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

Abstract

In a Hilbert space $\mathcal {H}$, given $A:\mathcal {H} \rightrightarrows \mathcal {H}$ 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

$$\text{(RN)} \quad \left\{ \begin{array}{l} v(t)\in A(x(t)),\\ \lambda (t) \dot x(t) + \dot v(t) + v(t) =0. \end{array}\right. $$

The term $\lambda (t) \dot x(t)$ acts as a Levenberg–Marquardt 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.