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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. [ABSTRACT FROM AUTHOR]

In this paper, we prove the continuity of maximum-entropy basis functions using variational analysis techniques. The use of information-theoretic variational principles to derive basis functions is a recent development. In this setting, data approximation is viewed as an inductive inference problem, with the basis functions being synonymous with a discrete probability distribution, and the polynomial reproducing conditions acting as the linear constraints. For a set of distinct nodes {xi}in=1 in ℝd, the convex approximation of a function u(x) is uh(x) = ∑ in=1 pi(x)ui, where {pi}in=1 are nonnegative basis functions, and uh(x) must reproduce affine functions ∑ in=1 pi(x)= 1, ∑ in=1 pi(x)xi = x. Given these constraints, we compute pi(x) by minimizing the relative entropy functional (Kullback-Leibler distance), D(p||m) = ∑in=1 pi(x) ln(pi(x)/mi(x)), where mi(x) is a known prior weight function distribution. To prove the continuity of the basis functions, we appeal to the theory of epiconvergence. [ABSTRACT FROM AUTHOR]