Dynamical System Approach for Time-Varying Constrained Convex Optimization Problems

Optimization problems emerging in most of the real-world applications are dynamic, where either the objective function or the constraints change continuously over time. This article proposes projected primal–dual dynamical system approaches to track the primal and dual optimizer trajectories of an inequality constrained time-varying (TV) convex optimization problem with a strongly convex objective function. First, we present a dynamical system that asymptotically tracks the optimizer trajectory of an inequality constrained TV optimization problem. Later, we modify the proposed dynamics to achieve the convergence to the optimizer trajectory within a fixed time. The asymptotic and fixed-time convergence of the proposed dynamical systems to the optimizer trajectory is shown via the Lyapunov-based analysis. Finally, we consider the TV extended Fermat–Torricelli problem of minimizing the sum-of-squared distances to a finite number of nonempty, closed, and convex TV sets, to illustrate the applicability of the projected dynamical systems proposed in this article.