Combining fast inertial dynamics for convex optimization with Tikhonov regularization

In a Hilbert space setting H, we study the convergence properties as t→+∞ of the trajectories of the second-order differential equation(AVD)α,ϵx¨(t)+αtx˙(t)+∇Φ(x(t))+ϵ(t)x(t)=0, where ∇Φ is the gradient of a convex continuously differentiable function Φ:H→R, α is a positive parameter, and ϵ(t)x(t) is a Tikhonov regularization term, with ϵ(t) positive, and limt→∞⁡ϵ(t)=0. In this damped inertial system, the damping coefficient αt vanishes asymptotically, but not too quickly, a key property to obtain rapid convergence of the values. In the case ϵ(⋅)≡0, this dynamic has been highlighted recently by Su, Boyd, and Candès as a continuous version of the Nesterov accelerated gradient method. Depending on the speed of convergence of ϵ(t) to zero, we analyze the convergence properties of the trajectories of (AVD)α,ϵ. We obtain results ranging from the rapid convergence of Φ(x(t)) to min⁡Φ when ϵ(t) decreases rapidly to zero, up to the strong convergence of the trajectories to the element of minimum norm of the set of minimizers of Φ, when ϵ(t) tends slowly to zero. When ϵ(t)=1tr, the critical value of r separating the two above cases is r=2.