An Asymptotical Variational Principle Associated with the Steepest Descent Method for a Convex Function

The asymptotical limit of the trajectory defined by the continuous steepest descent method for a proper closed convex function f on a Hilbert space is characterized in the set of minimizers of f via an asymptotical variational principle of Brezis-Ekeland type. The implicit discrete analogue (prox method) is also considered.