Stochastic models for online optimization

Abstract: In this paper, we propose control-theoretic methods as tools for the design of online optimization algorithms that are able to address dynamic, noisy, and partially uncertain time-varying quadratic objective functions. Our approach introduces two algorithms specifically tailored for scenarios where the cost function follows a stochastic linear model. The first algorithm is based on a Kalman filter-inspired approach, leveraging state estimation techniques to account for the presence of noise in the evolution of the objective function. The second algorithm applies $\mathcal{H}_\infty$-robust control strategies to enhance performance under uncertainty, particularly in cases in which model parameters are characterized by a high variability. Through numerical experiments, we demonstrate that our algorithms offer significant performance advantages over the traditional gradient-based method and also over the optimization strategy proposed in arXiv:2205.13932 based on deterministic models.