  Mirror descent (MD) is a powerful first-order optimization technique that
subsumes several optimization algorithms including gradient descent (GD). In
this work, we develop a semi-definite programming (SDP) framework to analyze
the convergence rate of MD in centralized and distributed settings under both
strongly convex and non-strongly convex assumptions. We view MD with a
dynamical system lens and leverage quadratic constraints (QCs) to provide
explicit convergence rates based on Lyapunov stability. For centralized MD
under strongly convex assumption, we develop a SDP that certifies exponential
convergence rates. We prove that the SDP always has a feasible solution that
recovers the optimal GD rate as a special case. We complement our analysis by
providing the $O(1/k)$ convergence rate for convex problems. Next, we analyze
the convergence of distributed MD and characterize the rate using SDP. To the
best of our knowledge, the numerical rate of distributed MD has not been
previously reported in the literature. We further prove an $O(1/k)$ convergence
rate for distributed MD in the convex setting. Our numerical experiments on
strongly convex problems indicate that our framework certifies superior
convergence rates compared to the existing rates for distributed GD.
