Accelerated primal–dual methods for strongly convex objective functions in continuous and discrete time

In this paper, we introduce a “second-order primal” + “first-order dual” continuous-time dynamic for linearly constrained optimization problems, where the objective function is $\mu$-strongly convex. We consider a constant damping $2\sqrt{\mu }$ for the second-order ordinary differential equation in the primal variable, following Nesterov’s acceleration for strongly convex optimization. A positive constant scaling is applied to the primal variable, while a positive increasing scaling function is applied to the dual variable. We prove that the proposed dynamic achieves a fast convergence rate for both the objective residual and the feasibility violation, with the decay rate potentially reaching $O\left(e^{-\sqrt{\mu }t}\right)$. Additionally, we show that the dynamic is robust under small perturbations. By discretizing the proposed continuous-time dynamic, we develop an accelerated linearized augmented Lagrangian method for strongly convex composite optimization with linear constraints, where the objective function has a nonsmooth + smooth composite structure. The proposed algorithm achieves a fast convergence rate that matches the one of the continuous-time dynamic. We also consider an inexact version of the proposed algorithm, which can be viewed as a discrete version of the perturbed continuous-time dynamic. Numerical results are provided to verify the practical performances.