Autoconjugate representers for linear monotone operators |
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Authors: | Heinz H Bauschke Xianfu Wang Liangjin Yao |
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Institution: | 1. Department of Mathematics, Irving K. Barber School, University of British Columbia Okanagan, Kelowna, BC, V1V 1V7, Canada
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Abstract: | Monotone operators are of central importance in modern optimization and nonlinear analysis. Their study has been revolutionized
lately, due to the systematic use of the Fitzpatrick function. Pioneered by Penot and Svaiter, a topic of recent interest
has been the representation of maximal monotone operators by so-called autoconjugate functions. Two explicit constructions
were proposed, the first by Penot and Zălinescu in 2005, and another by Bauschke and Wang in 2007. The former requires a mild
constraint qualification while the latter is based on the proximal average. We show that these two autoconjugate representers
must coincide for continuous linear monotone operators on reflexive spaces. The continuity and the linearity assumption are
both essential as examples of discontinuous linear operators and of subdifferential operators illustrate. Furthermore, we
also construct an infinite family of autoconjugate representers for the identity operator on the real line. |
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