C² Interior Barrier Uν in Optimization

• C² Interior Barrier Uν is a twice continuously differentiable potential designed to enforce strict interior feasibility in constrained optimization and sphere packing.
• It combines a logarithmic barrier with quadratic regularization to guarantee smooth gradients and robust convergence under discrete iterative updates.
• The method connects barrier-to-KKT consistency with geometric rigidity, supporting reliable numerical implementations in both nonlinear programming and periodic sphere packing.

C² Interior Barrier $U_\nu$

The C² interior barrier $U_\nu$ is a twice continuously differentiable potential function designed to enforce strict interior feasibility in constrained optimization and geometric packing problems, particularly in high-dimensional periodic sphere packings and nonlinear programming. Through a combination of logarithmic barrier terms and quadratic regularization, $U_\nu$ achieves both analytic smoothness (C²) and stability under discrete iterative update schemes. This construction enables rigorous convergence analysis, reliable numerical implementation, barrier-to-KKT consistency, and direct connections with geometric rigidity properties in contact frameworks.

1. Mathematical Definition and Regularity

$U_\nu$ is constructed on the slab of strictly feasible slack variables. For sphere packing, each slack

$s_{ij, t}(x, B) = \|x_i - x_j - t\|^2 - 4$

is maintained above a safety margin $\delta > 0$. The C² barrier is then defined as

$$U_\nu(x, B) = \sum_{i < j} \sum_{t \in S} \;\varphi_\nu(s_{ij, t}(x, B))$$

with the potential function

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

Here, $\nu > 0$ controls the barrier strength; the quadratic term regularizes the raw log-barrier.

The augmented quadratic term guarantees $U_\nu$ is globally C². Its gradient is Lipschitz on the region $\{s_{ij, t} \in [\delta, S]\}$. Specifically, Lemma 2.1 of (Alpay et al., 25 Sep 2025) provides explicit bounds for the Lipschitz constant, critical for ensuring discrete stability and second-order method compatibility.

2. Role in Periodic Sphere Packing Models

In periodic sphere packing, one aims to minimize the cell volume $V(B)$ under the non-overlap constraints $s_{ij, t} \geq 0$. $U_\nu$ serves as a barrier term in the augmented objective

$\tilde{\Phi}_\nu(x, B) = V(B) + U_\nu(x, B),$

which ensures strict interior feasibility as the algorithm evolves sphere positions $x$ and lattice basis $B$.

A strict feasibility safeguard requires each $s_{ij, t}(x, B) \geq \delta$. The log-barrier $-\nu \log s$ diverges as $s \to 0$, making violations highly penalized, while the quadratic term smooths the landscape near $s = \delta$. This guarantees trajectories do not approach the feasibility boundary, preserving contact structure and enabling robustness in numerical updates.

3. Discrete-Time Dynamics and Barrier-KKT Consistency

The damped velocity-Verlet scheme (Eq. 12, (Alpay et al., 25 Sep 2025)) is used for discrete updates: $$\begin{aligned} v^{k+\frac{1}{2}} &= v^k - \frac{\eta\Delta t}{2} v^k - \frac{\Delta t}{2} \nabla U_\nu(x^k, B^k), \ x^{k+\frac{1}{2}} &= x^k + \Delta t\, v^{k+\frac{1}{2}}, \ v^{k+1} &= (1-\frac{\eta\Delta t}{2}) v^{k+\frac{1}{2}} - \frac{\Delta t}{2}\, \nabla U_\nu(x^{k+\frac{1}{2}}, B^k), \ x^{k+1} &= x^{k+\frac{1}{2}} \end{aligned}$$ Explicit damping and step size criteria (e.g., $0 < \eta\Delta t < 2$, $\Delta t^2 \leq 1/2$) guarantee Lyapunov energy descent.

Minimizers of $V(B) + U_\nu(x, B)$ converge to KKT points of the original packing as $\nu \downarrow 0$, provided the active contacts are nondegenerate (LICQ). The limiting Lagrange multipliers for contact constraints are

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

which remain bounded and are shown to approach valid KKT values as per Theorem 3.1 in (Alpay et al., 25 Sep 2025).

