Weak Versus Strong Convergence of a Regularized Newton Dynamic for Maximal Monotone Operators

International audienceIn a Hilbert space H, given A : H H a general maximal monotone operator whose solution set is assumed to be non-empty, and λ(·) a time-dependent positive regularization parameter, we analyze, when t → +∞, the weak versus strong convergence properties of the trajectories of the Regularized Newton dynamic (RN) v(t) ∈ A(x(t)), λ(t) ˙ x(t) + ˙ v(t) + v(t) = 0. The term λ(t) ˙ x(t) acts as a Levenberg-Marquard regularization of the continuous Newton dynamic associated with A, which makes (RN) a well-posed system. The coefficient λ(t) is allowed to tend to zero as t → +∞, which makes (RN) asymptotically close to the Newton continuous dynamic. As a striking property, when λ(t) does not converge too rapidly to zero as t → +∞ (with λ(t) = e −t as the critical size), Attouch and Svaiter showed that each trajectory generated by (RN) converges weakly to a zero of A. By adapting Baillon's counterexample, we show a situation where A is the gradient of a smooth convex function, and there is a trajectory of the corresponding system (RN) that does not converge strongly. On the positive side, under certain particular assumptions about the operator A, or on the regularization parameter λ(·), we show the strong convergence when t → +∞ of the (RN) trajectories