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Zbl 1003.90030
Atteia, Marc; Ra\" issouli, Mustapha
Self dual operators on convex functionals; geometric mean and square root of convex functionals. (English)
[J] J. Convex Anal. 8, No.1, 223-240 (2001). ISSN 0944-6532

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 $\text{Conv}(X).$ \par Afterwards, we present an algorithm which associates, under convenient assumptions, a self-dual operator to a given operator and its dual.'' \par 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.
[Heinrich von Weizsäcker (Kaiserslautern)]

MSC 2000:: *90C25 Convex programming
46A20 Duality theory of topological linear spaces
26B25 Convexity and generalizations (several real variables)
52A05 Convex sets without dimension restrictions (convex geometry)

Keywords: convex functional; square root; convex geometric mean; algorithm; quadratic operators; self-dual operator

Cited in: Zbl 1256.90035 Zbl 1222.49025 Zbl 1062.47026 Zbl 1130.49304 Zbl 1023.47011

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