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 $D \subseteq \mathbb{R}^n$0 as the vector of these monomials. The key lifting variable is $D \subseteq \mathbb{R}^n$1.

The canonical polyhedral–SDP relaxation seeks

$D \subseteq \mathbb{R}^n$2

where $D \subseteq \mathbb{R}^n$3 is a polyhedral cone (e.g., entrywise nonnegativity for DNN relaxations), and $D \subseteq \mathbb{R}^n$4 enforces the consistency constraints $D \subseteq \mathbb{R}^n$5 whenever $D \subseteq \mathbb{R}^n$6. 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 $D \subseteq \mathbb{R}^n$7: $D \subseteq \mathbb{R}^n$8 where $D \subseteq \mathbb{R}^n$9, $f_i$0, and $f_i$1 encodes linear equality constraints. The augmented Lagrangian is

$f_i$2

parametrized by dual variables $f_i$3 and penalty $f_i$4.

The variable $f_i$5 is eliminated via proximal mappings, and each ALM iteration centers on the minimization of a convex function $f_i$6 over $f_i$7: $f_i$8

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 $f_i$9 with $2\tau \ge \max_i\{\deg f_i\}$0, reducing the number of unknowns and constraints from $2\tau \ge \max_i\{\deg f_i\}$1 to $2\tau \ge \max_i\{\deg f_i\}$2. The nonconvex subproblem

$2\tau \ge \max_i\{\deg f_i\}$3

is addressed via projected gradient steps on the manifold $2\tau \ge \max_i\{\deg f_i\}$4.

• Convex-lifting phase: Once progress in the low-rank objective stalls or the rank is insufficient, a single projected gradient step is performed on $2\tau \ge \max_i\{\deg f_i\}$5 in the original convex feasible set: $2\tau \ge \max_i\{\deg f_i\}$6 where $2\tau \ge \max_i\{\deg f_i\}$7. This corrects for infeasibility, escapes spurious stationary points, and automatically updates the factorization rank via eigendecomposition of $2\tau \ge \max_i\{\deg f_i\}$8.

The projection onto $2\tau \ge \max_i\{\deg f_i\}$9 uses the closed form

$x = (x_0; w) \in \mathbb{R}^{n+1}$0

Projection onto the polyhedral set $x = (x_0; w) \in \mathbb{R}^{n+1}$1 (enforcing possible $x = (x_0; w) \in \mathbb{R}^{n+1}$2 constraints) leverages

$x = (x_0; w) \in \mathbb{R}^{n+1}$3

where $x = (x_0; w) \in \mathbb{R}^{n+1}$4 is an averaging operator over "index-sum" classes to enforce consistency and normalization, and $x = (x_0; w) \in \mathbb{R}^{n+1}$5 applies entrywise nonnegativity, reducing cost to linear in the size of $x = (x_0; w) \in \mathbb{R}^{n+1}$6.

4. Exploiting Facial Structures and Dual Certificate Recovery

Facial reduction is systematically applied by considering the exposed faces of the semidefinite cone defined by $x = (x_0; w) \in \mathbb{R}^{n+1}$7 and $x = (x_0; w) \in \mathbb{R}^{n+1}$8. Any feasible point admits the representation

$x = (x_0; w) \in \mathbb{R}^{n+1}$9

where rows of $x_0 = 1$0 span $x_0 = 1$1. Restricting $x_0 = 1$2 to this subspace sustains feasibility and tightens the relaxation.

Dual certificate recovery for KKT optimality is achieved by

$x_0 = 1$3

which satisfies $x_0 = 1$4 for the computed $x_0 = 1$5, ensuring complementarity and obviating the need for solving large linear systems beyond the initial inversion of $x_0 = 1$6.

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_0 = 1$7 with index set $x_0 = 1$8;
• Consistency via $x_0 = 1$9;
• Constraints represented via localizing matrices $2\tau$0, and additional auxiliary variables $2\tau$1.

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 $2\tau$2 converge to a KKT point of the polyhedral–SDP problem, even with inexact subproblem solutions.
• Partial-smoothness property: The indicator $2\tau$3 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 $2\tau$4 for first-order updates and one $2\tau$5 eigendecomposition, with effective practical scaling approaching linearity in the number of nonzero constraints for moderate $2\tau$6.

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 ($2\tau$7) for dimensions up to $2\tau$8 ($2\tau$9) and $\bar{f}_i(x) = x_0^{2\tau-\deg f_i} f_i(w/x_0).$0 ($\bar{f}_i(x) = x_0^{2\tau-\deg f_i} f_i(w/x_0).$1).

Empirically recommended hyperparameters include:

• Initial penalty $\bar{f}_i(x) = x_0^{2\tau-\deg f_i} f_i(w/x_0).$2, adapting if primal residuals greatly exceed dual;
• Initial factorization rank $\bar{f}_i(x) = x_0^{2\tau-\deg f_i} f_i(w/x_0).$3;
• Barzilai–Borwein steps and nonmonotone line search in the low-rank phase;
• Projected-gradient stepsize $\bar{f}_i(x) = x_0^{2\tau-\deg f_i} f_i(w/x_0).$4, commonly $\bar{f}_i(x) = x_0^{2\tau-\deg f_i} f_i(w/x_0).$5, 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).