Combining fast inertial dynamics for convex optimization with Tikhonov regularization

In a Hilbert space setting mathcal H, we study the convergence properties as t → + ∞ of the trajectories of the second-order differential equation beginequation* mbox(AVD)α, ε quad quad ddotx(t) + fracαt dotx(t) + nabla Phi (x(t)) + ε (t) x(t) =0, endequation* where nablaPhi is the gradient of a convex continuously differentiable function Phi: mathcal H → mathbb R, α is a positive parameter, and ε (t) x(t) is a Tikhonov regularization term, with limt → ∞ε (t) =0. In this damped inertial system, the damping coefficient fracαt vanishes asymptotically, but not too quickly, a key property to obtain rapid convergence of the values. In the case ε (·) equiv 0, this dynamic has been highlighted recently by Su, Boyd, and Cand`es as a continuous version of the Nesterov accelerated method. Depending on the speed of convergence of ε (t) to zero, we analyze the convergence properties of the trajectories of mbox(AVD)α, ε. We obtain results ranging from the rapid convergence of Phi (x(t)) to min Phi when ε (t) decreases rapidly to zero, up to the strong ergodic convergence of the trajectories to the element of minimal norm of the set of minimizers of Phi, when ε (t) tends slowly to zero.