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About the Maximal Monotonicity of the Generalized Sum of Two Maximal Monotone Operators

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Abstract

We give several regularity conditions, both closedness and interior point type, that ensure the maximal monotonicity of the generalized sum of two strongly-representable monotone operators, and we extend some recent results concerning on the same problem.

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References

  1. 1.

    Bauschke, H.H.: Fenchel duality, Fitzpatrick functions and the extension of firmly nonexpansive mappings. Proc. Am. Math. Soc. 135(1), 135–139 (2007)

  2. 2.

    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(3–4), 499–523 (2006)

  3. 3.

    Borwein, J.M.: Maximal monotonicity via convex analysis. J. Convex Anal. 13, 561–586 (2006)

  4. 4.

    Borwein, J.M.: Maximality of sums of two maximal monotone operators in general Banach space. Proc. Am. Math. Soc. 135(12), 3917–3924 (2007)

  5. 5.

    Borwein, J.M., Lewis, A.S.: Partially finite convex programming, part I: quasi relative interiors and duality theory. Math. Program. 57(1), 15–48 (1992)

  6. 6.

    Boţ, R.I.: Conjugate Duality in Convex Optimization. Springer (2010)

  7. 7.

    Boţ, R.I., Csetnek, E.R.: An application of the bivariate inf-convolution formula to enlargments of monotone operators. Set-Valued Anal. 16, 983–997 (2008)

  8. 8.

    Boţ, R.I., Csetnek, E.R., Wanka, G.: A new condition for maximal monotonicity via representative functions. Nonlinear Anal. 67, 2390–2402 (2007)

  9. 9.

    Boţ, R.I., Grad, S.-M., Wanka, G.: Maximal monotonicity for the precomposition with a linear operator. SIAM J. Optim. 17(4), 1239–1252 (2006)

  10. 10.

    Boţ, R.I., Grad, S.-M., Wanka, G.: Weaker constraint qualifications in maximal monotonicity. Numer. Funct. Anal. Optim. 28(1–2), 27–41 (2007)

  11. 11.

    Burachik, R.S., Svaiter, B.F.: Maximal monotonicity, conjugation and duality product. Proc. Am. Math. Soc. 131(8), 2379–2383 (2003)

  12. 12.

    Burachik, R.S., Svaiter, B.F.: Maximal monotone operators, convex functions and a special family of enlargements. Set-Valued Anal. 10(4), 297–316 (2002)

  13. 13.

    Csetnek, R.E.: Overcoming the failure of the classical generalized interior-point regularity conditions in convex optimization. Applications of the duality theory to enlargements of maximal monotone operators, Dissertation. http://archiv.tu-chemnitz.depub/2009/0202/data/dissertation.csetnek.pdf

  14. 14.

    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 (1988)

  15. 15.

    Gossez, J.-P.: Opérateurs monotones non linéaires dans les espaces de Banach non réflexifs. J. Math. Anal. Appl. 34, 371–395 (1971)

  16. 16.

    Holmes, R.B.: Geometric Functional Analysis and its Applications. Springer, Berlin (1975)

  17. 17.

    Martínez-Legaz, J.E., Théra, M.: A convex representation of maximal monotone operators. J. Nonlinear Convex Anal. 2(2), 243–247 (2001)

  18. 18.

    Martínez-Legaz, J.E., Svaiter, B.F.: Monotone operators representable by l.s.c. convex functions. Set-Valued Anal. 13(1), 21–46 (2005)

  19. 19.

    Marques Alves, M., Svaiter, B.F.: Bronsted–Rockafellar property and maximality of monotone operators representable by convex functions in non-reflexive Banach spaces. J. Convex Anal. 15(4), 693–706 (2008)

  20. 20.

    Marques Alves, M., Svaiter, B.F.: A new old class of maximal monotone operators. J. Convex Anal. 16(3–4), 881–890 (2009)

  21. 21.

    Marques Alves, M., Svaiter, B.F.: On Gossez type (D) maximal monotone operators. J. Convex Anal. 17(3–4), 1077–1088 (2010)

  22. 22.

