  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.
