  This paper presents a comprehensive convergence analysis for the mirror
descent (MD) method, a widely used algorithm in convex optimization. The key
feature of this algorithm is that it provides a generalization of classical
gradient-based methods via the use of generalized distance-like functions,
which are formulated using the Bregman divergence. Establishing convergence
rate bounds for this algorithm is in general a non-trivial problem due to the
lack of monotonicity properties in the composite nonlinearities involved. In
this paper, we show that the Bregman divergence from the optimal solution,
which is commonly used as a Lyapunov function for this algorithm, is a special
case of Lyapunov functions that follow when the Popov criterion is applied to
an appropriate reformulation of the MD dynamics. This is then used as a basis
to construct an integral quadratic constraint (IQC) framework through which
convergence rate bounds with reduced conservatism can be deduced. We also
illustrate via examples that the convergence rate bounds derived can be tight.
