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. Key words. arithmetic average, arithmetic mean, convex analysis, convex function, epiconvergence, epigraphical average, epi-topology, essential smoothness, essential strict convexity, Fenchel conjugate, harmonic mean, Legendre function, Moreau envelope, proximal average, proximal mapping, subdifferential operator AMS subject classifications. Primary, 90C25; Secondary, 26A51, 26B25, 26E60, 46C05, 47H05, 52A41 DOI. 10.1137/070687542

• Type: Journal
• Year: 2008
• Where: SIAMJO
• Authors: Heinz H. Bauschke, Rafal Goebel, Yves Lucet, Xianfu Wang
• Keywords: Convex Function, Proximal Average, Proximal Mapping, SIAMJO 2008