Online Primal-Dual Methods with Measurement Feedback for Time-Varying Convex Optimization

TL;DR

This paper develops feedback-based online primal-dual algorithms for time-varying convex optimization, enabling adaptive, distributed control of dynamic systems with convergence guarantees and robustness to model mismatches.

Contribution

It introduces a novel online primal-dual method incorporating system feedback, allowing for distributed implementation and handling model uncertainties in time-varying convex optimization.

Findings

Algorithms achieve convergence in dynamic regret.

Q-linear convergence under regularized Lagrangian design.

Robustness to measurement noise and model mismatches.

Abstract

This paper addresses the design and analysis of feedback-based online algorithms to control systems or networked systems based on performance objectives and engineering constraints that may evolve over time. The emerging time-varying convex optimization formalism is leveraged to model optimal operational trajectories of the systems, as well as explicit local and network-level operational constraints. Departing from existing batch and feed-forward optimization approaches, the design of the algorithms capitalizes on an online implementation of primal-dual projected-gradient methods; the gradient steps are, however, suitably modified to accommodate feedback from the system in the form of measurements - hence, the term "online optimization with feedback." By virtue of this approach, the resultant algorithms can cope with model mismatches in the algebraic representation of the system states and outputs, they avoid pervasive measurements of exogenous inputs, and they naturally lend themselves to a distributed implementation. Under suitable assumptions, analytical convergence claims are established in terms of dynamic regret. Furthermore, when the synthesis of the feedback-based online algorithms is based on a regularized Lagrangian function, Q-linear convergence to solutions of the time-varying optimization problem is shown.