Abstract

The sum of (nonlinear) maximal monotone operators is reconsidered from the Yosida approximation and graph-convergence point of view. This leads to a new concept, called variational sum, which coincides with the classical (pointwise) sum when the classical sum happens to be maximal monotone. In the case of subdifferentials of convex lower semicontinuous proper functions, the variational sum is equal to the subdifferential of the sum of the functions. A general feature of the variational sum is to involve not only the values of the two operators at the given point but also their values at nearby points.

Author information

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

H. Attouch

Laboratoire d’Analyse Convexe, Université Montpellier II, F-34095 Montpellier Cedex 5

J.-B. Baillon

Université Lyon I, F-69622 Villeurbanne Cedex.

M. Théra

LACO, URA 1586, Université de Limoges, F-87060 Limoges Cedex.

Keywords

  • maximal monotone operator
  • Yosida approximation
  • convex function
  • subdifferential operator
  • set-convergence
  • graph-convergence
  • epi-convergence
  • Mosco-convergence
  • ρ-Hausdorff distance
  • graph-distance
  • epi-distance
  • inf-convolution
  • Fenchel duality
  • Brezis-Crandall-Pazy condition

References

  1. H. Attouch: Convergence de fonctions convexes, des sous-différentiels et semi-groupes associés, C.R. Acad. Sci. Paris 284, 539-542, 1977.Google Scholar
  2. H. Attouch: Convergence de fonctionnelles convexes, Journées d’Analyse Non Linéaire, Proceedings, Besançon, France, édité par Ph. Bénilan et J. Robert, Lecture Notes in Mathematics 665, 1977.https://doi.org/10.1007/BFb0061795Google Scholar
  3. H. Attouch: Familles d’opérateurs maximaux monotones et mesurabilité, Ann. Mat. Pura Appl.,(4) 120, 35-111, 1979.https://doi.org/10.1007/BF02411939Google Scholar
  4. H. Attouch: On the maximality of the sum of two maximal monotone operators, Nonlinear Analysis, Theory, Methods & Applications, Vol. 5, n0 2, pp 143-147, 1981.https://doi.org/10.1016/0362-546X(81)90039-0Google Scholar
  5. H. Attouch: Variational convergence for functions and operators, Applicable Maths. Series, Pitman, London, 1984.Google Scholar
  6. H. Attouch, D. Azé, R. Wets: Convergence of convex-concave saddle functions, Annales de l’Institut H. Poincaré, Analyse non linéaire, 5, 532-572, 1988.Google Scholar
  7. H. Attouch, J.-B. Baillon, M. Théra: Somme variationelle d’opérateurs maximaux monotones, C. R. Acad. Sci. Paris, 317, n0 5, 485-490, 1993.Google Scholar
  8. H. Attouch, G. Beer: On the convergence of subdifferentials, Arkiv der Mathematik, Vol 60, 389-400, 1993.https://doi.org/10.1007/BF01207197Google Scholar
  9. H. Attouch, R. Lucchetti, R. Wets: The topology of the ρ-Hausdorff distance, Annali Mat. Pura. Appl., IV, Vol.CLX, 303-320, 1991.https://doi.org/10.1007/BF01764131Google Scholar
  10. H. Attouch, A. Moudafi, H. Riahi: Quantitative stability analysis for maximal monotone operators and semigroups of contractions, Nonlinear Anal. Vol. 21, n 0 9, 497-723, 1993.https://doi.org/10.1016/0362-546X(93)90065-ZGoogle Scholar
  11. H. Attouch, M. Théra: Convergences en analyse multivoque et unilatérale, MATAPLI, Bulletin de liaison n0 36, 23-40, 1993.Google Scholar
  12. H. Attouch, J.-L. Ndoutoume, M. Théra: Epigraphical convergence of functions and convergence of their derivatives in Banach spaces, Séminaire d’Analyse Convexe de Montpellier, Vol. 20, exposé 9, 1990.Google Scholar
