Optimization Letters

, Volume 4, Issue 4, pp 473–490 | Cite as

Fifty years of maximal monotonicity

  • Jonathan M. BorweinEmail author
Review Article


Maximal monotone operator theory is about to turn (or just has turned) 50. I intend to briefly survey the history of the subject. I shall try to explain why maximal monotone operators are both interesting and important—culminating with a description of the remarkable progress made during the past decade.


Maximal monotonicity Convex analysis Fitzpatrick function Minty surjectivity theorem Optimization 


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  1. 1.
    Ahn, B.-H.: Computation of Market Equilibria for Policy Analysis : The Project Independence Evaluation System (PIES) Approach. Models demand especially via (anti) monotone operators. Garland Publishing, New York (1979)Google Scholar
  2. 2.
    Asplund E.: A monotone convergence theorem for sequences of nonlinear mappings. Proc. Symp. Pure Math. 18, 1–9 (1970)MathSciNetGoogle Scholar
  3. 3.
    Asplund E., Rockafellar R.T.: Gradients of convex functions. Trans. Am. Math. Soc. 139, 443–467 (1969)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Bahn, O., Haurie, A., Zachary, D.S.: Mathematical modelling and simulation models in energy systems. Les Cahier de GERAD, G-2004-41, (2004)Google Scholar
  5. 5.
    Bartz S., Bauschke H.H., Borwein J.M., Reich S., Wang X.: Fitzpatrick functions, cyclic monotonicity and Rockafellar’s antiderivative. Nonlinear Anal. 66, 1198–1223 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bauschke H.H., Borwein J.M.: Maximal monotonicity of dense type, local maximal monotonicity, and monotonicity of the conjugate are all the same for continuous linear operators. Pacific J. Math. 189, 1–20 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Bauschke H.H., Wang X.: The kernel average for two convex functions and its application to extensions and representation of monotone operators. Trans. Am. Math. Soc 361, 5947–5965 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Borwein J.M.: Maximal monotonicity via convex analysis. J. Conv. Anal. 13, 561–586 (2006)zbMATHGoogle Scholar
  9. 9.
    Borwein, J.M., Lewis, A.S.: Convex Analysis and Nonlinear Optimization. Theory and Examples, CMS Books, vol. 3. Springer, Berlin (2000). Second extended edition (2005)Google Scholar
  10. 10.
    Borwein J.M., Noll D.: Second order differentiability of convex functions in Banach spaces. Trans. Am. Math. Soc. 342, 43–82 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Borwein J.M., Reich S., Shafrir I.: Krasnoselski-Mann iterations in normed spaces. Canad. Math. Bull. 35, 21–28 (1992)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Borwein, J.M., Vanderwerff, J.D.: Convex Functions: Constructions, Characterizations and Counterexamples. Encyclopedia of Mathematics and Applications, vol. 109. Cambridge University Press, UK (2010)Google Scholar
  13. 13.
    Borwein J.M., Zhu Q.: Techniques of Variational Analysis, CMS Books, vol. 20. Springer, Berlin (2005)Google Scholar
  14. 14.
    Brézis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert (French). North-Holland Mathematics Studies, No. 5. Notas de Matemática (50). North-Holland Publishing Co., Amsterdam, London; Elsevier, New York (1973)Google Scholar
  15. 15.
    Brézis H., Haraux A.: Image d’une somme d’opŕateurs monotones et applications. (English summary). Israel J. Math. 23, 165–186 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Browder F.E.: Fixed point theory and nonlinear problems. Bull. Am. Math. Soc. 9, 1–39 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Browder F.E., Hess P.: Nonlinear mappings of monotone type in Banach spaces. J. Funct. Anal. 11, 251–294 (1972)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Burachik, R.S., Iusem, A.N.: Set-Valued Mappings and Enlargements of Monotone Operators. Springer Optimization and Its Applications, vol. 8. Springer, Berlin (2008)Google Scholar
  19. 19.
    Combettes P.L., Pesquet J.-C.: A Douglas-Rachford splitting approach to nonsmooth convex variational signal recovery. IEEE J. Select. Topics Signal Process. 1, 564–574 (2007)CrossRefGoogle Scholar
  20. 20.
    Deimling K.: Nonlinear Functional Analysis. Springer, Berlin (1985)zbMATHGoogle Scholar
  21. 21.
    Eckstein J., Bertsekas D.P.: On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55, 293–318 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Fitzpatrick S.: Representing monotone operators by convex functions. Proc. Centre Math. Anal. Austral. Nat. Univ. 20, 59–65 (1988)MathSciNetGoogle Scholar
  23. 23.
    Fitzpatrick S., Phelps R.R.: Bounded approximants to monotone operators on Banach spaces. Ann. Inst. Henri Poincaré, Analyse non linéaire 9, 573–595 (1992)zbMATHMathSciNetGoogle Scholar
  24. 24.
    Fort M.K. Jr: Points of continuity of semi-continuous functions. Publ. Math. Debrecen 2, 100–102 (1951)zbMATHMathSciNetGoogle Scholar
  25. 25.
