Asymptotics for Some Proximal-like Method Involving Inertia and Memory Aspects

AbstractGiven a Hilbert space H and a closed convex function Φ:$H\rightarrow{\mathbb{R}} \cup \{+\infty\}$, we consider the inertial proximal algorithm $$ x_{n+1}-x_n-\alpha_n(x_n-x_{n-1})+\beta_n\partial\Phi(x_{n+1})\ni 0, \qquad \qquad \qquad \qquad({\mathcal{A}}) $$where (αn) and (βn) are nonnegative sequences. The notation $\partial \Phi$ stands for the subdifferential of Φ in the sense of convex analysis. This algorithm can be viewed as the implicit discretization of a continuous gradient system involving a memory term. We give conditions that ensure that a suitable discrete energy decreases to $\inf\Phi$ as n→ + ∞. When Φ has a unique minimum, the question of the convergence of (xn) is solved. In the case of multiple minima, it is proved that if $\left(\prod_{k=1}^n \alpha_k\right)\not \in l^1$ and if a suitable geometric condition on the set argmin Φ is fulfilled, then non stationary sequences of (A) cannot converge.