  13. H. Attouch, R. Wets: Isometries for the Legendre-Fenchel transform, Transactions of the American Mathematical Society, 296, 33-60, 1986.https://doi.org/10.1090/S0002-9947-1986-0837797-XGoogle Scholar
  14. H. Attouch, R. Wets: Quantitative stability of variational systems : I. The epigraphical distance, Transactions of the American Mathematical Society, 328, n0 2, 695-729 1991.https://doi.org/10.1090/S0002-9947-1991-1018570-0Google Scholar
  15. H. Attouch, R. Wets: Epigraphical analysis, Analyse non linéaire, eds. H. Attouch, J.P. Aubin, F. Clarke, I. Ekeland, 73-100, Gauthier-Villars, Paris, 1989.https://doi.org/10.1016/s0294-1449(17)30036-7Google Scholar
  16. J-.P. Aubin, I. Ekeland: Applied Nonlinear Analysis, Wiley, New-York 1984.Google Scholar
  17. J.-P. Aubin, H. Frankowska: Set-valued Analysis, Birkhäuser, 1990.Google Scholar
  18. J.-P. Aubin, R. Wets: Stable approximations of set-valued maps, Ann. Inst. Henri Poincaré, Vol 5, n0 6, 519-535, 1988.https://doi.org/10.1016/s0294-1449(16)30334-1Google Scholar
  19. D. Azé: Rapidité de convergence pour une somme de fonctions convexes, Publication Avamac, Université de Perpignan, Vol. I, 1986.Google Scholar
  20. D. Azé, J.-P. Penot: Recent Quantitative Results about the Convergence of sets and Functions, Functional analysis and approximation, P.-L. Papini ed., Pitagora Editrice, Bologna, 90-110, 1988.Google Scholar
  21. D. Azé, J.-P. Penot: Operations on convergent families of sets and functions, Optimization, 21, 4, 521-534, 1990.https://doi.org/10.1080/02331939008843576Google Scholar
  22. J.-B. Baillon, Ph. Clément: Examples of unbounded imaginary powers of operators, J. Func. Anal., 100, 419-434, 1991.https://doi.org/10.1016/0022-1236(91)90119-PGoogle Scholar
  23. H. T. Banks, S. Reich, I. G. Rosen: An approximation theory for the identification of nonlinear distributed parameter systems. W.P. Lefschetz Center, Brown Univ. Providence, 1988.Google Scholar
  24. G. Beer: Conjugate convex functions and the epidistance topology, Proc. Amer. Math. Soc., 108, 117-128, 1990.https://doi.org/10.1090/S0002-9939-1990-0982400-8Google Scholar
  25. G. Beer: Topologies on Closed and Closed Convex Sets, Kluwer, 1993.https://doi.org/10.1007/978-94-015-8149-3Google Scholar
  26. G. Beer, R. Lucchetti: Convex Optimization and the Epi-distance Topology, Trans. Amer. Math. Soc. 327, 795-813,1991.https://doi.org/10.1090/S0002-9947-1991-1012526-XGoogle Scholar
  27. G. Beer, M. Théra: Attouch-Wets convergence and a differential operator for convex functions, to appear in Proc. Amer. Math. Soc. 1993.https://doi.org/10.1090/S0002-9939-1994-1204368-5Google Scholar
  28. Ph. Benilan: Une remarque sur la convergence des semi-groupes non linéaires, C. R. Acad. Sci. 272, 1182-1184, 1971.Google Scholar
  29. H. Brézis: Opérateurs maximaux monotones et semi groupes de contractions dans les espaces de Hilbert, North Holland, 1973.Google Scholar
  30. H. Brézis: Cours Université Paris 6, 1978.Google Scholar
  31. H. Brézis, M.G. Crandall, A. Pazy: Perturbations of nonlinear maximal monotone sets in Banach spaces, Comm. pure and Appl. Math., 23, 123-144, 1970.https://doi.org/10.1002/cpa.3160230107Google Scholar
  32. H. Brézis, A. Pazy: Convergence and approximation of nonlinear semi-groups in Banach spaces, J. Funct. Anal., n0 9, 63-74, 1972.https://doi.org/10.1016/0022-1236(72)90014-6Google Scholar
  33. F. Browder: Nonlinear operators and nonlinear equations of evolution. Proc. Symp. Pure Math. 18, Part 2, Amer. Math. Soc., 1976.https://doi.org/10.1090/pspum/018.2Google Scholar
