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Zbl 1215.26010
Johnstone, Jennifer A.; Koch, Valentin R.; Lucet, Yves
Convexity of the proximal average. (English)
[J] J. Optim. Theory Appl. 148, No. 1, 107-124 (2011). ISSN 0022-3239; ISSN 1573-2878/e

Let $\lambda \in \lbrack 0,1]$ and $\mu >0$. The proximal average of the lower semicontinuous proper convex functions $f_{0},f_{1}:\mathbb{R}^{d}\to]-\infty ,+\infty ]$ was defined by {\it H. H. Bauschke, E. Matoušková} and {\it S. Reich} [Nonlinear Anal., Theory Methods Appl., Ser. A 56, No. 5, 715--738 (2004; Zbl 1059.47060)] by $\mathcal{P}_{\mu }(f_{0},f_{1};\lambda )(\xi )=\inf_{(1-\lambda)y_{0}+\lambda y_{1}=\xi }\{(1-\lambda )f_{0}(y_{0})+\lambda f_{1}(y_{1})+\frac{(1-\lambda )\lambda }{2\mu }||y_{0}-y_{1}||^{2}\}$. The authors prove that this function is separately convex in $\mu $ and $\lambda $, and give examples of convex quadratic functions $f_{0}$ and $f_{1}$ showing that it is not necessarily convex in any of the pairs $(\xi ,\lambda ),$ $(\lambda, \mu )$, $(\xi ,\mu )$ and $(f_{0},f_{1})$. They also propose some interpolation algorithms for plotting proximal averages, and present computational experience to show their efficiency in terms of computational time and image file size.
[Juan-Enrique Mart\'inez-Legaz (Barcelona)]

MSC 2000:: *26B25 Convexity and generalizations (several real variables)
52A41 Convex functions and convex programs (convex geometry)

Keywords: convex analysis; convexity; proximal average; interpolation

Citations: Zbl 1059.47060

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