
Journal of Convex Analysis 16 (2009), No. 3, 673686 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 (setvalued operators with linear graphs). An important tool in our study are Fitzpatrick functions. The results obtained partially extend work on linear and at most singlevalued 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 [ Fulltextpdf (149 KB)] for subscribers only. 