Second-Order Optimization via Quiescence
<|reference_start|>Second-Order Optimization via Quiescence: Second-order optimization methods exhibit fast convergence to critical points, however, in nonconvex optimization, these methods often require restrictive step-sizes to ensure a monotonically decreasing objective function. In the presence of highly nonlinear objective functions with large Lipschitz constants, increasingly small step-sizes become a bottleneck to fast convergence. We propose a second-order optimization method that utilizes a dynamic system model to represent the trajectory of optimization variables as an ODE. We then follow the quasi-steady state trajectory by forcing variables with the fastest rise time into a state known as quiescence. This optimization via quiescence allows us to adaptively select large step-sizes that sequentially follow each optimization variable to a quasi-steady state until all state variables reach the actual steady state, coinciding with the optimum. The result is a second-order method that utilizes large step-sizes and does not require a monotonically decreasing objective function to reach a critical point. Experimentally, we demonstrate the fast convergence of this approach for optimizing nonconvex problems in power systems and compare them to existing state-of-the-art second-order methods, including damped Newton-Raphson, BFGS, and SR1.<|reference_end|>
@article{agarwal2024second-order,
    title={Second-Order Optimization via Quiescence},
    author={Aayushya Agarwal, Larry Pileggi, Ronald Rohrer},
    journal={arXiv preprint arXiv:2410.08033},
    year={2024},
    archivePrefix={arXiv},
    eprint={2410.08033},
    primaryClass={math.OC cs.SY eess.SY}
}