Finite-Time Convergence of Continuous-Time Optimization Algorithms via Differential Inclusions

Abstract

In this paper, we propose two discontinuous dynamical systems in continuous time with guaranteed prescribed finite-time local convergence to strict local minima of a given cost function. Our approach consists of exploiting a Lyapunov-based differential inequality for differential inclusions, which leads to finite-time stability and thus finite-time convergence with a provable bound on the settling time. In particular, for exact solutions to the aforementioned differential inequality, the settling-time bound is also exact, thus achieving prescribed finite-time convergence. We thus construct a class of discontinuous dynamical systems, of second order with respect to the cost function, that serve as continuous-time optimization algorithms with finite-time convergence and prescribed convergence time. Finally, we illustrate our results on the Rosenbrock function.

Presented at workshop "Beyond First Order Methods in Machine Learning" of NeurIPS 2019

References in corpus (6)

• Discontinuous Dynamical Systems: A tutorial on solutions, nonsmooth analysis, and stability
• Deep Learning Theory Review: An Optimal Control and Dynamical Systems Perspective
• ADMM and Accelerated ADMM as Continuous Dynamical Systems
• A Nonsmooth Dynamical Systems Perspective on Accelerated Extensions of ADMM
• An Optimal Control View of Adversarial Machine Learning
• Connecting Lyapunov Control Theory to Adversarial Attacks

Cited by in corpus (2)

• Fixed-Time Extremum Seeking
• Finite-time Koopman Identifier: A Unified Batch-online Learning Framework for Joint Learning of Koopman Structure and Parameters