The Proximal Average: Basic Theory

Abstract

The recently introduced proximal average of two convex functions is a convex function with many useful properties. In this paper, we introduce and systematically study the proximal average for finitely many convex functions. The basic properties of the proximal average with respect to the standard convex-analytical notions (domain, Fenchel conjugate, subdifferential, proximal mapping, epi-continuity, and others) are provided and illustrated by several examples.2000 Mathematics Subject Classification: Primary 90C25; Secondary 26A51, 26B25, 26E60, 46C05, 47H05, 52A41.

Firmly nonexpansive and Kirszbraun-Valentine extensions: a constructive approach via monotone operator theory

“…We refer the reader to [2,3,4,5,7] for further information on the proximal average, to [9,11,16,18,23] and the references therein for results on the Fitzpatrick function, and to [20,21,26] for the basic theory of Convex Analysis. The notation A for the maximal monotone extension of a monotone operator A : X ⇒ X will be used from now on.…”

The kernel average for two convex functions and its application to the extension and representation of monotone operators

“…Monotone operators play an important role in modern optimization and analysis; see, e.g., [19,20]. The Fitzpatrick function [10] associated with A has recently turned out to be a very effective tool for analyzing A; see, e.g., [1,2,3,4,5,7,8,9,10,11,12,13,14,15,16,17,21,22,23].…”

Autoconjugate representers for linear monotone operators

“…While this proximalaveraged based construction is more involved [4,5,7], it has the advantage of not imposing a constraint qualification. Fact 1.7 (See [9].)…”