Mathematical optimization is one of the cornerstones of modern engineering research and practice. Yet, throughout 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. While this new perspective is insightful in itself, liberating optimization

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... hods from specific numerical and algorithmic aspects 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 review several research streams that have been pursued in this direction, including extremum seeking and pertinent methods from model predictive and process control. However, our focus lies on recent methods under the name of "feedback-based optimization". This research stream studies control designs that directly implement optimization algorithms in closed loop with physical systems. Such ideas are finding widespread application in the design and retrofit of control protocols for communication networks and electricity grids. In addition to an overview over continuous-time dynamical systems for optimization, our particular emphasis lies on closed-loop stability as well as the enforcement of physical and operational constraints in closed-loop implementations. We further illustrate these methods in the context of classical problems, namely congestion control in communication networks and optimal frequency control in electricity grids, and we highlight one potential future application in the form of autonomous reserve dispatch in power systems.