Abstract
This work presents a collection of useful properties of the Moreau envelope for finite-dimensional, proper, lower semicontinuous, convex functions. In particular, gauge functions and piecewise cubic functions are investigated and their Moreau envelopes categorized. Characterizations of convex Moreau envelopes are established; topics include strict convexity, strong convexity, and Lipschitz continuity.
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Thank you to the anonymous referee for providing this reference.
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Acknowledgements
The authors thank the anonymous referee for the many useful comments and suggestions made to improve this manuscript. Chayne Planiden was supported by UBC University Graduate Fellowship and by Natural Sciences and Engineering Research Council of Canada. Xianfu Wang was partially supported by a Natural Sciences and Engineering Research Council of Canada Discovery Grant.
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Planiden, C., Wang, X. (2019). Proximal Mappings and Moreau Envelopes of Single-Variable Convex Piecewise Cubic Functions and Multivariable Gauge Functions. In: Hosseini, S., Mordukhovich, B., Uschmajew, A. (eds) Nonsmooth Optimization and Its Applications. International Series of Numerical Mathematics, vol 170. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-11370-4_5
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