SNO: A Polynomial Framework for Graph Optimization via Spectral Relaxations and Moment-SOS Hierarchies

We present SNO (Spectral Neighborhood Optimization), a computationally tractable framework for polynomial optimization on graph-structured problems. SNO combines moment-SOS hierarchies with spectral semidefinite programming methods, building upon the constant trace property (CTP) theory of Mai, Lasserre, and Magron (Math. Prog. Comp., 2023).

Key Innovation:
The framework introduces polynomial-based connectivity penalties specifically designed for compatibility with Sum-of-Squares (SOS) relaxations. Traditional graph connectivity measures use $\det(L)$ (polynomial degree $n$, computationally infeasible) or $\log\det(L)$ (non-polynomial, incompatible with SOS). SNO replaces these with trace-based polynomial penalties:

$$P_{\text{conn}}(X) = -\sum_{r=1}^{R} \alpha_r \text{tr}(L(X)^r)$$

where $L(X) = D(X) - X$ is the graph Laplacian. This formulation maintains polynomial structure with controlled degree (typically $R \leq 4$) while providing rigorous spectral interpretation: $\text{tr}(L^r) = \sum_{i=1}^{n} \lambda_i(L)^r$ penalizes small algebraic connectivity $\lambda_{n-1}(L)$.

Theoretical Contributions:
- Sphere lifting transformations inducing constant trace property with $\text{tr}(P_{m+1,k}M_k(y)P_{m+1,k}) = (1+R^2)^k$
- Explicit boundedness conditions: $\inf_{X \in \mathcal{F}} f_{\text{SNO}}(X) > -\infty$ with $|f_{\text{SNO}}(X)| \leq M$
- Convergence guarantees with explicit $O(k^{-1/c})$ rates under Nie-Schweighofer theory
- Spectral dual formulation: $-\tau_k = \inf_{z \in \mathbb{R}^{m_k}} \{\alpha_k \lambda_1(C_k - \mathcal{A}_k^*(z)) + b_k^\top z\}$ solvable by first-order methods

Applications:
The framework is validated on canonical problems: Traveling Salesman Problem (TSP) with degree constraints $X\mathbf{1} = 2\mathbf{1}$, form nonnegativity testing $\min_{\|x\|_2=1} p(x)$ for homogeneous polynomials $p \in \mathbb{R}[x]_{2d}$, and copositivity verification $\min_{\|y\|_2^2=1, y \geq 0} y^\top A y$.

Computational Advantages:
- Complexity: $O(T \cdot \omega_k^2)$ per iteration vs. $O(\omega_k^3)$ for interior-point methods, where $\omega_k = \binom{m+k}{k}$
- Demonstrated 5-25× speedup over MOSEK/SDPT3 for equality-constrained problems (Mai et al., 2023)
- First-order spectral methods scale to $\omega_k \approx 10^5$ vs. $\omega_k < 10^4$ for interior-point

Keywords:
Polynomial optimization, Moment-SOS hierarchy, Spectral relaxations, Graph connectivity, Combinatorial optimization, Semidefinite programming, Constant trace property, Traveling salesman problem, Laplacian matrix, Algebraic connectivity