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Self-Dual Smooth Approximations of Convex Functions via the Proximal Average

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Book cover Fixed-Point Algorithms for Inverse Problems in Science and Engineering

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 49))

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.

Author information

Authors and Affiliations

  1. Department of Mathematics, Irving K. Barber School, University of British Columbia, Kelowna, B.C., V1V 1V7, Canada

    Heinz H. Bauschke

Authors
  1. Heinz H. Bauschke
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  2. Sarah M. Moffat
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  3. Xianfu Wang
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Corresponding author

Correspondence to Heinz H. Bauschke .

Editor information

Editors and Affiliations

  1. Okanagan Campus, Department of Mathematics and Statistic, University of British Columbia, Kelowna, V1V 1V7, British Columbia, Canada

    Heinz H. Bauschke

  2. School of Mathematics & Statistics, Division of Information Technology, University of South Australia, Mawson Lakes Blvd., Mawson Lakes, 5095, South Australia, Australia

    Regina S. Burachik

  3. Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Place Jussieu 4, Paris, 75005, France

    Patrick L. Combettes

  4. Dept. Physics, Lab. Atomic & Solid State, Cornell University, Clark Hall, Ithaca, 14853-2501, New York, USA

    Veit Elser

  5. Institut für Numerische und Angewandte M, Universität Göttingen, Lotzestr. 16-18, Göttingen, 37073, Germany

    D. Russell Luke

  6. Faculty of Mathematics, Dept. Combinatorics & Optimization, University of Waterloo, Waterloo, N2L 3G1, Ontario, Canada

    Henry Wolkowicz

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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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