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Decompositions and representations of monotone operators with linear graphs

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Title: Decompositions and representations of monotone operators with linear graphs
Author: Yao, Liangjin
Degree Master of Science - MSc
Program Interdisciplinary Studies
Copyright Date: 2007
Publicly Available in cIRcle 2008-11-24
Subject Keywords Maximal montone operator; Anixymmetric operator; Decomposition; Subdifferential; Semicontinuous convex function; Matrix; Fitzpatrick function; Fenchel conjugate; Linear relations
Abstract: We consider the decomposition of a maximal monotone operator into the sum of an antisymmetric operator and the subdifferential of a proper lower semicontinuous convex function. This is a variant of the well-known decomposition of a matrix into its symmetric and antisymmetric part. We analyze in detail the case when the graph of the operator is a linear subspace. Equivalent conditions of monotonicity are also provided. We obtain several new results on auto-conjugate representations including an explicit formula that is built upon the proximal average of the associated Fitzpatrick function and its Fenchel conjugate. These results are new and they both extend and complement recent work by Penot, Simons and Zălinescu. A nonlinear example shows the importance of the linearity assumption. Finally, we consider the problem of computing the Fitzpatrick function of the sum, generalizing a recent result by Bauschke, Borwein and Wang on matrices to linear relations.
URI: http://hdl.handle.net/2429/2807

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