Mathematical Programming

, Volume 148, Issue 1–2, pp 49–88 | Cite as

Applications of convex analysis within mathematics

  • Francisco J. Aragón ArtachoEmail author
  • Jonathan M. Borwein
  • Victoria Martín-Márquez
  • Liangjin Yao
Full Length Paper Series B


In this paper, we study convex analysis and its theoretical applications. We first apply important tools of convex analysis to Optimization and to Analysis. We then show various deep applications of convex analysis and especially infimal convolution in Monotone Operator  Theory. Among other things, we recapture the Minty surjectivity theorem in Hilbert space, and present a new proof of the sum theorem in reflexive spaces. More technically, we also discuss autoconjugate representers for maximally monotone operators. Finally, we consider various other applications in mathematical analysis.


Convex function Chebyshev set Fenchel conjugate Monotone operator Fitzpatrick function Autoconjugate representer 

Mathematics Subject Classification (2000)

Primary 47N10 90C25 Secondary 47H05 47A06 47B65 



The authors are grateful to the three anonymous referees for their pertinent and constructive comments. The authors also thank Dr. Hristo S. Sendov for sending them the manuscript [60]. The authors were all partially supported by various Australian Research Council grants.


  1. 1.
    Alber, Y., Butnariu, D.: Convergence of Bregman projection methods for solving consistent convex feasibility problems in reflexive Banach spaces. J. Optim. Theory Appl. 92, 33–61 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Alimohammady, M., Dadashi, V.: Preserving maximal monotonicity with applications in sum and composition rules. Optim. Lett. 7, 511–517 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Asplund, E.: Averaged norms. Israel J. Math. 5, 227–233 (1967)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Attouch, H., Riahi, H., Thera, M.: Somme ponctuelle d’operateurs maximaux monotones [Pointwise sum of maximal monotone operators]. Well-posedness and stability of variational problems. Serdica. Math. J. 22, 165–190 (1996)MathSciNetGoogle Scholar
  5. 5.
    Bac̆ák, M., Borwein, J.M.: On difference convexity of locally Lipschitz functions. Optimization 60, 961–978 (2011)Google Scholar
  6. 6.
    Bauschke, H.H., Borwein, J.M., Wang, X., Yao, L.: Construction of pathological maximally monotone operators on non-reflexive Banach spaces. Set-Valued Var. Anal. 20, 387–415 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)CrossRefzbMATHGoogle Scholar
  8. 8.
    Bauschke, H.H., Wang, X.: The kernel average for two convex functions and its applications to the extension and representation of monotone operators. Trans. Am. Math. Soc. 36, 5947–5965 (2009)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Bauschke, H.H., Wang, X., Yao, L.: Monotone linear relations: maximality and Fitzpatrick functions. J. Convex Anal. 16, 673–686 (2009)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Bauschke, H.H., Wang, X., Yao, L.: Autoconjugate representers for linear monotone operators. Math. Program. Ser. B 123, 5–24 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Bauschke, H.H., Wang, X., Yao, L.: General resolvents for monotone operators: characterization and extension, In: Biomedical Mathematics: Promising Directions in Imaging, pp. 57–74. Therapy Planning and Inverse Problems, Medical Physics Publishing (2010)Google Scholar
  12. 12.
    Borwein, J.M.: A generalization of Young’s \(\ell ^p\) inequality. Math. Inequal. Appl. 1, 131–136 (1998)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Borwein, J.M.: Maximal monotonicity via convex analysis. J. Convex Anal. 13, 561–586 (2006)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Borwein, J.M.: Maximality of sums of two maximal monotone operators in general Banach space. Proc. Am. Math. Soc. 135, 3917–3924 (2007)CrossRefzbMATHGoogle Scholar
  15. 15.
    Borwein, J.M.: Fifty years of maximal monotonicity. Optim. Lett. 4, 473–490 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Borwein, J.M., Bailey, D.H.: Mathematics by Experiment: Plausible Reasoning in the 21st Century, 2nd edn. A.K. Peters Ltd, Natick (2008)Google Scholar
  17. 17.
    Borwein, J.M., Bailey, D.H., Girgensohn, R.: Experimentation in Mathematics: Computational Paths to Discovery. A.K. Peters Ltd, Natick (2004). ISBN 1-56881-211-6Google Scholar
  18. 18.
