Online Optimization With Integral Action: An Optimal Algorithm For A Time Varying Quadratic Cost Function

In this paper, we develop an online optimization algorithm with integral action for solving online optimization problems characterized by quadratic cost functions with a linearly varying optimal point. The proposed algorithm can be formulated as an uncertain linear feedback system incorporating a discrete time double integrator. By leveraging established results on the optimal gain margin, we demonstrate that the proposed algorithm achieves the optimal convergence rate compared to other methods including a discrete time double integrator applied to quadratic cost functions in finite-dimensional spaces.