Convergence of Projected Primal-Dual Dynamics with Applications in Data Centers

Abstract This paper studies the convergence of projected primal-dual dynamics under mild conditions on the (general) optimization problem. In particular, we do not require strict convexity of the objective function nor uniqueness of the optimizer. By regarding the inequality constraints as hard constraints, we construct a suitable primal-dual dynamics in the complementarity formalism. We establish pointwise asymptotic stability of the set primal-dual optimizers by a suitable invariance principle involving two different Lyapunov functions. In addition, we show how these results can be applied for online optimization in data centers.