Journal of Optimization Theory and Applications

, Volume 171, Issue 3, pp 757–784 | Cite as

Maximal Monotone Inclusions and Fitzpatrick Functions

  • J. M. Borwein
  • J. DuttaEmail author


In this paper, we study maximal monotone inclusions from the perspective of gap functions. We propose a very natural gap function for an arbitrary maximal monotone inclusion and will demonstrate how naturally this gap function arises from the Fitzpatrick function, which is a convex function, used to represent maximal monotone operators. This allows us to use the powerful strong Fitzpatrick inequality to analyse solutions of the inclusion. We also study the special cases of a variational inequality and of a generalized variational inequality problem. The associated notion of a scalar gap is also considered in some detail. Corresponding local and global error bounds are also developed for the maximal monotone inclusion.


Maximal monotone operator Monotone inclusions Variational inequality Fitzpatrick function Gap functions  Error bounds 

Mathematics Subject Classification

90C30 49J52 



We are thankful to the anonymous referees for their constructive suggestions which has improved the presentation of the paper and also bringing to our notice the references [12] and [17]. We would also like to thank Poonam Kesarwani for her help with the MATLAB.


  1. 1.
    Fukushima, M.: Equivalent differentiable optimization and descent methods for asymmetric variational inequalities. Math. Prog. 53, 99–110 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Facchinei, F., Pang, J.-S.: Finite Dimensional Variational Inequalities and Complementarity Problems, vol. 1 and 2. Springer, Berlin (2003)zbMATHGoogle Scholar
  3. 3.
    Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Borwein, J.M., Vanderwerff, J.D.: Convex Functions : Constructions, Characterizations and Counterexamples. Cambridge University Press, Cambridge (2010)CrossRefzbMATHGoogle Scholar
  5. 5.
    Borwein, J.M., Lewis, A.S.: Convex Analysis and Nonlinear Optimization: Theory and Examples. Springer, Berlin (2000) (2nd Edn, 2006)Google Scholar
  6. 6.
    Fitzpatrick, S.: Representing monotone operators by convex functions, Workshop/Miniconference on Functional Analysis and Optimization (Canberra 1988). In: Proceedings of Center Mathematical Analysis, Australian National University 20, Australian National University, Canberra, (1988), pp. 59–65Google Scholar
  7. 7.
    Crouzeix, J.-P.: Pseudomonotone variational inequality problems: existence of solutions. Math. Prog. 78, 305–314 (1997)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Aussel, D., Dutta, J.: On gap functions for multivalued Stampacchia variational inequalities. J. Optim. Theory Appl. 149, 513–527 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Borwein, J.M.: Generalized linear complementarily problems treated without fixed-point theory. J. Optim. Theory Appl. 43, 343–356 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Frank, M., Wolfe, P.: An algorithm for quadratic programming. Naval Res. Logist. Q. 3, 95–110 (1956)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Alves, M.M., Svaiter, B.F.: Bronsted-Rockafellar property and the maximality of monotone operators representable by convex functions in non-reflexive Banach spaces. J. Convex Anal. 15, 693–706 (2008)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Monteiro, R.D.C., Svaiter, B.F.: On the complexity of the hybrid extragradient method for the iterates and ergodic mean. SIAM J. Optim. 20, 2755–2787 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings: A View from Variational Analysis. Springer Monographs in Mathematics, Berlin (2009)CrossRefzbMATHGoogle Scholar
  14. 14.
    Borwein, J.M., Zhuang, D.M.: Verifiable necessary and sufficient conditions for openness and regularity of set-valued and single-valued maps. J. Math. Anal. Appl. 134, 441–459 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Borwein, J.M.: Minimal cuscos and subgradients of Lipschitz functions. In: Baillon, J-B., Thera, M. (eds.): Fixed Point Theory and its Applications, Pitman Lecture Notes in Mathematics. Longman, Essex (1991)Google Scholar
  16. 16.
    Aragon Atacho, F.J., Geoffroy, M.H.: Characterization of Metric Regularity of Subdifferentials. Preprint (2014)Google Scholar
  17. 17.
    Bauschke, H.H., McLaren, D.A., Sendov, H.S.: Fitzpatrick functions, inequalities, examples and remarks on a problem by S. Fitzpatrick. J. Convex Anal. 13, 499–523 (2006)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Nesterov, Y., Scrimali, L.: Solving strongly monotone variational and quasi-variational inequalities. Discrete Contin. Dyn. Syst. 31, 1383–1396 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Komiya, H.: Elementary proof of Sion’s minimax theorem. Kodai Math. J. 11, 5–7 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Shapiro, A., Nemirovski, A.: Duality of Linear Conic Problems, preprint, Optimization-online (2003)Google Scholar
  21. 21.
    Bazaara, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms, 3rd edn. Wiley, Hoboken (2006)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.CARMAUniversity of NewcastleCallaghanAustralia
  2. 2.Economics Group, Department of Humanities and Social ScienceIndian Institute of Technology KanpurKanpurIndia

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