Polynomial Optimization via Lasserre's SDP Hierarchy

• Polynomial optimization via Lasserre's SDP hierarchy is a method that systematically approximates the global minimum of a polynomial using convex semidefinite programming relaxations.
• It leverages moment matrices and sum-of-squares certificates to provide explicit degree bounds that guarantee finite convergence under generic regularity conditions.
• The approach enables efficient extraction of optimizers via flat truncation and extends to various cases including gradient, grid, and binary optimization problems.

Polynomial optimization via Lasserre's semidefinite programming (SDP) hierarchy refers to the systematic approximation of the global minimum of a real polynomial, possibly constrained by polynomial equalities and/or inequalities, by a sequence of convex SDP relaxations based on truncated moment sequences and sum-of-squares (SOS) certificates. The correctness, rate of convergence, and explicit degree bounds for finite convergence of this hierarchy are governed by deep interrelations between algebraic geometry, real algebraic optimization, and nonlinear programming optimality theory.

1. Foundations of Lasserre's SDP (Moment–SOS) Hierarchy

Given a polynomial optimization problem over $\mathbb{R}^n$:

$$\min \{ f(x) : g_1(x) = 0, \, \ldots, \, g_n(x) = 0 \}$$

with $$f, g_1, \ldots, g_n \in \mathbb{R}[x_1, \ldots, x_n]$$, the solution set (i.e., real zero locus of the constraints) is assumed to be finite. The Lasserre hierarchy introduces a sequence of truncated moment variables $y = (y_\alpha)_{|\alpha| \leq 2k}$ corresponding to the multi-indices $\alpha$, and defines the linear Riesz functional $L_y(p) = \sum p_\alpha y_\alpha$ for polynomials $p(x)$.

The $k$th-level relaxation replaces nonconvex feasible sets with convex constraints:

• Moment matrix $M_k(y) \succeq 0$ (where $[M_k(y)]_{\alpha,\beta} = y_{\alpha+\beta}$, $$\min \{ f(x) : g_1(x) = 0, \, \ldots, \, g_n(x) = 0 \}$$0),
• Localizing matrix constraints $$\min \{ f(x) : g_1(x) = 0, \, \ldots, \, g_n(x) = 0 \}$$1 for each $$\min \{ f(x) : g_1(x) = 0, \, \ldots, \, g_n(x) = 0 \}$$2 ($$\min \{ f(x) : g_1(x) = 0, \, \ldots, \, g_n(x) = 0 \}$$3),
• Normalization $$\min \{ f(x) : g_1(x) = 0, \, \ldots, \, g_n(x) = 0 \}$$4.

The relaxation value is

$$\min \{ f(x) : g_1(x) = 0, \, \ldots, \, g_n(x) = 0 \}$$5

which satisfies $$\min \{ f(x) : g_1(x) = 0, \, \ldots, \, g_n(x) = 0 \}$$6 (where $$\min \{ f(x) : g_1(x) = 0, \, \ldots, \, g_n(x) = 0 \}$$7 is the true global minimum).

2. Finite Convergence and Effective Degree Bounds

Finite convergence refers to the termination of the hierarchy at some finite level $$\min \{ f(x) : g_1(x) = 0, \, \ldots, \, g_n(x) = 0 \}$$8 such that $$\min \{ f(x) : g_1(x) = 0, \, \ldots, \, g_n(x) = 0 \}$$9. The work "Exactness and Effective Degree Bound of Lasserre's Relaxation for Polynomial Optimization over Finite Variety" establishes explicit degree bounds for this phenomenon under generic geometric conditions:

• (A1) No Solution at Infinity: The homogeneous system defined by the leading (highest degree) parts $$f, g_1, \ldots, g_n \in \mathbb{R}[x_1, \ldots, x_n]$$0 must have no common (nontrivial) complex root at infinity. Formally, for each constraint $$f, g_1, \ldots, g_n \in \mathbb{R}[x_1, \ldots, x_n]$$1, consider its homogenization $$f, g_1, \ldots, g_n \in \mathbb{R}[x_1, \ldots, x_n]$$2 and require the system $$f, g_1, \ldots, g_n \in \mathbb{R}[x_1, \ldots, x_n]$$3 for all $$f, g_1, \ldots, g_n \in \mathbb{R}[x_1, \ldots, x_n]$$4 has only the trivial solution.
• (A2) Nonsingularity at Real Minimizers: All real optimal solutions $$f, g_1, \ldots, g_n \in \mathbb{R}[x_1, \ldots, x_n]$$5 must be nonsingular for the constraint Jacobian $$f, g_1, \ldots, g_n \in \mathbb{R}[x_1, \ldots, x_n]$$6 (i.e., $$f, g_1, \ldots, g_n \in \mathbb{R}[x_1, \ldots, x_n]$$7 is of full rank $$f, g_1, \ldots, g_n \in \mathbb{R}[x_1, \ldots, x_n]$$8).

Main Theorem (Explicit Bound):

Under (A1)-(A2), the Lasserre hierarchy attains the exact minimum at

$$f, g_1, \ldots, g_n \in \mathbb{R}[x_1, \ldots, x_n]$$9

meaning $y = (y_\alpha)_{|\alpha| \leq 2k}$0 for all $y = (y_\alpha)_{|\alpha| \leq 2k}$1 (Hua et al., 2021).

