Fixed-Time Convergence in Continuous-Time Optimization: A Fractional Approach

Continuous-time optimization is now an active research field and many valuable results have been proposed. As is known, a fast convergence speed is always desirable for an optimization algorithm and thus the fixed-time convergence in continuous-time optimization is considered in this letter. Different from existing fixed-time schemes, a novel fixed-time gradient method with a fractional adaptive gain is proposed, whose convergence time is determined by the first positive zero of a Mittag-Leffler function and independent of initial conditions. Furthermore, to avoid the possible singularity, two non-singular fixed-time gradient methods are proposed. Finally, all the results are extended to the second-order algorithm, whose convergence time is proven to be independent of both initial conditions and the shape of the target function.