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

Abstract

Abstract. We provide and analyze an average for two convex functions, based on a kernel function. It covers several known averages such as the arithmetic average, epigraphical average, and the proximal average. When applied to the Fitzpatrick function and the conjugate of Fitzpatrick function associated with a monotone operator, our average produces an autoconjugate (also known as selfdual Lagrangian) which can be used for finding an explicit maximal monotone extension of the given monotone operator. This completely settles one of the open problems posed by Fitzpatrick in the setting of reflexive Banach spaces.

Fifty years of maximal monotonicity

“…Proof Associate a monotone T to P, as in Proposition 2, and let T be any maximal extension of T as is assured by Zorn's lemma (actually recent work [7,Sect. 5] lets this be done more constructively).…”

The Proximal Average: Basic Theory

“…Unfortunately, if the functions f 1 and f 2 lack coercivity, then the epigraphical average fails to be helpful: for instance, if f 1 and f 2 are two distinct linear functions, then their epigraphical average is identically equal to −∞, and hence of little use. The proximal average, first introduced in [6] in the context of fixed point theory and recently studied in [4,5,7,10,15] from various viewpoints, avoids the mentioned difficulties and possesses numerous properties that are attractive to Convex Analysts.…”

What Shape Is Your Conjugate? A Survey of Computational Convex Analysis and Its Applications

“…New Convex Analysis transforms, like the kernel average [27], have been recently introduced that require the extension of current algorithms. While some transforms like the Moreau envelope and the Fenchel conjugate can be computed efficiently for convex and nonconvex functions, the efficient computation for others is limited to convex functions e.g.…”