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This paper is concerned with convergence analysis for the mirror descent (MD)method, a well-known algorithm in convex optimization. An analysis frameworkvia integral quadratic constraints (IQCs) is constructed to analyze theconvergence rate of the MD method with strongly convex objective functions inboth continuous-time and discrete-time. We formulate the problem of findingconvergence rates of the MD algorithms into feasibility problems of linearmatrix inequalities (LMIs) in both schemes. In particular, in continuous-time,we show that the Bregman divergence function, which is commonly used as aLyapunov function for this algorithm, is a special case of the class ofLyapunov functions associated with the Popov criterion, when the latter isapplied to an appropriate reformulation of the problem. Thus, applying thePopov criterion and its combination with other IQCs, can lead to convergencerate bounds with reduced conservatism. We also illustrate via examples that theconvergence rate bounds derived can be tight.

Convergence Rate Bounds for the Mirror Descent Method: IQCs and the Bregman Divergence