---
title: 'RiNNAL-POP: Scalable Algorithm for POP Relaxations'
url: https://www.emergentmind.com/topics/rinnal-pop-algorithmic-framework
type: topic
---

# RiNNAL-POP: Scalable Algorithm for POP Relaxations

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 [2512.06359].

## 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 [2512.06359].

## 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 [2512.06359].

## 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 [2512.06359].

Source: https://www.emergentmind.com/topics/rinnal-pop-algorithmic-framework