RiNNAL-POP: Scalable Algorithm for POP Relaxations

• RiNNAL-POP is a framework that uses low‐rank augmented Lagrangian methods to solve large-scale polyhedral semidefinite and moment–SOS relaxations for polynomial optimization problems.
• It reformulates the relaxation via tailored projection schemes and splitting techniques, unifying SDP, DNN, RLT, and SOS approaches in a conic programming setup.
• Empirical studies show 5×–100× runtime improvements and high solution accuracy (KKT residual <10⁻⁶) across benchmarks with high-dimensional problem instances.

The RiNNAL-POP algorithmic framework is a low-rank augmented Lagrangian method (ALM) designed to solve large-scale polyhedral semidefinite programming (SDP) relaxations and moment–sum-of-squares (SOS) relaxations of polynomial optimization problems (POPs). By exploiting low-rank factorization, tailored projection schemes, and hidden facial structures in the conic relaxations, RiNNAL-POP achieves improved scalability and solution accuracy for high-dimensional and highly constrained POP instances, substantially outperforming prior state-of-the-art solvers on benchmark problems (Hou et al., 6 Dec 2025).

1. Problem Formulation and Polyhedral–SDP Relaxation

Consider the general POP of the form

$$\zeta^* = \min_{w \in D}\ \{ f_0(w) \mid f_i(w) = 0,\ i=1,\dots,m \},$$

where $D \subseteq \mathbb{R}^n$ is a conic feasibility domain and each $f_i$ is a real multivariate polynomial. The relaxation process proceeds in two standard steps: homogenization and lifting.

• Homogenization: For given even order $2\tau \ge \max_i\{\deg f_i\}$, set $x = (x_0; w) \in \mathbb{R}^{n+1}$, $x_0 = 1$, and define the degree-$2\tau$ homogenization

$\bar{f}_i(x) = x_0^{2\tau-\deg f_i} f_i(w/x_0).$

• Lifting: Let $\mathcal{A} \subset \mathbb{N}^{n+1}$ index all degree-$\tau$ monomials, and define $u^{\mathcal{A}}(x)$ as the vector of these monomials. The key lifting variable is $$X = u^{\mathcal{A}}(x) u^{\mathcal{A}}(x)^\top \in \mathbb{S}^{|\mathcal{A}|}$$.

The canonical polyhedral–SDP relaxation seeks

$$\zeta^{\rm relax} = \min_{X \in \mathbb{S}^{\mathcal{A}}_+ \cap \mathcal{P}^{\mathcal{A}} \cap \mathcal{L}^{\mathcal{A}}} \left\{ \langle Q^0, X \rangle \,\big|\, \langle H^0, X \rangle = 1;\, A X = 0;\, \langle Q^i, X \rangle = 0 \right\},$$

where $\mathcal{P}^{\mathcal{A}}$ is a polyhedral cone (e.g., entrywise nonnegativity for DNN relaxations), and $\mathcal{L}^{\mathcal{A}}$ enforces the consistency constraints $X_{\alpha,\beta} = X_{\gamma,\delta}$ whenever $\alpha + \beta = \gamma + \delta$. The relaxation unifies various standard hierarchies—standard SDP, diagonally dominant (DNN), RLT, and SOS—under a general conic program (Hou et al., 6 Dec 2025).

2. Augmented Lagrangian Splitting and Algorithmic Structure

The polyhedral–SDP relaxation is reformulated in splitting form over primal variables $(X, Y)$: $$\min_{X,Y} \; \langle Q^0, X \rangle + \delta_{F \cap \mathbb{S}_+}(X) + \delta_P(Y)\quad \text{s.t.}\ X - Y = 0,\ Q(X) = b,$$ where $F = \{X: AX = 0\}$, $$P = \mathcal{P}^{\mathcal{A}} \cap \mathcal{L}^{\mathcal{A}} \cap \{\langle H^0, X \rangle = 1\}$$, and $Q(X)$ encodes linear equality constraints. The augmented Lagrangian is

$$L_\sigma(X, Y; y, W) = \langle Q^0, X \rangle - \langle y, Q(X) - b \rangle - \langle W, X - Y \rangle + \tfrac{\sigma}{2}\|Q(X)-b\|^2 + \tfrac{\sigma}{2}\|X-Y\|^2,$$

parametrized by dual variables $(y, W)$ and penalty $\sigma$.

The variable $Y$ is eliminated via proximal mappings, and each ALM iteration centers on the minimization of a convex function $\phi(X)$ over $X \in F \cap \mathbb{S}_+$: $$\phi(X) = \langle Q^0, X \rangle + \frac{\sigma}{2}\|Q(X) - b - \sigma^{-1}y\|^2 + \frac{\sigma}{2}\|X - \sigma^{-1}W - \Pi_P(X - \sigma^{-1}W)\|^2.$$

3. Low-rank Algorithmic Steps and Projection Schemes

RiNNAL-POP employs a hybrid two-phase strategy in every ALM subproblem:

• Low-rank phase: The primal matrix is factorized as $X = RR^\top$ with $R \in \mathbb{R}^{|\mathcal{A}| \times r}$, reducing the number of unknowns and constraints from $O(n^2)$ to $O(nr)$. The nonconvex subproblem

$\min_{R: AR = 0}\ \phi(RR^\top)$

is addressed via projected gradient steps on the manifold $\{R: AR = 0\}$.

