Dual-NCP Architectures Overview

• Dual-NCP architectures are specialized frameworks that integrate multi-contact nonlinear complementarity problem formulations with dual-core hardware for enhanced simulation and AI inference.
• The CANAL and SubADMM solvers illustrate trade-offs, where CANAL achieves superlinear local convergence at higher complexity and SubADMM offers superior parallel efficiency.
• Dual connectivity designs in these systems improve reliability by mitigating single-point failures and optimizing resource usage for resilient communications.

Dual-NCP Architectures encompass specialized computational and algorithmic structures for multi-contact nonlinear complementarity problems (NCPs), as well as hardware-software hybrids for dual connectivity and heterogeneously optimized dual-core designs. This article focuses on rigorous definitions, mathematical frameworks, scheduling and tuning methodologies, reliability analyses, and empirical trade-offs for Dual-NCP architectures—primarily referencing advanced robotic simulation methods (Lee et al., 24 Feb 2025), high-throughput AI processor designs (Zhao et al., 2021), and resilient communication protocols under correlated failures (Ganjalizadeh et al., 2019).

1. Mathematical Foundations of Multi-Contact Dual-NCP Architectures

Multi-contact NCPs arise fundamentally in physical simulation with stiff, densely coupled constraints—such as robot manipulation, locomotion, and granular interaction. The velocity-level NCP is expressed: $$\text{Find } (\hat v, \lambda) \text{ s.t. } A\hat v = b + J^T\lambda; \quad (J\hat v,\,\lambda)\in\mathcal{S}_c$$ where $A \in \mathbb{R}^{n \times n}$ (discrete dynamics), $J$ (constraint Jacobian), and $\mathcal{S}_c$ (contact set) encode hard complementarity, spring-damper, and frictional constraints. Augmented Lagrangian approaches recast constraints with slack $z = J\hat v$ and dual multipliers $u$, formulating the problem: $$\mathcal{L}(\hat v,z,u) = \tfrac{1}{2}\hat v^T A \hat v - b^T\hat v + g(z) + u^T(J\hat v-z) + \tfrac{\beta}{2}\|J\hat v-z\|^2$$ with iterations over primal $(\hat v, z)$ and dual $u$ variables. This structure forms the backbone for advanced solver variants (Lee et al., 24 Feb 2025).

2. Cascaded Newton-Based and Subsystem-Based Dual NCP Solvers

CANAL: Cascaded Newton-Based Augmented Lagrangian

The Cascaded Newton-based Augmented Lagrangian (CANAL) method introduces cone complementarity and adaptive penalization. Each Newton update solves $r(\hat v) = \nabla h(\hat v)$ for the convex surrogate $h(\hat v)$, using proximity operators for $\lambda_i$, fully analytic $3\times3$ generalized Hessians $H$, and a safeguarded exact line search. Dual/penalty updates enforce constraint residual shrinkage; adaptive $\beta$ escalation mitigates non-convergence.

SubADMM: Subsystem-Based ADMM

Subsystem-based Alternating Direction Method of Multipliers (SubADMM) decomposes the multibody problem into $N$ subsystems, performing parallel updates of $\hat v_j$ per subsystem and $(Z_i,\lambda_i)$ per contact constraint. Fast small-block linear solves and closed-form contact projections facilitate linear scaling with core count. Adaptive $\beta$ and convergence checks balance primal and dual residuals.

Comparison Table: CANAL vs SubADMM (from (Lee et al., 24 Feb 2025))

Solver	Iter. to $10^{-6}$	Time [ms]	Final Residual
CANAL	10	0.35	$10^{-8}$
SubADMM	100	0.12	$10^{-5}$

CANAL achieves superlinear local convergence and high accuracy but at the cost of global factorization complexity. SubADMM enables order-of-magnitude better parallel efficiency and memory scaling, but requires more iterations and tuning of penalty parameters.

