The Proximal Point Algorithm Revisited

In this paper, we consider the proximal point algorithm for the problem of finding zeros of any given maximal monotone operator in an infinite-dimensional Hilbert space. For the usual distance between the origin and the operator’s value at each iterate, we put forth a new idea to achieve a new result on the speed at which the distance sequence tends to zero globally, provided that the problem’s solution set is nonempty and the sequence of squares of the regularization parameters is nonsummable. We show that it is comparable to a classical result of Brézis and Lions in general and becomes better whenever the proximal point algorithm does converge strongly. Furthermore, we also reveal its similarity to Güler’s classical results in the context of convex minimization in the sense of strictly convex quadratic functions, and we discuss an application to an ϵ-approximation solution of the problem above.