2016/02/05 by Attouch, Hedy, Chbani, Zaki · 2 citations
#FOS: Mathematics #Optimization and Control (math.OC)
paper · doi:10.48550/arxiv.1602.01973
In a Hilbert space setting \mathcal H, we study the convergence properties as t → + ∞ of the trajectories of the second-order differential equation (AVD)α, ε x(t) + \fracαt x(t) + ∇ Φ(x(t)) + ε(t) x(t) =0, where ∇Φ is the gradient of a convex continuously differentiable function Φ: \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(⋅) ≡ 0, this dynamic has been highlighted recently by Su, Boyd, and Candès 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 (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 ergodic convergence of the trajectories to the element of minimal norm of the set of minimizers of Φ, when ε(t) tends slowly to zero.