Fractional differential equation approach for convex optimization with convergence rate analysis

Abstract

A fractional differential equation (FDE) based algorithm for convex optimization is presented in this paper, which generalizes ordinary differential equation (ODE) based algorithm by providing an additional tunable parameter $$\alpha \in (0,1]$$. The convergence of the algorithm is analyzed. For the strongly convex case, the algorithm achieves at least the Mittag-Leffler convergence, while for the general case, the algorithm achieves at least an $$O(1/t^\alpha )$$ convergence rate. Numerical simulations show that the FDE based algorithm may have faster or slower convergence speed than the ODE based one, depending on specific problems.