    Borwein, J.M., Fitzpatrick, S.: Local boundedness of monotone operators under minimal hypotheses. Bull. Aust. Math. Soc. 39, 439–441 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Borwein, J.M., Burachik, R.S., Yao, L.: Conditions for zero duality gap in convex programming. J. Nonlinear Convex Anal. (in press).
  20. 20.
    Borwein, J.M., Hamilton, C.: Symbolic convex analysis: algorithms and examples. Math. Program. 116, 17–35 (2009). Maple packages SCAT and CCAT available at
  21. 21.
    Borwein, J.M., Lewis, A.S.: Convex Analyis andd Nonsmooth Optimization, 2nd edn. Springer, New Yrok (2005)Google Scholar
  22. 22.
    Borwein, J.M., Vanderwerff, J.D.: Convex Functions. Cambridge University Press, Cambridge (2010)zbMATHGoogle Scholar
  23. 23.
    Borwein, J.M., Yao, L.: Structure theory for maximally monotone operators with points of continuity. J. Optim. Theory Appl. 157, 1–24 (2013). (Invited paper)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Borwein, J.M., Yao, L.: Recent progress on monotone operator theory. Infin. Prod. Oper. Appl. Contemp. Math. (in press).
  25. 25.
    Borwein, J.M., Zhu, Q.J.: Techniques of Variational Analysis, CMS Books in Mathematics/Ouvrages de Mathmatiques de la SMC, 20. Springer, New York (2005)Google Scholar
  26. 26.
    Boţ, R.I., Grad, S., Wanka, G.: Duality in Vector Optimization. Springer, New York (2009)zbMATHGoogle Scholar
  27. 27.
    Boţ, R.I., Wanka, G.: A weaker regularity condition for subdifferential calculus and Fenchel duality in infinite dimensional spaces. Nonlinear Anal. 64, 2787–2804 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Burachik, R.S., Iusem, A.N.: Set-Valued Mappings and Enlargements of Monotone Operators, vol. 8. Springer, New York (2008)Google Scholar
  29. 29.
    Brezis, H., Crandall, G., Pazy, P.: Perturbations of nonlinear maximal monotone sets in Banach spaces. Commun. Pure Appl. Math. 23, 123–144 (1970)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Diestel, J.: Geometry of Banach spaces. Springer, New York (1975)zbMATHGoogle Scholar
  31. 31.
    Day, M.M.: Normed Linear Spaces, 3rd edn. Springer, New York (1973)CrossRefzbMATHGoogle Scholar
  32. 32.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Fitzpatrick, S.: Representing monotone operators by convex functions. In: Workshop/Miniconference on Functional Analysis and Optimization (Canberra 1988), Proceedings of the Centre for Mathematical Analysis, vol. 20, pp. 59–65. Australian National University, Canberra, Australia (1988)Google Scholar
  34. 34.
    Ghoussoub, N.: Self-dual Partial Differential Systems and Their Variational Principles. Springer Monographs in Mathematics. Springer, New York (2009)Google Scholar
  35. 35.
    Gossez, J.-P.: On the range of a coercive maximal monotone operator in a nonreflexive Banach space. Proc. Am. Math. Soc. 35, 88–92 (1972)CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Hiriart-Urruty, J.-B., Moussaoui, M., Seeger, A., Volle, M.: Subdifferential calculus without qualification conditions, using approximate subdifferentials: a survey. Nonlinear Anal. 24, 1727–1754 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    Hiriart-Urruty, J.-B., Phelps, R.: Subdifferential calculus using \(\varepsilon \)-subdifferentials. J. Funct. Anal. 118, 154–166 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    Hörmander, L.: Sur la fonction d’appui des ensembles convexes dans un espace localement convexe. Arkiv för Matematik 3, 181–186 (1955)CrossRefzbMATHGoogle Scholar
  39. 39.
    Klee, V.: Convexity of Chebysev sets. Mathematische Annalen 142, 292–304 (1961)CrossRefzbMATHMathSciNetGoogle Scholar
  40. 40.
    Kohsaka, F., Takahashi, W.: Existence and approximation of fixed points of firmly nonexpansivetype mappings in Banach spaces. SIAM J. Optim. 19, 824–835 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  41. 41.
    Lindenstrauss, J.: On nonseparable reflexive Banach spaces. Bull. Am. Math. Soc. 72, 967–970 (1966)CrossRefzbMATHMathSciNetGoogle Scholar
  42. 42.
    Maréchal, P.: A convexity theorem for multiplicative functions. Optim. Lett. 6, 357–362 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  43. 43.
    Minty, G.: Monotone (nonlinear) operators in a Hilbert space. Duke Math. J. 29, 341–346 (1962)CrossRefzbMATHMathSciNetGoogle Scholar
  44. 44.
    Moreau, J.J.: Fonctions convexes en dualité, Faculté des Sciences de Montpellier, Séminaires de Mathématiques Université de Montpellier, Montpellier (1962)Google Scholar
