The Resolvent Average of Monotone Operators: Dominant and Recessive Properties

Abstract

Within convex analysis, a rich theory with various applications has been evolving since the proximal average of convex functions was first introduced over a decade ago. When one considers the subdifferential of the proximal average, a natural averaging operation of the subdifferentials of the averaged functions emerges. In the present paper we extend the reach of this averaging operation to the framework of monotone operator theory in Hilbert spaces, transforming it into the resolvent average. The theory of resolvent averages contains many desirable properties. In particular, we study a detailed list of properties of monotone operators and classify them as dominant or recessive with respect to the resolvent average. As a consequence, we recover a significant part of the theory of proximal averages. Furthermore, we shed new light on the proximal average and present novel results and desirable properties the proximal average possesses which have not been previously available.

An abstract proximal point algorithm

“…Set F := n∈N F ix(T n ) = ∅. For x ∈ X, let (x n ) be defined by (8). Let (γ n ) be a sequence of positive real numbers such that ∞ n=0 γ 2 n = ∞.…”

“…Assume that (T n ) is jointly (P 2 ) with respect to (γ n ). Let x ∈ X and (x n ) be given by (8). Then the sequence d(xn,xn+1) γn is nonincreasing.…”

“…Assume that the family (T n ) is jointly (P 2 ) with respect to (γ n ) (in particular, (T n ) may be jointly firmly nonexpansive) and that F := n∈N F ix(T n ) = ∅. Let x ∈ X and (x n ) be given by (8).…”

“…This uniqueness was shown by the work of Kohlenbach [29], Kohlenbach and Oliva [37, Section 4.1] and Briseid [14] to guarantee the extraction of a rate of convergence, relative to some other piece of quantitative information. In Section 4, therefore, we define a general notion of uniformity applicable to our families of mappings (extending the similar notion given in [8] in the context of Hilbert spaces). One then shows that the concrete algorithms have corresponding "uniform" cases that are subsumed into this definition, e.g.…”

On the asymptotic behaviour of the Aragón Artacho–Campoy algorithm

“…Because J C γ is a convex combination of the resolvents J C and P {w} , we see that C γ is nothing but a resolvent average of C and N {w} . See [3] for a detailed study of resolvent averages. We note that if C is σ C -monotone, i.e.,…”