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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
##### Keywords:
convex analysis; convexity; proximal average; interpolation
##### Software:
Scilab; CSHEP2D; na13; na24
Full Text:
##### References:
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