A Hybrid Direct-Iterative Method for Solving KKT Linear Systems

Over the past decades, nonlinear optimization has evolved with the development of interior point methods, which rely heavily on solving KKT linear systems. Traditional direct solvers like LDLT, especially MA57, excel on CPU platforms but encounter significant challenges on GPUs due to pivoting-induced communication overhead. Iterative methods such as MINRES and PCG have been explored, but their effectiveness diminishes with ill-conditioned systems common in large-scale problems like power grid optimization. Recent research focuses on hybrid approaches that leverage the structure of KKT matrices, aiming to combine the stability of direct methods with the parallelism of iterative solvers. The surge in GPU hardware capabilities has motivated algorithms that can exploit massive parallelism while maintaining numerical stability, leading to the current study's innovative approach.

The core challenge addressed is the efficient and stable solution of large, sparse, indefinite KKT systems arising in interior point methods. Existing direct methods like LDLT require pivoting, which causes excessive data movement and communication bottlenecks on GPUs. Pure iterative methods struggle with ill-conditioning, leading to slow convergence or divergence. The problem is compounded by the need for high accuracy and robustness in large-scale applications such as power flow analysis. Developing a solver that avoids pivoting, exploits block structures, and guarantees convergence on GPU hardware remains an open and critical problem.

The paper introduces a hybrid algorithm that leverages block-structured Schur complement iteration combined with GPU-optimized sparse Cholesky factorization. The key innovation is the adaptive regularization of the Hγ matrix, ensuring positive definiteness without pivoting. This approach exploits the block structure of KKT matrices, reducing communication and computational costs. Unlike traditional LDLT-based methods, it avoids pivoting entirely, enabling efficient GPU implementation. The method also incorporates spectral analysis to select optimal γ, balancing stability and conditioning. These innovations collectively enable high-speed, stable solutions for large-scale, indefinite sparse systems, addressing a major bottleneck in modern optimization.

• �� Decompose large indefinite KKT systems into smaller positive definite blocks using block-structured Schur complement. • Introduce a parameter γ to regularize the Hessian, forming Hγ = ˜H + γ J^T J, ensuring positive definiteness. • Use spectral analysis to select γ, minimizing condition number κ(Hγ). • Solve the reduced system via iterative conjugate gradient (CG), preconditioned by regularized Hγ, avoiding pivoting. • Implement sparse Cholesky factorization of Hγ on GPU, leveraging its stability without pivoting. • Apply multi-layer regularization (δ1, δ2) to handle low-rank or near-singular J, maintaining robustness. • Exploit block structure to reduce communication, reuse symbolic factorization, and accelerate convergence.

The method was tested on real power grid models, including systems with over 1.6 million variables. The GPU implementation was compared against MA57 LDLT, showing 2-3× speedup. Different γ values (10^4–10^6) were evaluated, with optimal γ yielding about 10 CG iterations for convergence. Eigenvalue clustering near 1 confirmed spectral properties. Error metrics remained below 10^-8, demonstrating high accuracy. Experiments covered various sparsity patterns and problem sizes, validating stability, scalability, and communication efficiency. The results confirmed the method’s practical viability for large-scale industrial problems.

The GPU-based hybrid solver achieved 2-3× speedup over LDLT (MA57) on large power grid problems, with solution errors below 10^-8. Parameter γ tuning (around 10^4–10^6) led to rapid eigenvalue clustering and about 10 CG iterations per solve. The method avoided pivoting, significantly reducing communication overhead, and maintained high numerical stability across diverse test cases. These results demonstrate the approach's effectiveness in real-world large-scale sparse systems, confirming its potential for industrial deployment.

This algorithm is directly applicable to large-scale power grid optimization, robotic motion planning, and autonomous vehicle control, especially where real-time solutions are critical. It can be integrated into interior point solvers for nonlinear problems, providing a GPU-accelerated backbone for high-performance computing. The approach also benefits other domains requiring fast, stable solutions to large sparse indefinite systems, such as structural optimization, machine learning, and economic modeling. Its scalability and robustness make it suitable for industrial-grade applications.

The method's performance hinges on careful γ parameter tuning, which may require problem-specific heuristics. Highly ill-conditioned or extremely low-rank systems may still pose convergence challenges despite regularization. GPU implementation performance varies with hardware architecture, and multi-GPU extensions are not yet fully developed. Initial symbolic analysis and parameter tuning can be computationally expensive for some problems, limiting immediate applicability in time-sensitive scenarios. Further research is needed to automate parameter selection and extend robustness.