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On monotone operator classes and the Borwein-Wiersma decomposition : with demonstrations using low dimensional examples and the construction of decompositions. Edwards, Mclean Robert


In Hilbert spaces, five classes of monotone operator of relevance to the theory of monotone operators, variational inequality problems, equilibrium problems, and differential inclusions are investigated. These are the classes of paramonotone, strictly monotone, 3-cyclic monotone, 3*-monotone (or rectangular, or *-monotone), and maximal monotone operators. Examples of simple operators with all possible combinations of class inclusion are given, which together with some additional results lead to an exhaustive knowledge of monotone class relationships for linear operators, linear relations, and for monotone operators in general. Many of the example operators considered are the sum of a subdifferential with a skew linear operator (and so are Borwein-Wiersma decomposable). Since for a single operator its Borwein-Wiersma decompositions are not unique, clean, essential, extended, and standardized decompositions are defined and the theory developed. In particular, every Borwein-Wiersma decomposable operator has an essential decomposition, and many sufficient conditions are given for the existence of a clean decomposition. Various constructive methods are demonstrated together which, given any Borwein-Wiersma decomposable operator, are able to obtain a decomposition, as long as the operator has starshaped domain. These methods are more accurate if a clean decomposition exists. The techniques used apply a variant of Fitzpatrick's Last Function, the theory of which is developed here, where this function is shown to consist of a Riemann integration and be equivalent to Rockafellar's antiderivative when applied to subdifferentials. Furthermore, a different saddle function representation for monotone operators is created using this function which has theoretical and numerical advantages over more classical representations.

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