Internal Model-Based Online Optimization

TL;DR

This paper introduces a model-based online optimization algorithm using control theory principles, achieving zero tracking error for quadratic problems and demonstrating superior performance over existing methods.

Contribution

It develops a novel control-inspired online optimization algorithm with proven convergence and robustness, extending to non-quadratic problems via approximate models.

Findings

Achieves zero tracking error in quadratic problems.

Demonstrates superior performance in numerical experiments.

Provides convergence analysis for convex and strongly convex cases.

Abstract

In this paper we propose a model-based approach to the design of online optimization algorithms, with the goal of improving the tracking of the solution trajectory (trajectories) w.r.t. state-of-the-art methods. We focus first on quadratic problems with a time-varying linear term, and use digital control tools (a robust internal model principle) to propose a novel online algorithm that can achieve zero tracking error by modeling the cost with a dynamical system. We prove the convergence of the algorithm for both strongly convex and convex problems. We further discuss the sensitivity of the proposed method to model uncertainties and quantify its performance. We discuss how the proposed algorithm can be applied to general (non-quadratic) problems using an approximate model of the cost, and analyze the convergence leveraging the small gain theorem. We present numerical results that showcase the superior performance of the proposed algorithms over previous methods for both quadratic and non-quadratic problems.