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Self dual operators on convex functionals; geometric mean and square root of convex functionals. (English) Zbl 1003.90030
From the authors’ abstract: “Let $\text{Conv}(X)$ be the set of the convex functionals defined on a linear space $X$, with values in $\Bbb{R} \cup \{ + \infty\}.$ In this paper we give an extension of the notion of duality for (convex) functionals to mappings which operate from $\text{Conv}(X) \times \text{Conv}(X)$ into $\text{Conv}(X).$ Afterwards, we present an algorithm which associates, under convenient assumptions, a self-dual operator to a given operator and its dual.” The algorithm can be understood as an adaptation of the classical Newton type procedure to compute the square root. The authors’ iterative method approximates in particular a square root of a convex functional, but the idea has a much wider scope as is illustrated by several geometric examples.

90C25Convex programming
46A20Duality theory of topological linear spaces
26B25Convexity and generalizations (several real variables)
52A05Convex sets without dimension restrictions (convex geometry)