  The vanilla fractional order gradient descent may oscillatively converge to a
region around the global minimum instead of converging to the exact minimum
point, or even diverge, in the case where the objective function is strongly
convex. To address this problem, a novel adaptive fractional order gradient
descent (AFOGD) method and a novel adaptive fractional order accelerated
gradient descent (AFOAGD) method are proposed in this paper. Inspired by the
quadratic constraints and Lyapunov stability analysis from robust control
theory, we establish a linear matrix inequality to analyse the convergence of
our proposed algorithms. We prove that the proposed algorithms can achieve
R-linear convergence when the objective function is $\textbf{L-}$smooth and
$\textbf{m-}$strongly-convex. Several numerical simulations are demonstrated to
verify the effectiveness and superiority of our proposed algorithms.