In SOS dual form, the optimality certificate is: $y = (y_\alpha)_{|\alpha| \leq 2k}$2 where $y = (y_\alpha)_{|\alpha| \leq 2k}$3 is an SOS (sum of at most two squares), and the degrees of $y = (y_\alpha)_{|\alpha| \leq 2k}$4 and $y = (y_\alpha)_{|\alpha| \leq 2k}$5 are both bounded by $y = (y_\alpha)_{|\alpha| \leq 2k}$6.

3. Algebraic–Geometric and Positivity Mechanisms

The proof utilizes projective algebraic geometry:

• The absence of solutions at infinity implies the constraints define a complete intersection zero-dimensional variety, giving rise to precise Hilbert series and basis properties ("H-basis") for the constraint ideal.
• For any $y = (y_\alpha)_{|\alpha| \leq 2k}$7 in the constraint ideal $y = (y_\alpha)_{|\alpha| \leq 2k}$8, there exist $y = (y_\alpha)_{|\alpha| \leq 2k}$9 of appropriate degree so that $\alpha$0 with $\alpha$1.
• Over the real points, nonsingularity at minimizers allows the construction of a sum-of-squares positivity certificate on the variety.

Using these facts, the difference $\alpha$2 is shown, modulo the ideal, to be a sum of squares (at most two squares) and constraints, allowing the explicit degree bound above.

4. Specialized Cases: Unconstrained, Gradient, and Binary Optimization

• Unconstrained minimization reduces to the case of $\alpha$3 equality constraints $\alpha$4, i.e., vanishing gradient. The explicit degree bound for finite convergence of the "gradient-type" Lasserre (SOS) relaxation is

$\alpha$5

yielding exact recovery of the global minimum for all $\alpha$6 under genericity and Hessian nonsingularity (Hua et al., 2021).

• Grid case ($\alpha$7) retrieves earlier degree bounds such as Laurent's $\alpha$8.
• Binary optimization problems over $\alpha$9 (e.g., quadratic or higher-order pseudo-Boolean problems) fit into this framework; for quadratic objectives, the degree bound becomes $L_y(p) = \sum p_\alpha y_\alpha$0, generalizable to higher degree (Hua et al., 2021).

5. Importance of Flat Truncation, Extraction, and Practicality

For finite convergence, it is necessary and sufficient (under suitable genericity) that the moment matrix $L_y(p) = \sum p_\alpha y_\alpha$1 exhibits "flat truncation," i.e., rank stability in $L_y(p) = \sum p_\alpha y_\alpha$2 and $L_y(p) = \sum p_\alpha y_\alpha$3 for some $L_y(p) = \sum p_\alpha y_\alpha$4, where $L_y(p) = \sum p_\alpha y_\alpha$5 (Nie, 2011). This property enables numerical extraction of all optimizers via spectral decomposition of $L_y(p) = \sum p_\alpha y_\alpha$6.

From an implementation viewpoint, no relaxation order above $L_y(p) = \sum p_\alpha y_\alpha$7 is required: SDPs of order $L_y(p) = \sum p_\alpha y_\alpha$8 are sufficient to recover $L_y(p) = \sum p_\alpha y_\alpha$9 and minimizers, dramatically reducing computational complexity in finite-variety regimes.

6. Broader Context: Varieties, Certificates, and Generalizations

The finite convergence and explicit degree bound results extend prior theory:

• For real varieties with finite solution sets, Lasserre's hierarchy converges in finitely many steps, independently of whether the complex variety is finite (Nie, 2012).
• Certificates based on genericity and critical point exactness can be generated for radical and non-radical ideal cases, and for both gradient-ideal and classical grid or binary settings.
• In the convex setting, under a saddle-point and strict Hessian condition (not strict convexity of $p(x)$0 itself), the hierarchy also enjoys finite convergence (Jeyakumar et al., 2013). More generally, convergence (possibly only asymptotic) is guaranteed under Archimedean or compactness conditions (Tacchi, 2020).

These results provide theoretical and practical guidelines for when semidefinite-based polynomial optimization yields exact answers after finitely many SDP steps.

7. Implications, Limitations, and Scope

• The explicit bounds apply generically in the coefficient space: the “no solution at infinity” and “nonsingularity” hypotheses are satisfied on a Zariski-open set of constraint tuples of fixed degrees.
• Non-generic or singular cases (e.g., solutions at infinity, singular minimizers, or non-reduced varieties) may exhibit only asymptotic convergence or may require more advanced Positivstellensatz certificates.
• The theoretical results apply to a substantial class of structured polynomial optimization problems, such as those arising in discrete, binary, grid, or generic gradient-vanishing contexts, and now provide sharp degree bounds absent in the previous literature.

The finite convergence at an explicit and computable relaxation order enables efficient global optimization in all scenarios where the variety defined by the constraints is finite and sufficiently regular [(Hua et al., 2021); (Nie, 2012); (Nie, 2011)].