An asymptotical variational principle associated with the steepest descent method for a convex function.

Let X be a real Hilbert space endowed with inner product 〈., .〉 and associated norm ‖.‖, and let f be a proper closed convex function on X. The paper considers the problem of minimizing f , that is, of finding infX f and some element in the optimal set S := Argmin f , this set assumed being non empty. Letting ∂f denote the subdifferential operator associated with f , we focus on the continuous steepest descent method associated with f , i.e., the differential inclusion