Spectral Projected Interior Trajectory Method

• The Spit method is an optimization framework that integrates barrier functions and spectral stabilization to ensure strict feasibility and convergence to KKT points.
• It employs a damped velocity–Verlet update with spectral nudges to maintain stability in high-dimensional, constrained geometries like sphere packings.
• The methodology bridges barrier dynamics with rigidity theory, offering precise control of geometric properties and energy descent guarantees for reproducibility.

The Spectral Projected Interior Trajectory (Spit) Method is a rigorously formulated optimization methodology combining barrier-augmented objectives, discrete-time dynamical schemes, and spectral rigidity principles. Developed in the context of high-dimensional sphere packing and generalized constrained optimization, Spit enforces strict interior feasibility while guaranteeing convergence to Karush-Kuhn-Tucker (KKT) points and enabling precise control of geometric and combinatorial properties, including periodic rigidity. The method is based on a unified Riemannian variational and spectral framework with a smooth interior barrier, energy descent principles, and a convergence-guaranteed discrete update. It is distinct from matrix-free approaches, which relax projection enforcement via ordinary differential equations; Spit’s primary contributions lie in its explicit barrier formulation, spectral stabilization, and energy-based feasibility preservation (Alpay et al., 25 Sep 2025).

1. Barrier-Based Geometric Modeling and Riemannian Variational Framework

Central to Spit is a barrier-based geometric model that rigorously prevents violation of packing constraints. For periodic sphere packings, the method introduces a $C^2$ interior barrier potential: $$U_\nu(x,B) = \sum_{i<j} \sum_{t \in S} \varphi_\nu(s_{ij, t}(x, B)),$$ where

$$\varphi_\nu(s) = -\nu \log s + \frac{\nu}{2\delta} (s - \delta)^2 \quad (s > 0),$$

and $s_{ij, t}(x, B) = \|x_i - x_j - t\|^2 - 4$ measures squared separation with respect to lattice translations. The logarithmic term ensures the function blows up as $s \to 0$, strictly enforcing feasibility, while the quadratic term imparts strong convexity near the safety margin ($s = \delta$), essential for theoretical guarantees. The $C^2$ smoothness and the existence of a Lipschitz gradient on the strictly feasible domain facilitate both practical stability and Lyapunov-based energy analysis.

2. Discrete-Time Dynamics: Velocity–Verlet-Type Update and Spectral Regularization

Spit formalizes and corrects the discrete-time update of system states via a damped velocity–Verlet scheme: $$\begin{aligned} v^{(k+1/2)} &= v^k - \frac{\eta \Delta t}{2} v^k - \frac{\Delta t}{2} \nabla U_\nu(x^k, B^k), \ x^{(k+1/2)} &= x^k + \Delta t \, v^{(k+1/2)}, \ v^{(k+1)} &= \left(1 - \frac{\eta \Delta t}{2}\right) v^{(k+1/2)} - \frac{\Delta t}{2} \nabla U_\nu(x^{(k+1/2)}, B^k), \ x^{(k+1)} &= x^{(k+1/2)}. \end{aligned}$$ Explicit step size ($\Delta t$) and damping ($\eta$) constraints ($0 < \eta\Delta t < 2$, $\Delta t^2 \leq 1/2$) guarantee stability. The update exploits the barrier's smoothness and is supplemented by "spectral nudges"—adjustments exploiting the connectivity (second eigenvalue $\lambda_2$) of the packing contact graph—to safeguard against near-degeneracy and improve the conditioning of optimization steps.

3. Barrier-to-KKT Consistency and Quadratic Regularization

A core theoretical guarantee is convergence of minimizers of the barrier-augmented objective to KKT points as $\nu \to 0$: $$\min_{x,B} \Phi_\nu(x,B) = V(B) - \nu \sum \log s_{ij,t}(x,B), \qquad \text{or} \qquad \tilde{\Phi}_\nu(x,B) = V(B) + U_\nu(x,B),$$ where $V(B)$ is the lattice volume. The quadratic term in $\varphi_\nu$ is critical for bounding the Lagrange multipliers,

$$\mu_{ij,t}^{(\nu)} = \frac{\nu}{s_{ij,t}} - \frac{\nu}{\delta}(s_{ij,t} - \delta),$$

guaranteeing smoothness and convexity near feasibility boundaries. The result, documented in “Log-barrier ⇒ KKT” and its corollary (Alpay et al., 25 Sep 2025), bridges barrier dynamics with standard constrained optimization theory.

