Journal of Convex Analysis 16 (2009), No. 3, 673--686
Copyright Heldermann Verlag 2009
Monotone Linear Relations: Maximality and Fitzpatrick Functions
Heinz H. Bauschke
Dept. of Mathematics, Irving K. Barber School, University of British Columbia, Okanagan, Kelowna, B.C. V1V 1V7, Canada
heinz.bauschke@ubc.ca
Xianfu Wang
Dept. of Mathematics, Irving K. Barber School, University of British Columbia, Okanagan, Kelowna, B.C. V1V 1V7, Canada
shawn.wang@ubc.ca
Liangjin Yao
Dept. of Mathematics, Irving K. Barber School, University of British Columbia, Okanagan, Kelowna, B.C. V1V 1V7, Canada
ljinyao@interchange.ubc.ca
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.
Keywords: Adjoint process, Fenchel conjugate, Fitzpatrick family, Fitzpatrick function, linear relation, maximal monotone operator, monotone operator, skew linear relation.
MSC: 47A06, 47H05; 26B25, 47A05, 49N15, 52A41, 90C25
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