Abstract

A general abstract duality result is proposed for equations which are governed by the sum of two operators (possibly multivalued). It allows to unify a large number of variational duality principles, including the Clarke-Ekeland least dual action principle and the Singer-Toland duality. Moreover, it offers a new duality approach to some central questions in the theory of variational inequalities and maximal monotone operators.

Author information

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

H. Attouch

Laboratoire d’Analyse Convexe, Université Montpellier II, 34095 Montpellier Cedex 5, France.

M. Théra

LACO, URA-CNRS 1586, Université de Limoges, 87060 Limoges Cedex, France

Keywords

  • Abstract duality principle
  • Clarke-Ekeland duality
  • Hamiltonian systems
  • Fenchel-Moreau-Rockafellar duality
  • Singer-Toland duality
  • Robinson normal mappings
  • maximal monotone operators
  • convex subdifferential operator
  • qualification conditions

References

  1. H. Attouch: Viscosity solutions of minimization problems, to appear in SIAM Journal on Optimization.Google Scholar
  2. H. Attouch, J.-B. Baillon and M. Théra: Variational sum of maximal monotone operators, Journal of Convex Analysis, 1 (1994) 1–29.https://doi.org/10.68381/jca01001Google Scholar
  3. H. Attouch, J.-B. Baillon and M. Théra: Somme variationnelle d’opérateurs maximaux monotones, C.R. Acad. Sci. Paris 317 Série I (1993) 485–490.Google Scholar
  4. H. Attouch and H. Brézis: Duality for the sum of convex functions in general Banach spaces, Aspects of Math. and its Applications. J. Barroso, Elsevier Science, Amsterdam (1986) 125–133.https://doi.org/10.1016/S0924-6509(09)70252-1Google Scholar
  5. H. Attouch, Z. Chbani and A. Moudafi: Recession operators and solvability of variational problems (1994) submitted.Google Scholar
  6. H. Attouch, H. Riahi and M. Théra: Somme Ponctuelle d’Opérateurs Maximaux Monotones, manuscrit, Juillet 1994.Google Scholar
  7. J.-P. Aubin: Mathematical Methods of Game and Economic Theory, North-Holland, Amsterdam, 1979.Google Scholar
  8. J.-P. Aubin: Variational inequalities revisited, IIASA working paper (1983) 83–43.Google Scholar
  9. D. Azé: Duality for the sum of convex functions in general normed spaces, to appear in Arkiv för Math.Google Scholar
  10. D. Azé: An example of stability for the minima of a sequence of DC functions: Homogenization for a class of nonlinear Sturm-Liouville problems, Mathematical Programming 41 (1988) 127–140.https://doi.org/10.1007/BF01580760Google Scholar
  11. H. Berestycki, H. Brézis: Sur certains problèmes de frontière libre, C. R. Acad. Sc. Paris 283 (1976) 1091–1094.Google Scholar
  12. J. M. Borwein and A. S. Lewis: Practical conditions for Fenchel Duality in Infinite Dimensions, Proceedings of the International Conference on Fixed Points and Applications, J.-B. Baillon and M. Théra Editors, Pitman Research Notes in Mathematics 252 (1990) 83–99.Google Scholar
  13. H. Brézis: Analyse Fonctionnelle, Théorie et Applications, Masson, 1987.Google Scholar
  14. H. Brézis: Periodic solutions of nonlinear vibrating strings and duality principles, Bull. Amer. Math. Soc. 4 no. 1 (1983) 411–425.Google Scholar
  15. H. Brézis, M.G. Crandall and A. Pazy: Perturbations of nonlinear maximal monotone sets in Banach spaces, Comm. pure and Appl. Math. 23 (1970) 123–144.https://doi.org/10.1002/cpa.3160230107Google Scholar
  16. H. Brézis, A. Haraux: Images d’une somme d’opérateurs monotones et applications, Israel J. Math. 23 no. 2 (1976) 165–186.https://doi.org/10.1007/BF02756796Google Scholar
  17. R. Cipolatti, A. Damlamian: Nonconvex duality for a nonlinear boundary problem: the plasma equilibrium, Contribution to nonlinear partial differential equations, Pitman Research Notes 89 (1983) Edited by Bardos, Damlamian, Diaz, Hernandez.Google Scholar
  18. F. H. Clarke: Periodic solutions to Hamiltonian inclusions, J. Diff. Eq. 40 (1981) 1–6.https://doi.org/10.1016/0022-0396(81)90007-3Google Scholar
  19. F. H. Clarke: A classical variational principle for Hamiltonian trajectories, Proc. Amer. Math. Soc. 76 (1979) 186–188.https://doi.org/10.1090/S0002-9939-1979-0534415-7Google Scholar
  20. F. H. Clarke, I. Ekeland: Hamiltonian trajectories with prescribed minimal period, Comm. Pure Appl. Math. 33 (1980) 103–116.https://doi.org/10.1002/cpa.3160330202Google Scholar
  21. F. H. Clarke, I. Ekeland: Nonlinear oscillations and boundary value problems for Hamiltonian systems, Arch. Rat. Mech. Anal. 78 (1982) 315–333.https://doi.org/10.1007/BF00249584Google Scholar
  22. R. Cominetti: Analyse du second ordre de problèmes d’optimisation non différentiable, Thèse de doctorat d’Etat, Université Blaise Pascal, Clermont-Ferrand, France, 1989.Google Scholar
  23. A. Damlamian: Application de la dualité non convexe à un problème linéaire à frontière libre C. R. Acad. Sc. Paris 286 (1978) 153–155.Google Scholar