3. Dual-Core Heterogeneous Processor Architectures for AI Inference

The dual-OPU architecture (Zhao et al., 2021) leverages two independently optimized cores: a channel-parallel c-core (for regular convolutions) and a pixel-parallel p-core (for depthwise/pointwise convolutions). Each core integrates homogeneous, fine-grained PE arrays with tailored memory hierarchies.

• c-core: Maximizes runtime PE efficiency for high channel count layers via large channel-parallel PE arrays, omitting line buffers.
• p-core: Optimized for spatially reusable pixel workloads, deploying extra LUT/FF for sliding window buffers and spatial tiling.

A load-balancing scheduling algorithm interleaves operations on layers from different input images and iteratively splits layer tiles to minimize two-batch latency. Design auto-tuning via branch-and-bound explores the space of PE counts and vector widths under multi-resource constraints.

Dual-Core Scheduling Table (from (Zhao et al., 2021))

PE Config	DSP / η_runtime	Throughput (fps)
P(128,9) (baseline)	577 / (59%)	264.6
C(128,12)+P(8,16) dual	832 / (70%)	358.4 (+35.4%)

Area-matched dual-core designs achieve 11% higher runtime PE efficiency and 31% higher throughput over single-core processors. For multi-network workloads, throughput gains average 11% versus state-of-the-art FPGA implementations.

4. Dual Connectivity Architectures: Reliability Under Correlated Failures

Dual connectivity (DC) architectures (Ganjalizadeh et al., 2019) in 5G URLLC (Ultra-Reliable Low Latency Communication) maintain parallel radio links via RAN-split and CN-split designs:

• RAN-split DC: Duplication/removal of packets at the PDCP layer of the Master gNB; prone to single-point failure and sensitive to correlated wireless shadowing.
• CN-split DC: Duplication endpoints in UE (UL) and UPF (DL); distributes risk, tolerates greater link/path length.

Correlation in failures (measured by Pearson coefficient $\rho$) inflates end-to-end packet error probability $P_{e2e}(\rho)$ superlinearly for RAN-split, linearly for CN-split. Even small $\rho\sim 10^{-3}$ mandates architecture selection; CN-split outperforms under shadowed conditions, especially as service distance and intermediate hops increase.

5. Empirical Performance, Trade-Offs, and Optimization

CANAL demonstrates order-of-magnitude higher accuracy ($10^{-8}$), while SubADMM attains superior computational speed and scalability with moderate accuracy ($10^{-5}-10^{-6}$). Scheduling complexity for dual-core overlay processors remains low at runtime despite increased compile-time work. Dual connectivity reliability inherently depends on single-point failure mitigation and correlation-aware network path selection.

Resource allocation trade-offs (DSP, LUT, BRAM) are essential for optimal throughput in dual-core PE designs. Adaptive penalty handling improves speed for ADMM, and parallelism is maximized at both subsystem and constraint levels.

Resource Usage Table (from (Zhao et al., 2021))

PE Array	Line Buf	Multipliers	LUT Total
P(64,9)	39,868	40,896	98,623
C(128,8)	0	72,704	104,453

6. Generalizability and Design Recommendations

The structural principles of dual-NCP architectures extend broadly to other sparsity-exploiting multibody systems (e.g., tendon graphs, deformable objects) and to differentiable simulation for contact inference. Cascade convexification in AL frameworks generalizes whenever prox operators exist for the NCP. SubADMM naturally adapts to any subsystem-partitionable multibody topology.

For communications, avoiding physical proximity-induced correlation (where $\rho \ge 10^{-3}$) dictates favoring CN-split DC architectures and seeking maximum path independence in the core network (Ganjalizadeh et al., 2019). For hardware, specialization at the core level for inference accelerators yields throughput gains that scale with heterogeneity of model workloads and layer types (Zhao et al., 2021).

In summary, Dual-NCP architectures synthesize mathematical rigor, algorithmic specialization, hardware optimization, and statistical reliability analysis to meet practical demands in high-accuracy simulation, real-time AI inference, and ultra-reliable network communications.