Optimization Algorithms as Robust Feedback Controllers

TL;DR

The article reframes optimization as a dynamical system and studies how to couple optimization dynamics with physical plants to realize robust, constraint-satisfying real-time operation. It surveys gradient, projected-gradient, and primal-dual saddle-point flows, analyzes stability (via singular perturbation and IQC/LMIs), and explores constraint handling through projection, dualization, and anti-windup strategies, including data-driven and model-free variants. A key contribution is outlining concrete online schemes for maintaining feasibility and convergence in uncertain, time-varying environments, with an application to real-time reserve dispatch in electricity grids that demonstrates practical maturity. The work highlights how feedback-based optimization can reduce model dependence, enhance robustness, and enable autonomous operation, while identifying important avenues for future research in nonconvex, time-varying, and large-scale settings.

Abstract

Mathematical optimization is one of the cornerstones of modern engineering research and practice. Yet, throughout all application domains, mathematical optimization is, for the most part, considered to be a numerical discipline. Optimization problems are formulated to be solved numerically with specific algorithms running on microprocessors. An emerging alternative is to view optimization algorithms as dynamical systems. Besides being insightful in itself, this perspective liberates optimization methods from specific numerical and algorithmic aspects and opens up new possibilities to endow complex real-world systems with sophisticated self-optimizing behavior. Towards this goal, it is necessary to understand how numerical optimization algorithms can be converted into feedback controllers to enable robust "closed-loop optimization". In this article, we focus on recent control designs under the name of "feedback-based optimization" which implement optimization algorithms directly in closed loop with physical systems. In addition to a brief overview of selected continuous-time dynamical systems for optimization, our particular emphasis in this survey lies on closed-loop stability as well as the robust enforcement of physical and operational constraints in closed-loop implementations. To bypass accessing partial model information of physical systems, we further elaborate on fully data-driven and model-free operations. We highlight an emerging application in autonomous reserve dispatch in power systems, where the theory has transitioned to practice by now. We also provide short expository reviews of pioneering applications in communication networks and electricity grids, as well as related research streams, including extremum seeking and pertinent methods from model predictive and process control, to facilitate high-level comparisons with the main topic of this survey.

Figures (21)

• Figure 1: Simple feedback-based gradient flow
• Figure 2: Illustrations for \ref{['ex:num_grad']} (top left: objective function; remaining panels: system trajectories for different control gains $\epsilon$)
• Figure 3: Gradient trajectories for a non-convex objective function. Trajectories under the Euclidean metric (left) and a generic variable metric (right) differ significantly. The critical points (i.e., the minima, maximum, and saddle-point) and their stability properties are unaffected by the choice of metric.
• Figure 4: Gradient trajectories for strongly convex, but ill-conditioned objective. The trajectories under the Euclidean metric (left) quickly approach a subspace on which the objective is almost flat, and then converge only slowly to the global optimizer. Trajectories under the "Newton metric" (right), approach the global optimizer isotropically, unaffected by the ill-conditioning of the objective.
• Figure 5: Left: Examples of tangent and normal cones of a set $\mathcal{X}$ (gray area is infeasible). Right: Projected vector field onto the tangent cone. See \ref{['fig:cstr_pgrad']} for the resulting projected gradient trajectory.

Theorems & Definitions (27)

• Example 1.1
• Example 1.2
• Theorem 1
• Example 2.1: Variable Metric and Newton Gradient Flow
• Remark 2.1: Discretization & Proximal-Point Algorithm
• Remark 2.2: Non-differentiable objective & subgradients
• Remark 2.3: Variable Metric and Projected Newton Gradient Flow
• Example 2.2: Saddle-Point Flow for Optimization with Equality Constraints
• Example 2.3: Projected Saddle-Point Flow for Optimization with Inequality Constraints
• Example 2.4