  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 methods 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.
