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Convexity of the proximal average. (English) Zbl 1215.26010
Let λ[0,1] and μ>0. The proximal average of the lower semicontinuous proper convex functions f 0 ,f 1 : d ]-,+] 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 𝒫 μ (f 0 ,f 1 ;λ)(ξ)=inf (1-λ)y 0 +λy 1 =ξ {(1-λ)f 0 (y 0 )+λf 1 (y 1 )+(1-λ)λ 2μ||y 0 -y 1 || 2 }. The authors prove that this function is separately convex in μ and λ, and give examples of convex quadratic functions f 0 and f 1 showing that it is not necessarily convex in any of the pairs (ξ,λ), (λ,μ), (ξ,μ) 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:
26B25Convexity and generalizations (several real variables)
52A41Convex functions and convex programs (convex geometry)
Software:
CSHEP2D
References:
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