    Pennanen, T.: Dualization of generalized equations of maximal monotone type. SIAM J. Optim. 10, 809–835 (2000)

  23. 23.

    Penot, J.P.: Is convexity useful for the study of monotonicity? In: Agarwal, R.P., O’Regan, D. (eds.) Nonlinear Analysis and Applications, vol. 1, 2, pp. 807–822. Kluwer, Dordrecht (2003)

  24. 24.

    Penot, J.P.: A representation of maximal monotone operators by closed convex functions and its impact on calculus rules. C. R. Math. Acad. Sci. Paris 338(11), 853–858 (2004)

  25. 25.

    Penot, J.P.: The relevance of convex analysis for the study of monotonicity. Nonlinear Anal. 58(7–8), 855–871 (2004)

  26. 26.

    Penot, J.P., Zălinescu, C.: Convex analysis can be helpful for the asymptotic analysis of monotone operators. Math. Program., Ser. B 116, 481–498 (2009)

  27. 27.

    Penot, J.P., Zălinescu, C.: Some problems about the representation of monotone operators by convex functions. ANZIAM J. 47, 1–20 (2005)

  28. 28.

    Rockafellar, R.T.: Conjugate duality and optimization. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, vol. 16. Society for Industrial and Aplied Mathematics, Philadelphia (1974)

  29. 29.

    Rockafellar, R.T.: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149, 75–88 (1970)

  30. 30.

    Rockafellar, R.T.: On the maximal monotonicity of subdifferential mappings. Pac. J. Math. 33(1), 209–216 (1970)

  31. 31.

    Simons, S.: From Hahn–Banach to Monotonicity. Springer, Berlin (2008)

  32. 32.

    Simons, S.: Minimax and Monotonicity. Springer, Berlin (1998)

  33. 33.

    Simons, S.: The range of a monotone operator. J. Math. Anal. Appl. 199, 176–201 (1996)

  34. 34.

    Simons, S.: Quadrivariate existence theorems and strong representability. arXiv:0809.0325v2 [math.FA]

  35. 35.

    Simons, S., Zălinescu, C.: Fenchel duality, Fitzpatrick functions and maximal monotonicity. J. Nonlinear Convex Anal. 6(1), 1–22 (2005)

  36. 36.

    Voisei, M.D.: The sum and chain rules for maximal monotone operators. Set-Valued Anal. 16, 461–476 (2008)

  37. 37.

    Voisei, M.D.: Calculus rules for maximal monotone operators in general Banach spaces. J. Convex Anal. 15(1), 73–85 (2008)

  38. 38.

    Voisei, M.D., Zălinescu, C.: Strongly-representable monotone operators. J. Convex Anal. 16(3–4), 1011–1033 (2009)

  39. 39.

    Voisei, M.D., Zălinescu, C.: Maximal monotonicity criteria for the composition and the sum under minimal interiority conditions. Math. Program., Ser. B 123, 265–283 (2010)

  40. 40.

    Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002)

  41. 41.

    Zălinescu, C.: Solvability results for sublinear functions and operators. Z. Oper.-Res. A-B 31(3), A79–A101 (1987)

  42. 42.

    Zălinescu, C.: A new proof of the maximal monotonicity of the sum using the Fitzpatrick function. In: Giannessi, F., Maugeri, A. (eds.) Variational Analysis and Applications, Nonconvex Optimization and its Applications, vol. 79, pp. 1159–1172. Springer, New York (2005)

  43. 43.

    Zălinescu, C.: A comparison of constraint qualifications in infinite-dimensional convex programming revisited. J. Aust. Math. Soc. Ser. B. 40, 353–378 (1999)

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Author information

Correspondence to Szilárd László.

Additional information

This work was supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0024.

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László, S., Burján-Mosoni, B. About the Maximal Monotonicity of the Generalized Sum of Two Maximal Monotone Operators. Set-Valued Var. Anal 20, 355–368 (2012). https://doi.org/10.1007/s11228-011-0202-z

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Keywords

  • Monotone operator
  • Strongly-representable operator
  • Representative function
  • Generalized sum

Mathematics Subject Classifications (2010)

  • 47H05
  • 46N10
  • 42A50