  34. Z. Chbani: Caractérisation de la convergence d’intégrales définies à partir d’opérateurs elliptiques par la convergence des coefficients, Séminaire d’Analyse Convexe de Montpellier, Vol 21, exposé 12, 1991.Google Scholar
  35. P. R. Chernoff: Product formulas, nonlinear semigroups, and addition of unbounded operators, Mem. Amer. Math. Soc., 140, 1-121, 1974.https://doi.org/10.1090/memo/0140Google Scholar
  36. V. Chiado Piat, G. Dal Maso, Defranceschi: G-convergence of monotone operators, Annales de l’Institut H. Poincaré, Analyse non linéaire, Vol 7, n0 3, 123-160, 1990.https://doi.org/10.1016/s0294-1449(16)30298-0Google Scholar
  37. Ph. Clément, P. Egberts: On the sum of two maximal monotone operators, Differential and Integral Equations, Vol. 3. n0 6, 1127-1138, 1990.https://doi.org/10.57262/die/1379101983Google Scholar
  38. M. Crandall: Semi-groups of nonlinear transformations in Banach spaces, Contributions to nonlinear functional analysis, E. Zarantonello ed., Acad. Press, 1971.https://doi.org/10.1016/B978-0-12-775850-3.50010-8Google Scholar
  39. G. Da Prato, P. Grisvard: Sommes d’opérateurs linéaires et équations différentielles opé rationnelles, J. Math. Pures Appl., 54, 305-387, 1975.Google Scholar
  40. A. Haraux: Nonlinear evolution equations- Global behavior of solutions, Lecture notes in Mathematics 841, Springer-Verlag, 1981.https://doi.org/10.1007/BFb0089606Google Scholar
  41. T. Kato: Perturbation theory for linear operators, Springer, Berlin, 1966.https://doi.org/10.1007/978-3-642-53393-8Google Scholar
  42. D. Kravvaritis, N. Papageorgiou: Multivalued perturbations of subdifferential type evolution equations in Hilbert spaces, J. of Dif. Equations, 76, 2, 238-255, 1988.https://doi.org/10.1016/0022-0396(88)90073-3Google Scholar
  43. F. Kubo: Conditional expectations and operations derived from network connections, J. of Math. Anal. and Appl. 80, 477-489, 1981.https://doi.org/10.1016/0022-247X(81)90117-7Google Scholar
  44. M. Lapidus: Formules de Trotter et calcul opérationnel de Feynman, Thèse d’ Etat, Université Paris VI, Juin 1986.Google Scholar
  45. B. Lemaire: Coupling optimization methods and variational convergence, trends in mathematical optimization. Series of Num. Math., K.H. Hoffmann, J.B. Hiriart-Urruty, C. Lemarechal, J. Zowe eds, Birkhäuser, Basel, Vol 84, 163-179, 1988.https://doi.org/10.1007/978-3-0348-9297-1_12Google Scholar
  46. M. Matzeu: Su un tipo di continuità dell’operatore subdifferenziale, Boll. U.M.I. 11,107-124, 1975.Google Scholar
  47. U. Mosco: Convergence of convex sets and of solutions of variational inequalities, Advances in Math. 3, 510-585, 1969.https://doi.org/10.1016/0001-8708(69)90009-7Google Scholar
  48. K. Mouallif: Convergence variationnelle et méthodes perturbées pour les problèmes d’optimisation et de point selle, Thèse d’État, Université de Liège, 1989.Google Scholar
  49. A. Moudafi: Perturbation de la méthode de l’inverse partiel. Publication Avamac 87- 05, Université de Perpignan, Vol. II, 1987.Google Scholar
  50. J.-L. Ndoutoume: Epi-convergence en calcul différentiel gé néralisé. Résultats de convergence et approximation, Thè se, Université de Perpignan, 1987.Google Scholar
  51. J.-L. Ndoutoume, M. Théra: Generalized second-order derivatives of convex functions in reflexive Banach spaces, to appear in J. of Austr. Math. Soc.Google Scholar
  52. N. Papageorgiou, A. Kandikalis: Convergence in approximation and nonsmooth analysis, J. Appr. Theory, 49, 41-54, 1987.https://doi.org/10.1016/0021-9045(87)90112-2Google Scholar
