Finite-time and Fixed-time Convergence in Continuous-time Optimization

It is known that the gradient method can be viewed as a dynamic system where various iterative schemes can be designed as a part of the closed loop system with desirable properties. In this paper, the finite-time and fixed-time convergence in continuous-time optimization are mainly considered. By the advantage of sliding mode control, a finite-time gradient method is proposed, whose convergence time is dependent on initial conditions. To make the convergence time robust to initial conditions, two different designs of fixed-time gradient methods are then provided. One is designed using the property of sine function, whose convergence time is dependent on the frequency of a sine function. The other one is designed using the property of Mittag-Leffler function, whose convergence time is determined by the first positive zero of a Mittag-Leffler function. All the results are extended to more general cases and finally demonstrated by some dedicated simulation examples.