Second-Order Dynamics with Hessian-Driven Damping for Linearly Constrained Convex Minimization
Simon K. Niederländer · SIAM Journal on Control and Optimization · 2021
In a real Hilbert space setting, we investigate the asymptotic properties of the solutions of a second-order differential system in view of linearly constrained convex minimization. The inertial dynamics are governed by a Hessian-driven damping term associated with the convex function to be minimized and potential effects induced by the linear constraints. We provide conditions on both the damping and the potential for which the solutions converge towards some feasible point of the convex minimization problem; this convergence is towards some minimizer provided that the solutions' initial data is specifically preselected. In addition, we present asymptotic estimates on the convergence rate of the solutions depending on the interaction between damping and potential effects. Our analysis is mainly based on energy-like arguments that capture the dissipative nature of the inertial dynamics by means of a Bregman distance. We complement our study with the fact that the second-order dynamics admit a first-order representation in terms of the Arrow--Hurwicz differential system. Numerical experiments further illustrate our theoretical findings.