Abstract
The proximal average of two convex functions has proven to be a useful tool in convex analysis. In this note, we express the Goebel self-dual smoothing operator in terms of the proximal average, which allows us to give a different proof of self duality. We also provide a novel self-dual smoothing operator. Both operators are illustrated by smoothing the norm.
AMS 2010 Subject Classification: Primary 26B25; Secondary 26B05, 65D10, 90C25
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Acknowledgements
The authors thank the two referees for their constructive comments. Heinz Bauschke was partially supported by the Natural Sciences and Engineering Research Council of Canada and by the Canada Research Chair Program. Sarah Moffat was partially supported by the Natural Sciences and Engineering Research Council of Canada. Xianfu Wang was partially supported by the Natural Sciences and Engineering Research Council of Canada.
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Bauschke, H.H., Moffat, S.M., Wang, X. (2011). Self-Dual Smooth Approximations of Convex Functions via the Proximal Average. In: Bauschke, H., Burachik, R., Combettes, P., Elser, V., Luke, D., Wolkowicz, H. (eds) Fixed-Point Algorithms for Inverse Problems in Science and Engineering. Springer Optimization and Its Applications(), vol 49. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-9569-8_2
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DOI: https://doi.org/10.1007/978-1-4419-9569-8_2
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