  53. G.B. Passty: The parallel sum of nonlinear monotone operators, Nonlinear Analysis, Theory, Methods & applications, Vol. 10, n0. 3, 215-227, 1986.https://doi.org/10.1016/0362-546X(86)90001-5Google Scholar
  54. J.-P. Penot: Topologies and convergences on the space of convex functions, Preprint. Univ. Pau.Google Scholar
  55. J.-P. Penot: The cosmic Hausdorff topology, the bounded-Hausdorff topology and continuity of polarity, Proc. Amer. Math. Soc. (to appear).Google Scholar
  56. S. Reich: Convergence and approximation of nonlinear semigroups, J. Math. Anal. Appl., 76, 77-83, 1980.https://doi.org/10.1016/0022-247X(80)90060-8Google Scholar
  57. H. Riahi: Quelques résultats de stabilité en analyse épigraphique : approche topologique et quantitative. Thèse, Université de Montpellier, 1989.Google Scholar
  58. R.T. Rockafellar: Conjugate duality and optimization, SIAM Coll. Regional Conference Series in Applied Mathematics n0 16, Philadelphia, 1976.Google Scholar
  59. R.T. Rockafellar: Maximal monotone relations and the second derivative of non-smooth functions, Annales de l’Institut H. Poincaré, Analyse non linéaire, 2, 510-585, 1985.https://doi.org/10.1016/s0294-1449(16)30401-2Google Scholar
  60. R.T. Rockafellar: On the Maximal Monotonicity of the Subdifferential Mapping, Pacific Journal of Mathematics, 33, 209-216, 1970.https://doi.org/10.2140/pjm.1970.33.209Google Scholar
  61. B. Simon: Quantum Mechanics for Hamiltonians defined as Quadratic Forms, Princeton, N. J. 1973.Google Scholar
  62. B. Simon: Schrödinger semigroups, Bull. Amer. Math., Soc., 7, 447-526, 1982.https://doi.org/10.1090/S0273-0979-1982-15041-8Google Scholar
  63. P. Shunmugaraj, D.V. Pai: On stability of approximate solutions of minimization problems, W.P. Indian Institute of Technology, Bombay, to appear in Numerical Functional Analysis and Optimization.Google Scholar
  64. Y. Sonntag: Convergence au sens de Mosco; théorie et applications à l’approximation des solutions d’inéquations. Thèse d’Etat. Université de Provence, 1982.Google Scholar
  65. M. Soueycatt: Analyse epi/hypo-graphique des problèmes de points selles, Thèse Université de Montpellier, 1991.Google Scholar
  66. A. Strauss: The nonlinear Schrödinger equation, in Contempory Developments in Continuum Mechanics and PDE edited by G. de la Penha and L. Mediros, 452-465, North Holland, Amsterdam, 1978.https://doi.org/10.1016/S0304-0208(08)70877-6Google Scholar
  67. R. M. Torrejon: Perturbation of nonlinear operators and their parallel sum, Nonlinear Analysis, Theory, Methods & Applications, Vol.10, n0 5, 483-490, 1986.https://doi.org/10.1016/0362-546X(86)90053-2Google Scholar
  68. P. Tossings: Convergence variationnelle et opérateurs maximaux monotones, W.P. n 0 90014, Université de Liège, 1990.Google Scholar
  69. H. P. Trotter: On the product of semigroups of operators, Proc. Amer. Math. Soc., 8, 545-551, 1959.https://doi.org/10.1090/S0002-9939-1959-0108732-6Google Scholar
  70. M. Tsukada: Convergence of best approximations in smooth Banach spaces, J. Approx. Theory, 40, 301-309, 1984.https://doi.org/10.1016/0021-9045(84)90003-0Google Scholar
  71. T. Zolezzi: On stability analysis in mathematical programming, North-Holland, Mathematical Programming Study, 21, 227-242, 1984.https://doi.org/10.1007/BFb0121220Google Scholar

Suggested citation

H. Attouch, J.-B. Baillon, M. Théra. “Variational Sum of Monotone Operators.” Journal of Convex Analysis 1 (1994), No. 1, 1–29. https://doi.org/10.68381/jca01001

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 1994. Rights now held by Banach Press LLC.