4. Stability, Rigidity, and Variational Structure

Prestress stability is formalized: if there exists an equilibrium contact stress $\omega$ such that the quadratic form

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

is positive definite on nontrivial motions (modulo rigid-body translations and lattice symmetries), the packing is periodically infinitesimally rigid. This result connects the barrier approach to classical rigidity theory and validates the stability of the geometric network under infinitesimal perturbations (Alpay et al., 25 Sep 2025).

The paper establishes that Lyapunov functions combining $U_\nu$ and discrete velocity/step variables descend monotonically under the given update rules: $$\mathcal{E}^k \equiv U_\nu(x^k, B^k) + \frac{1}{2} \|v^k\|^2 + \frac{\gamma}{2}\|x^k - x^{k-1}\|^2,$$ with $\gamma$ chosen according to $\Delta t$ to ensure $\mathcal{E}^{k+1} \leq \mathcal{E}^k$.

5. Interior Barrier Methods in Nonlinear Programming

In nonlinear programming contexts, "C² Interior Barrier $U_\nu$" designates smooth barriers such as the log-barrier (and smoothed variants) used in path-following interior-point algorithms (Neuenhofen, 2018). Twice continuous differentiability is essential for Newton-type updates and for guaranteeing quadratic convergence near solutions.

Logarithmic barriers in the merit function prevent iterates from reaching constraint boundaries: $$-\tau_e \cdot \mathbf{1}^\top \left( \log(x - x_{\text{lower}}) + \log(x_{\text{upper}} - x) \right).$$ Combined with quadratic penalization and modified augmented Lagrangian updates, this leads to robust convergence and feasibility maintenance. The trust-funnel mechanism further discourages infeasible stationary points by penalizing constraint violation with additional C² barrier terms (Neuenhofen, 2018).

6. Hessian-Vector Formulas and Implementation

Explicit Hessian-vector formulas facilitate smoothness and curvature estimation, crucial for explicit time step and stability controls. For sphere packing,

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

where $r_c = x_i - x_j - Bz$, $s_c = \|r_c\|^2 - 4$ (Alpay et al., 25 Sep 2025). Similar formulations extend to cross-terms and cell basis variations.

Reproducibility stubs recommend concrete parameter values (e.g., $\delta$, $\nu$, $\eta$, $\Delta t$) and explicit initialization schemes, enabling reliable empirical verification and direct transfer to simulation environments in low-dimensional or moderate-scale high-dimensional settings (Alpay et al., 25 Sep 2025).

7. Connections to Regularity Theory and Fully Nonlinear PDEs

The concept of an interior C² barrier generalizes the role of uniform second derivative estimates in analytic regularity theory. In scalar curvature problems and $\sigma_2$-Hessian equations, interior C² estimates serve as analytic barriers, preventing second derivative blow-up and ensuring regularity of convex or suitably controlled admissible solutions (Guan et al., 2017, Qiu, 2017). The barrier property directly underpins existence and uniqueness proofs in geometric PDEs and isometric embedding problems, where convexity or weakened conditions on higher symmetric functions (e.g., $\sigma_3(D^2u) \geq -A$) enable robust interior regularity results.

Summary Table: Mathematical Formulations and Properties of $U_\nu$

Feature / Formula	Description	Reference
$$\varphi_\nu(s) = -\nu\log s + (\nu/(2\delta))(s - \delta)^2$$	C² potential, smooths log-barrier for strict feasibility	(Alpay et al., 25 Sep 2025)
$U_\nu(x,B) = \sum \varphi_\nu(s_{ij,t})$	Barrier on sphere packing contact constraints	(Alpay et al., 25 Sep 2025)
$$\mathcal{E}^k = U_\nu(x^k,B^k) + \frac{1}{2}\|v^k\|^2 + (\gamma/2)\|x^k-x^{k-1}\|^2$$	Discrete Lyapunov energy function	(Alpay et al., 25 Sep 2025)
$-\tau_e \cdot \sum \log(\cdot)$	Logarithmic barrier term in NLP merit function	(Neuenhofen, 2018)

The C² interior barrier $U_\nu$ thus provides a unified analytic and algorithmic device for enforcing robust strict feasibility, facilitating convergence analysis, enabling practical computation, and ensuring geometric stability across diverse contexts including sphere packing, nonlinear programming, and fully nonlinear differential equations.