Time-varying optimization problems arise in a va riety of engineering applications. The available information about how the problem changes in time dictates the types of algorithms that are applicable to a particular problem as well as the types of convergence guarantees that may be proven. In this paper, we study dynamic gradient-feedback algorithms for time-varying optimization in discrete time. By casting the design of such algorithms as an output regulation problem for dynamical systems, we provide necessary and sufficient conditions for the existence of a gradient-feedback algorithm that asymptotically tracks a critical trajectory of the optimization problem. When these conditions hold, we provide a design procedure to construct such an algorithm. As a fundamental limitation, we show that any algorithm that asymptotically tracks a critical trajectory needs to contain an internal model of the temporal variation, which we refer to as the internal model principle of time-varying optimization.
Bianchin et al. (Wed,) studied this question.