Monotone Linear Relations: Maximality and Fitzpatrick Functions

2008/05/28 by Heinz H. Bauschke, Bauschke, Heinz H., Xianfu Wang +3 · 7 citations
Computer Science · Mathematics · #Optimization and Variational Analysis #Point processes and geometric inequalities #Mathematical Inequalities and Applications

paper · pdf · doi:10.48550/arxiv.0805.4256

Abstract

We analyze and characterize maximal monotonicity of linear relations (set-valued operators with linear graphs). An important tool in our study are Fitzpatrick functions. The results obtained partially extend work on linear and at most single-valued operators by Phelps and Simons and by Bauschke, Borwein and Wang. Furthermore, a description of skew linear relations in terms of the Fitzpatrick family is obtained. We also answer one of Simons problems by showing that if a maximal monotone operator has a convex graph, then this graph must actually be affine.

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• Monotone operators and "bigger conjugate" functions
• For maximally monotone linear relations, dense type, negative-infimum type, and Fitzpatrick-Phelps type all coincide with monotonicity of the adjoint
• The sum of a maximally monotone linear relation and the subdifferential of a proper lower semicontinuous convex function is maximally monotone

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