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.
| Identifer | oai:union.ndltd.org:LACETR/oai:collectionscanada.gc.ca:BVAU.2429/2807 |
| Date | 05 1900 |
| Creators | Yao, Liangjin |
| Publisher | University of British Columbia |
| Source Sets | Library and Archives Canada ETDs Repository / Centre d'archives des thèses électroniques de Bibliothèque et Archives Canada |
| Language | English |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
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