Finite-Time Optimization via Scaled Gradient-Momentum Flows

TL;DR

This paper introduces a scaled gradient-momentum framework for continuous-time optimization that guarantees global finite-time convergence through a novel state-dependent scaling mechanism.

Contribution

It presents a new scaled gradient-momentum approach enabling classical dynamics to achieve finite-time convergence, with explicit conditions linking function properties and stability.

Findings

The framework achieves global finite-time convergence.

Explicit conditions connect gradient-dominance to stability.

Numerical experiments confirm theoretical predictions.

Abstract

In this paper, we develop a scaled gradient-momentum framework for continuous-time optimization that achieves global finite-time convergence. A state-dependent scaling mechanism is introduced to enable classical dynamics, such as Heavy-Ball-type and proportional-integral (PI)-type flows, to attain finite-time convergence. We establish explicit conditions that bridge the gradient-dominance property of the objective function and finite-time stability of the proposed scaled dynamics. Numerical experiments validate the theoretical results.