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.

Author information

Contact details are reproduced from the original publication and may be historical.

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

Keywords

  • Adjoint process
  • Fenchel conjugate
  • Fitzpatrick family
  • Fitzpatrick function
  • linear relation
  • maximal monotone operator
  • monotone operator
  • skew linear relation

Mathematics Subject Classification

47A06, 47H05; 26B25, 47A05, 49N15, 52A41, 90C25

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Suggested citation

H. H. Bauschke, X. Wang, L. Yao. “Monotone Linear Relations: Maximality and Fitzpatrick Functions.” Journal of Convex Analysis 16 (2009), No. 3&4, 673–686. https://doi.org/10.68381/jca16039

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Published by Heldermann Verlag, 2009. Rights now held by Banach Press LLC.