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Convexity of the proximal average. (English) Zbl 1215.26010
Let \(\lambda \in [ 0,1]\) 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 H. H. Bauschke, E. Matoušková and 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.

MSC:
26B25 Convexity of real functions of several variables, generalizations
52A41 Convex functions and convex programs in convex geometry
Software:
Scilab; CSHEP2D; na13; na24
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