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Convexity of the proximal average. (English) Zbl 1215.26010
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.

26B25Convexity and generalizations (several real variables)
52A41Convex functions and convex programs (convex geometry)
na24; na13; CSHEP2D
Full Text: DOI
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