  In this paper, we develop a unified framework for analyzing the tracking
error and dynamic regret of inexact online optimization methods under a variety
of settings. Specifically, we leverage the quadratic constraint approach from
control theory to formulate sequential semidefinite programs (SDPs) whose
feasible points naturally correspond to tracking error bounds of various
inexact online optimization methods including the inexact online gradient
descent (OGD) method, the online gradient descent-ascent method, the online
stochastic gradient method, and the inexact proximal online gradient method. We
provide exact analytical solutions for our proposed sequential SDPs, and obtain
fine-grained tracking error bounds for the online algorithms studied in this
paper. We also provide a simple routine to convert the obtained tracking error
bounds into dynamic regret bounds. The main novelty of our analysis is that we
derive exact analytical solutions for our proposed sequential SDPs under
various inexact oracle assumptions in a unified manner.
