalable ADMM or first-order methods, making the solution of large-level relaxations tractable (Ma et al., 2019).

6. Limitations, Worst-Case Behavior, and Integrality Gaps

While the Lasserre hierarchy is theoretically exact in the limit (and often finitely exact in practice), pathological formulations exist. For 0/1 polynomial optimization, integrality gaps at level $n-1$ can persist only in the presence of "single-vertex cutting" (SVC) constraints or degree-$n$ polynomials. These cases are rare for naturally arising combinatorial problems, where the hierarchy typically closes the gap rapidly (Kurpisz et al., 2015, Mastrolilli, 2013).

The hierarchy's worst-case convergence rate is tightly connected to the extremal behavior of associated orthogonal polynomials, with no possible general improvement beyond $O(1/r^2)$ (Klerk et al., 2018, Gribling et al., 1 May 2025). For unconstrained or non-Archimedean cases, convergence is not guaranteed (Nie, 2012).

7. Summary Table of Key Results

Setting / Extension	Key Properties	References
General compact $K$	Monotone convergence to $f^*$; strong duality via redundant ball constraint	(Josz et al., 2014)
Quantitative rates ($[-1,1]^n$, sphere)	$O(1/r^2)$ convergence; rates matched by lower bounds via orthogonal polynomials	(Klerk et al., 2019, Klerk et al., 2018, Slot, 2021)
Tight & sparse hierarchies	Finite convergence under CQ, SC, SOSC or via KKT multipliers; multi-ordered for sparsity	(Nie, 2017, Josz et al., 2017)
Noncommutative, complex, SOCP variants	Generalized to noncommutative, complex, and mixed SDP/SOCP; hyponormality, block-diag	(Chuong et al., 2017, Josz et al., 2017, Klep et al., 2024)
Integrality gaps and limits	Pathological (but rare) gaps at $n-1$ for 0/1 cases; combinatorial moment analysis	(Kurpisz et al., 2015, Mastrolilli, 2013)

8. Practical Impact and Applications

The Lasserre hierarchy underpins state-of-the-art global polynomial optimization solvers, exact algorithms for polynomial systems, and critical relaxations in combinatorial and power systems optimization. Its performance for structured or large-scale instances is contingent on exploiting sparsity, block structure, and advanced algorithmic variants. Recent work shows that it is possible to achieve global optimality for systems with thousands of variables through symmetry reduction and customized hierarchies (Josz et al., 2017).

The spectrum of extensions (upper bound hierarchies, noncommutative domains, complex variable frameworks, and bilevel normality conditions) reflects the continual deepening of the theory, with convergence, duality, and finite exactness now being understood to a high degree under refined geometric conditions (Slot, 2021, Wang, 2024).