Second-Order Optimization via Quiescence

TL;DR

This paper introduces a novel second-order optimization method that employs a quiescence-based approach to adaptively use large step-sizes, enabling faster convergence on nonconvex problems without requiring monotonic decrease of the objective.

Contribution

The paper presents a new second-order optimization technique using a dynamic system model and quiescence to improve convergence speed on nonconvex problems.

Findings

Demonstrates fast convergence on power system optimization problems

Outperforms existing second-order methods like BFGS and Newton-Raphson

Does not require objective function to decrease monotonically

Abstract

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.