  24. I. Ekeland: Dual variational methods in nonconvex optimization and differential equations, Mathematical Theories of Optimization, Edited by J.P Cecconi and T. Zolezzi, Lecture Notes in Math. 979 (1983) Springer Verlag 108–120.https://doi.org/10.1007/BFb0066251Google Scholar
  25. I. Ekeland: Legendre duality in non convex optimization and the calculus of variations, SIAM J. Control and Optimization 15 (1977) 890–905.https://doi.org/10.1137/0315058Google Scholar
  26. S. Gowda, M. Teboulle: A comparison of constraint qualification in infinite dimensional convex programming, to appear in SIAM Control and Optimization.Google Scholar
  27. J.-B. Hiriart-Urruty: Generalized differentiability, duality and optimization for problems dealing with the difference of convex functions, in “Convexity and Duality in Optimization” Lecture Notes in Economics and Mathematical Systems 256 (1986) 37–70.https://doi.org/10.1007/978-3-642-45610-7_3Google Scholar
  28. V. Jeyakumar: Duality and infinite optimization, Nonlinear Analysis, Theory, Methods and Applications 15 no. 12 (1990) 1111–1122.https://doi.org/10.1016/0362-546X(90)90047-KGoogle Scholar
  29. P. J. Laurent: Approximation et Optimisation, Herman, Paris, 1972.Google Scholar
  30. J. Mawhin, M. Willem: Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences 74 Springer-Verlag 1989.https://doi.org/10.1007/978-1-4757-2061-7Google Scholar
  31. U. Mosco: Mosco Dual variational inequalities, J. Math. Anal. Appl. 40 (1972) 202—206.https://doi.org/10.1016/0022-247X(72)90043-1Google Scholar
  32. R. R. Phelps: Lectures on Maximal Monotone Operators Lecture given at Pragues/Paseky Summer School, Czech Republic, August 15–28, 1993.Google Scholar
  33. S. M. Robinson: Normal maps induced by linear transformations, Mathematics of Operations Research 17 no. 1 (1992) 691–714.https://doi.org/10.1287/moor.17.3.691Google Scholar
  34. S. M. Robinson: Sensitivity analysis of variational inequalities by normal-map techniques, In : F. Giannessi and A. Maugeri, eds., Variational Inequalities and Network Equilibrium Problems, Plenum Publishing Corp., New York, 1995, 257–269.https://doi.org/10.1007/978-1-4899-1358-6_20Google Scholar
  35. B. Rodrigues, S. Simons: Conjugate functions and subdifferentials in non normed situation for operators with complete graphs, Nonlinear Anal. 12 no. 10 (1988) 1069–1078.https://doi.org/10.1016/0362-546X(88)90102-2Google Scholar
  36. R. T. Rockafellar: Conjugate duality and optimisation, Regional Conference Series in Applied Mathematics. 16 Philadelphia, 1976.Google Scholar
  37. R. T. Rockafellar: Convex Analysis, Princeton, 1970.https://doi.org/10.1515/9781400873173Google Scholar
  38. R. T. Rockafellar: On the maximal monotonicity of subdifferential mappings, Pacific J. Math. 33 (1970) 209–216.https://doi.org/10.2140/pjm.1970.33.209Google Scholar
  39. R. T. Rockafellar: On the maximal monotonicity of the sum of nonlinear monotone operators, Trans. Amer. Math. Soc. 149 (1970) 75–88.https://doi.org/10.1090/S0002-9947-1970-0282272-5Google Scholar
  40. I. Singer: A Fenchel-Rockafellar type duality theorem for maximization, Bull. Australian Math. Soc. 29 (1979) 1983–198.https://doi.org/10.1017/S0004972700010844Google Scholar
  41. R. Temam: Remarks on a free boundary value problem arising in plasma physics, Communication in P.D.E 2 (1977) 563–586.https://doi.org/10.1080/03605307708820039Google Scholar
  42. J. Toland: Duality in nonconvex optimization, J. Math. Anal. Appl. 66 (1978) 399–415.https://doi.org/10.1016/0022-247X(78)90243-3Google Scholar
  43. J. Toland: A duality principle for nonconvex optimization and the calculus of variation, Arch. Rational Mech. Anal. 71 (1979) 41–61.https://doi.org/10.1007/BF00250669Google Scholar
  44. M. Volle: Some applications of the Attouch-Brézis condition to closedness criterion, optimization and duality, Sémin. Anal. Convexe, Univ. Sci. Tech. Languedoc (1990) Exposé no. 16.Google Scholar
  45. M. Volle: Sur quelques formules de dualité convexe et non convexe, Set-Valued Analysis, Special Issue on Set Convergence in Nonlinear Analysis and Optimization, edited by H. Attouch and M. Théra, 2 no. 1 & 2 (1994) 369–379.https://doi.org/10.1007/BF01027112Google Scholar
  46. C. Zalinescu: Solvability results for sublinear functions and operators, Zeitschrift für Operations Research 31 (1987) A79–A101.https://doi.org/10.1007/BF01259338Google Scholar
  47. F. Zhou: Quelques notes sur la dualité de la somme des fonctions convexes, manuscrit, 1994.Google Scholar

Suggested citation

H. Attouch, M. Théra. “A General Duality Principle for the Sum of Two Operators.” Journal of Convex Analysis 3 (1996), No. 1, 1–24. https://doi.org/10.68381/jca03001

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 1996. Rights now held by Banach Press.