4. Prestress Stability, Rigidity Theory, and Spectral Properties

Spit establishes connections to rigidity theory by proving that strict prestress stability, defined via an equilibrium stress $\omega$ with a quadratic form

$$Q_\omega(u, A) = \sum_{(i,j,t) \in \mathcal{A}} \omega_{ij,t} [n_{ij,t}^\top(u_i - u_j - Ar_{ij,t})]^2,$$

implies periodic infinitesimal rigidity. That is, positive definiteness of $Q_\omega$ over nontrivial motions ensures that feasible infinitesimal motions are trivial (i.e., congruent with problem symmetries), guaranteeing a packing’s geometric stability and uniqueness. This criterion links the combinatorial structure of the packing's graph to its continuous dynamical properties.

5. Lyapunov Energy Descent, Feasibility Projection, and Convergence Guarantees

Energy descent along trajectories is enforced by a Lyapunov function: $$\mathcal{E}^k = U_\nu(x^k, B^k) + \frac{1}{2}\|v^k\|^2 + \frac{\gamma}{2}\|x^k - x^{k-1}\|^2,$$ with appropriate choice of $\gamma$ (e.g., $\gamma = 1/\Delta t^2 - (\cdots)/2$) ensuring nonnegativity. Proposition “Discrete (unprojected) Lyapunov descent” offers monotonic decrease of $\mathcal{E}$ for each update under step-size and damping constraints, foundational for convergence proofs.

Strict feasibility is maintained by an energy nonexpansive feasibility projection: $$\min_{y} M_{x,B}(y) \quad \text{subject to} \quad s_{ij,t}(x,B) + \langle \nabla_x s_{ij,t}(x,B), y - x\rangle \geq \delta,$$ where $M_{x,B}(y)$ majorizes $U_\nu$ plus quadratic stabilization terms. A joint projection routine also exists for simultaneously updating $x$ and lattice basis $B$. The “Joint nonexpansiveness” lemma shows that these projections do not increase $\mathcal{E}$.

Under regularity assumptions (strict local minimizer $x^*$ and projected Hessian $\nabla_{xx}^2 U_\nu(x^*, B^*) \geq mI$ for $m>0$), the method achieves local linear convergence with step-size $\Delta t < 2/\sqrt{m}$ and damping $0 < \eta \Delta t < 2$.

6. Practical Implementation and Reproducibility Guidelines

For algorithmic reproducibility, explicit formulas for Hessian–vector products are provided, e.g.,

$$[\nabla_{xx}^2 U_\nu p]_i = \sum_{j,z} [4\varphi'' r_c (r_c^\top (p_i - p_j)) + 2 \varphi' (p_i - p_j)],$$

per sphere coordinate block, with similar formulas for mixed and lattice blocks. These formulas enable power-iteration schemes for spectral estimation of Lipschitz and curvature constants.

A reproducibility stub is documented for low dimensions ($n=2$, $N=32,64$ spheres) specifying $\delta = 10^{-3}$, $\nu = 10^{-2}$, $\eta \Delta t \in [0.8, 1.2]$, and appropriate $\Delta t$. The finite shift set $S$ (e.g., $Bz$ with $\|z\|_\infty \leq 1$) is prescribed. Logging strategies include monitoring $\mathcal{E}^k$, minimal slack, and $\lambda_2(G^k)$ for the contact graph. Practical guidance includes x-only updates with periodic joint projection and activation of spectral nudges based on graph connectivity.

7. Relation to Matrix-Free ODE-Based Interior Point Methods

A notable methodological comparison exists with matrix-free interior point trajectories based on augmented Lagrangian ODEs (Qian et al., 28 Dec 2024). Both Spit and matrix-free methods generate convergent interior trajectories and can be discretized. However, Spit’s approach requires explicit projections and Hessian-based updates, potentially producing ill-conditioned matrices near boundaries. In contrast, matrix-free methods deploy a scaling matrix $U$ and relax constraint enforcement, avoiding expensive matrix inverses and operating solely with matrix–vector products. This distinction makes Spit suited for scenarios demanding geometric control and explicit feasibility, whereas matrix-free ODE-based algorithms may excel in large-scale or ill-conditioned regimes, for example in linearly constrained convex programming.

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The Spectral Projected Interior Trajectory method constitutes a rigorous convergence-guaranteed optimization strategy, distinguished by $C^2$ barrier enforcement, spectral stabilization, geometric rigidity analysis, and explicit energy descent control. Combined with precise formulas and reproducibility protocols, it enables strict feasibility preservation and robust convergence for complex constrained problems in high dimensions.