  45. 45.
    Moreau, J.J.: Fonctions à valeurs dans \([-\infty ,+\infty ]\); notions algébriques, Faculté des Sciences de Montpellier, Séminaires de Mathématiques, Université de Montpellier, Montpellier (1963)Google Scholar
  46. 46.
    Moreau, J.J.: Étude locale d’une fonctionnelle convexe, Faculté des Sciences de Montpellier, Séminaires de Mathématiques Université de Montpellier, Montpellier (1963)Google Scholar
  47. 47.
    Moreau, J.J.: Sur la function polaire d’une fonctionelle semi-continue supérieurement. Comptes Rendus de l’Académie des Sciences 258, 1128–1130 (1964)zbMATHMathSciNetGoogle Scholar
  48. 48.
    Moreau, J.J.: Proximité et dualité dans un espace hilbertien. Bulletin de la Société Mathématique de France 93, 273–299 (1965)zbMATHMathSciNetGoogle Scholar
  49. 49.
    Moreau, J.J.: Fonctionnelles convexes , pp. 1–108. Séminaire Jean Leray, College de France, Paris (1966–1967). Available at
  50. 50.
    Moreau, J.J.: Convexity and duality. In: Functional Analysis and Optimization, pp. 145–169. Academic Press, New York (1966)Google Scholar
  51. 51.
    Penot, J.-P.: The relevance of convex analysis for the study of monotonicity. Nonlinear Anal. 58, 855–871 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  52. 52.
    Penot, J.-P., Zălinescu, C.: Some problems about the representation of monotone operators by convex functions. Aust. N. Z. Ind. Appl. Math. J. 47, 1–20 (2005)zbMATHGoogle Scholar
  53. 53.
    Phelps, R.R.: Convex Functions, Monotone Operators and Differentiability, 2nd edn. Springer, New York (1993)zbMATHGoogle Scholar
  54. 54.
    Phelps, R.R., Simons, S.: Unbounded linear monotone operators on nonreflexive Banach spaces. J. Nonlinear Convex Anal. 5, 303–328 (1998)zbMATHMathSciNetGoogle Scholar
  55. 55.
    Rockafellar, R.T.: Extension of Fenchel’s duality theorem for convex functions. Duke Math. J. 33, 81–89 (1966)CrossRefzbMATHMathSciNetGoogle Scholar
  56. 56.
    Rockafellar, R.T.: On the maximal monotonicity of subdifferential mappings. Pac. J. Math. 33, 209–216 (1970)CrossRefzbMATHMathSciNetGoogle Scholar
  57. 57.
    Rockafellar, R.T.: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149, 75–88 (1970)CrossRefzbMATHMathSciNetGoogle Scholar
  58. 58.
    Rockafellar, R.T., Wets, R.J.-B.: Variational analysis. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 317. Springer, Berlin (1998) (3rd Printing, 2009)Google Scholar
  59. 59.
    Rudin, R.: Functional Analysis, 2nd edn. McGraw-Hill, New York (1991)zbMATHGoogle Scholar
  60. 60.
    Sendov, H.S., Zitikis, R.: The shape of the Borwein-Affleck-Girgensohn function generated by completely monotone and Bernstein functions. J. Optim. Theory Appl. (in press).
  61. 61.
    Simons, S.: Minimax and Monotonicity. Springer, New York (1998)Google Scholar
  62. 62.
    Simons, S.: From Hahn–Banach to Monotonicity. Springer, New York (2008)zbMATHGoogle Scholar
  63. 63.
    Simons, S., Zălinescu, C.: A new proof for Rockafellar’s characterization of maximal monotone operators. Proc. Am. Math. Soc. 132, 2969–2972 (2004)CrossRefzbMATHGoogle Scholar
  64. 64.
    Simons, S., Zǎlinescu, C.: Fenchel duality, Fitzpatrick functions and maximal monotonicity. J. Nonlinear Convex Anal. 6, 1–22 (2005)zbMATHMathSciNetGoogle Scholar
  65. 65.
    Sucheston, L.: Banach limits. Am. Math. Mon. 74, 308–311 (1967)CrossRefzbMATHMathSciNetGoogle Scholar
  66. 66.
    Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific Publishing, New Jersey (2002)CrossRefzbMATHGoogle Scholar
  67. 67.
    Zeidler, E.: Nonlinear Functional Analysis and its Applications II/A: Linear Monotone Operators. Springer, New York (1990)CrossRefzbMATHGoogle Scholar
  68. 68.
    Zeidler, E.: Nonlinear Functional Analysis and its Applications II/B: Nonlinear Monotone Operators. Springer, New York (1990)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2013

Authors and Affiliations

  • Francisco J. Aragón Artacho
    • 1
    Email author
  • Jonathan M. Borwein
    • 1
    • 2
  • Victoria Martín-Márquez
    • 3
  • Liangjin Yao
    • 1
  1. 1.Centre for Computer Assisted Research Mathematics and Its Applications (CARMA)University of NewcastleCallaghanAustralia
  2. 2.King Abdul-Aziz UniversityJeddahSaudi Arabia
  3. 3.Departamento de Análisis Matemático, Facultad de MatemáticasUniversidad de SevillaSevillaSpain

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