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

TL;DR

This paper introduces discontinuous continuous-time optimization algorithms that guarantee finite-time convergence to local minima using Lyapunov-based differential inequalities, with proven bounds on convergence time.

Contribution

It presents a novel class of second-order discontinuous dynamical systems that ensure prescribed finite-time convergence in continuous-time optimization.

Findings

Guaranteed finite-time local convergence to minima.

Lyapunov-based differential inequalities ensure convergence bounds.

Validated on Rosenbrock function.

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.