• Convex-lifting phase: Once progress in the low-rank objective stalls or the rank is insufficient, a single projected gradient step is performed on $X$ in the original convex feasible set: $$X \leftarrow \Pi_{F \cap \mathbb{S}_+}(\widehat{X} - t\nabla \phi(\widehat{X})),$$ where $\widehat{X} = RR^\top$. This corrects for infeasibility, escapes spurious stationary points, and automatically updates the factorization rank via eigendecomposition of $X$.

The projection onto $F \cap \mathbb{S}_+$ uses the closed form

$$\Pi_{F \cap \mathbb{S}_+}(G) = \Pi_{\mathbb{S}_+}(J G J),\qquad J = I - A^\top(AA^\top)^{-1}A.$$

Projection onto the polyhedral set $P$ (enforcing possible $O(n^{2\tau})$ constraints) leverages

$\Pi_P = \Pi_{N^A} \circ \Pi_{^A},$

where $\Pi_{^A}$ is an averaging operator over "index-sum" classes to enforce consistency and normalization, and $\Pi_{N^A}$ applies entrywise nonnegativity, reducing cost to linear in the size of $X$.

4. Exploiting Facial Structures and Dual Certificate Recovery

Facial reduction is systematically applied by considering the exposed faces of the semidefinite cone defined by $AX = 0$ and $X \succeq 0$. Any feasible point admits the representation

$$AX=0 \;\Leftrightarrow\; \exists \Theta \succeq 0:\ X = U^\top \Theta U,$$

where rows of $U$ span $\ker(A)$. Restricting $X$ to this subspace sustains feasibility and tightens the relaxation.

Dual certificate recovery for KKT optimality is achieved by

$S = J\nabla \phi(X)J,$

which satisfies $XS = 0$ for the computed $X$, ensuring complementarity and obviating the need for solving large linear systems beyond the initial inversion of $AA^\top$.

5. Extension to Moment–Sum-of-Squares Hierarchies

The RiNNAL-POP framework generalizes to moment–SOS relaxations, such as the Lasserre hierarchy, by casting these relaxations in the same splitting form:

• Moment matrices $X \in \mathbb{S}_+^{A_\tau}$ with index set $A_\tau$;
• Consistency via $\mathcal{L}^{A_\tau}$;
• Constraints represented via localizing matrices $M_{h_j}(X) \succeq 0$, and additional auxiliary variables $Y^{(j)} \succeq 0$.

The ALM subproblem then includes one low-rank/convex-lifting phase per matrix block, and projections are extended accordingly, maintaining efficiency and scalability for large-scale moment–SOS relaxations (Hou et al., 6 Dec 2025).

6. Theoretical Guarantees: Convergence and Complexity

Rigorous theoretical results for the ALM under the RiNNAL-POP framework are established:

• Global ALM convergence: With mild boundedness and Slater conditions, the iterates $(X^k, y^k, W^k)$ converge to a KKT point of the polyhedral–SDP problem, even with inexact subproblem solutions.
• Partial-smoothness property: The indicator $\Psi(X) = \delta_{F \cap \mathbb{S}_+}(X)$ is partly smooth relative to its manifold, aiding local analysis and convergence.
• Finite-step rank identification: Under a nondegeneracy condition, the algorithm identifies the rank of solution matrices in finite steps.
• Complexity: Each ALM iteration costs $O(|\mathcal{A}| r^2 + |\mathcal{A}|^{2\tau})$ for first-order updates and one $O(|\mathcal{A}|^3)$ eigendecomposition, with effective practical scaling approaching linearity in the number of nonzero constraints for moderate $r$.

7. Empirical Performance and Practical Implementation

Extensive numerical experiments on benchmark POPs—including StQP, BIQ, MBP, MQKP, BQM, KM, matrix/tensor copositivity, and nonnegative tensor factorization—demonstrate empirical superiority to SDPNAL+, with typical runtime improvements of 5×–100×, recovery of low-rank solutions, and high solution accuracy ($\mathrm{KKT\;res}<10^{-6}$) for dimensions up to $n \approx 6000$ ($\tau=1$) and $n \approx 80$ ($\tau=2$).

Empirically recommended hyperparameters include:

• Initial penalty $\sigma_0 \approx 1$, adapting if primal residuals greatly exceed dual;
• Initial factorization rank $r_0 = \min\{200, \lceil|\mathcal{A}|/5\rceil\}$;
• Barzilai–Borwein steps and nonmonotone line search in the low-rank phase;
• Projected-gradient stepsize $t \leq 2\delta/L$, commonly $t = 1/\sigma$, in the convex phase;
• Early termination of the low-rank phase upon objective stalling, followed by a single convex-lifting correction.

Collectively, these methodological and computational advances yield a robust, scalable framework for the solution of large-scale polyhedral–SDP and moment–SOS relaxations in polynomial optimization (Hou et al., 6 Dec 2025).