    Fortin, M., Glowinski, R. (eds): Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems. North-Holland Publishing Co., Amsterdam (1983)zbMATHGoogle Scholar
  26. 26.
    Franssen, H.T.: Towards project independence: energy in the coming decade. Prepared for the Joint Committee on Atomic Energy, United States Congress, OSTI ID: 7283709. Library of Congress, Congressional Research Service, Ocean and Coastal Resources Project (1975)Google Scholar
  27. 27.
    Glowinski, R., Le Tallec, P.: Augmented Lagrangian and Operator-splitting Methods in Nonlinear Mechanics. SIAM, USA (1989)Google Scholar
  28. 28.
    Gossez J.-P.: Opérateurs monotones non linaires dans les espaces de Banach non réflexifs. J. Math. Anal. Appl. 34, 371–376 (1971)zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Gulevich N.M.: Fixed points of nonexpansive mappings. J. Math. Sci. 79, 765–815 (1996)CrossRefMathSciNetGoogle Scholar
  30. 30.
    Kenderov P.S.: Semi-continuity of set-valued monotone mappings. Fund. Math. 88, 61–69 (1975)zbMATHMathSciNetGoogle Scholar
  31. 31.
    Kenderov P.S.: Monotone operators in Asplund spaces. C.R. Acad. Bulgare Sci. 30, 963–964 (1977)zbMATHMathSciNetGoogle Scholar
  32. 32.
    Kartsatos A.G., Skrypnik I.V.: On the eigenvalue problem for perturbed nonlinear maximal monotone operators in reflexive Banach spaces. Trans. Am. Math. Soc. 358, 3851–3881 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Mignot F.: Contrôle dans les inéquations variationelles elliptiques. J. Funct. Anal. 22, 130–185 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Minty G.J.: Monotone Networks. Proc. Roy Soc. Lond. 257, 194–212 (1960)zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Minty G.J.: Monotone (nonlinear) operators in Hilbert space. Duke Math. J. 29, 341–346 (1962)zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Minty G.J.: On the monotonicity of the gradient of a convex function. Pacific J. Math. 14, 243–247 (1964)zbMATHMathSciNetGoogle Scholar
  37. 37.
    Minty, G.J.: On some aspects of the theory of monotone operators. In: Theory and Application of Monotone Operators (Proceedings of NATO Advanced Study Institute, Venice, 1968), pp. 67–82 (1969)Google Scholar
  38. 38.
    Musev, B., Ribarska, N.: On a question of J. Borwein and H. Wiersma. Preprint (2009)Google Scholar
  39. 39.
    Phelps, R.R.: Convex Functions, Monotone Operators and Differentiability. Lecture Notes in Mathematics, vol. 1364. Springer, Berlin (1989). Second edition 1993Google Scholar
  40. 40.
    Preiss D., Phelps R.R., Namioka I.: Smooth Banach spaces, weak Asplund spaces and monotone or usco mappings. Israel J. Math. 72, 257–279 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Rockafellar R.T.: On the maximal monotonicity of subdifferential mappings. Pacific J. Math. 33, 209–216 (1970)zbMATHMathSciNetGoogle Scholar
  42. 42.
    Rockafellar R.T.: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149, 75–88 (1970)zbMATHMathSciNetGoogle Scholar
  43. 43.
    Rockafellar R.T.: Convex Analysis. Princeton University Press, USA (1970)zbMATHGoogle Scholar
  44. 44.
    Rockafellar R.T.: On the virtual convexity of the domain and range of a nonlinear maximal monotone operator. Math. Ann. 185, 81–90 (1970)zbMATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    Rockafellar R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  46. 46.
    Simons S.: Minimax and Monotonicity. Lecture Notes in Mathematics, vol. 1693. Springer, Berlin (1998)Google Scholar
  47. 47.
    Simons S.: Minimax and Monotonicity. Lecture Notes in Mathematics, vol. 1693, 2nd edition: Renamed From Hahn-Banach to Monotonicity. Springer, Berlin (2008)Google Scholar
  48. 48.
    Simons, S.: Banach SSD spaces and classes of monotone sets. http://arxiv/org/abs/0908.0383v2 posted 26 August (2009)
  49. 49.
    Spingarn J.E.: Partial inverse of a monotone operator. Appl. Math. Optim. 10, 247–265 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  50. 50.
    Tseng P.: Applications of a splitting algorithm to decomposition in convex programming and variational inequalities. SIAM J. Control Optim. 29, 119–138 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  51. 51.
    Zarantonello, E.H.: Projections on Convex Sets in Hilbert Space and Spectral Theory I Projections on Convex Sets. In: Contributions to Nonlinear Functional Analysis. Academic Press, Dublin (1971)Google Scholar
  52. 52.
    Zarantonello, E.H. (eds): Contributions to Nonlinear Functional Analysis. Academic Press, Dublin 237–341 (1971)zbMATHGoogle Scholar
  53. 53.
    Zarantonello E.H.: Dense single-valuedness of monotone operators. Israel J. Math. 15, 158–166 (1973)zbMATHCrossRefMathSciNetGoogle Scholar
  54. 54.
    Zeidler E.: Nonlinear Functional Analysis and its Applications: Part 2 A: Linear Monotone Operators and Part 2 B: Nonlinear Monotone Operators. Springer, Berlin (1990)Google Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Centre for Computer Assisted Research Mathematics and its Applications (CARMA)University of NewcastleNewcastleAustralia

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