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Bounded (Hausdorff) Convergence: Basic Facts and Applications

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Variational Analysis and Applications

Part of the book series: Nonconvex Optimization and Its Applications ((NOIA,volume 79))

Abstract

We present a survey of some uses of a remarkable convergence on families of sets or functions. We evoke some of its applications and stress some calculus rules. The main novelty lies in the use of a notion of “firm” (or uniform) asymptotic cone to an unbounded subset of a normed space. This notion yields criteria for the study of boundedness properties.

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References

  1. S. Adly, E. Ernst and M. Théra, Stability of the solution set of non-coercive variational inequalities, Commun. Contemp. Math. 4 (2002), 145–160.

    Article  MATH  MathSciNet  Google Scholar 

  2. A. Agadi and J.-P. Penot, New asymptotic cones and usual tangent cones, submitted.

    Google Scholar 

  3. H. Attouch, Variational Convergence for Functions and Operators, Pitman, Boston, (1984).

    MATH  Google Scholar 

  4. H. Attouch, D. Azé and G. Beer, On some inverse stability problems for the epigraphical sum, Nonlinear Anal. 16 (1991), 241–254.

    Article  MATH  MathSciNet  Google Scholar 

  5. H. Attouch, R. Lucchetti and R. J.-B. Wets, The topology of the ρ-Hausdorff distance, Ann. Mat. Pura Appl. (4) 160 (1991), 303–320.

    Article  MATH  MathSciNet  Google Scholar 

  6. H. Attouch, A. Moudafi and H. Riahi, Quantitative stability analysis for maximal monotone operators and semi-groups of contractions, Nonlinear Anal. 21 (1993), 697–723.

    Article  MATH  MathSciNet  Google Scholar 

  7. H. Attouch, J. Ndoutoume and M. Théra, Epigraphical convergence of functions and convergence of their derivatives in Banach spaces, Sém. Anal. Convexe 20 (1990), Exp. No. 9, 45 pp.

    Google Scholar 

  8. H. Attouch and R. J.-B. Wets, Isometries for the Legendre-Fenchel transform, Trans. Amer. Math. Soc. 296 (1986), 33–60.

    Article  MATH  MathSciNet  Google Scholar 

  9. H. Attouch and R. J.-B. Wets, Epigraphical analysis, Ann. Inst. H. Poincaré Anal. Non Linéaire 6(suppl.) (1989), 73–100.

    MathSciNet  Google Scholar 

  10. H. Attouch and R. J.-B. Wets, Quantitative stability of variational systems: I. The epigraphical distance, Trans. Amer. Math. Soc. 328 (1991), 695–729.

    Article  MATH  MathSciNet  Google Scholar 

  11. H. Attouch and R. J.-B. Wets, Quantitative stability of variational systems. II. A framework for nonlinear conditioning, SIAM J. Optim. 3 (1993), 359–381.

    Article  MATH  MathSciNet  Google Scholar 

  12. H. Attouch and R. J.-B. Wets, Quantitative stability of variational systems. III: ε-approximate solutions, Math. Programming 61A (1993), 197–214.

    Article  MATH  MathSciNet  Google Scholar 

  13. J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Basel, (1990).

    MATH  Google Scholar 

  14. A. Auslender, How to deal with the unboundedness in optimization: theory and algorithms, Math. Programming ser. B 31 (1997), 3–19.

    Article  MathSciNet  Google Scholar 

  15. A. Auslender and M. Teboulle, Asymptotic cones and functions in optimization and variational inequalities, Springer, New York, 2002.

    Google Scholar 

  16. D. Azé, An inversion theorem for set-valued maps, Bull. Aust. Math. Soc. 37, No.3 (1988), 411–414.

    MATH  Google Scholar 

  17. D. Azé and C.C. Chou, On a Newton type iterative method for solving inclusions, Math. Oper. Res. 20, No.4 (1995), 790–800.

    MATH  MathSciNet  Google Scholar 

  18. D. Azé and J.-P. Penot, Recent quantitative results about the convergence of convex sets and functions, Functional analysis and approximation (Bagni di Lucca, 1988), Pitagora, Bologna, (1989), 90–110.

    Google Scholar 

  19. D. Azé and J.-P. Penot, Operations on convergent families of sets and functions, Optimization 21 (1990), 521–534.

    MATH  MathSciNet  Google Scholar 

  20. D. Azé and J.-P. Penot, Qualitative results about the convergence of convex sets and convex functions, Optimization and nonlinear analysis (Haifa, 1990), Longman Sci. Tech., Harlow, (1992), 1–24.

    Google Scholar 

  21. D. Azé and J.-P. Penot, On the dependence of fixed point sets of pseudo-contractive multimappings. Applications to differential inclusions, submitted.

    Google Scholar 

  22. D. Azé and A. Rahmouni, On primal dual stability in convex optimization, J. Convex Anal. 3 (1996), 309–329.

    MATH  MathSciNet  Google Scholar 

  23. G. Beer, Conjugate convex functions and the epi-distance topology, Proc. Amer. Math. Soc. 108 (1990), 117–126.

    Article  MATH  MathSciNet  Google Scholar 

  24. G. Beer, Topologies on Closed and Convex Sets, Kluwer, Dordrecht, (1993).

    MATH  Google Scholar 

  25. G. Beer and R. Lucchetti, Convex optimization and the epi-distance topology, Trans. Amer. Math. Soc. 327 (1991), 795–813.

    Article  MATH  MathSciNet  Google Scholar 

  26. G. Beer and R. Lucchetti, The epi-distance topology: continuity and stability results with applications to convex optimization problems, Math. Oper. Res. 17 (1992), 715–726.

    MATH  MathSciNet  Google Scholar 

  27. L. Contesse and J.-P. Penot, Continuity of the Fenchel correspondence and continuity of polarities, J. Math. Anal. Appl. 156 (1991), 305–328.

    Article  MATH  MathSciNet  Google Scholar 

  28. J. Daneš and J. Durdill, A note on the geometric characterization of differentiability, Comm. Math. Univ. Carolin. 17 (1976), 195–204.

    MATH  Google Scholar 

  29. J.-P. Dedieu, Cône asymptote d’un ensemble non convexe. Application à l’optimisation, C. R. Acad. Sci. Paris 287 (1977), 501–503.

    MathSciNet  Google Scholar 

  30. A. Dontchev and T. Zolezzi, Well-posed Optimization Problems, Lecture Notes in Maths 1543, Springer-Verlag, Berlin, (1993).

    MATH  Google Scholar 

  31. J. Durdill, On the geometric characterization of differentiability I, Comm. Math. Univ. Carolin. 15 (1974), 521–540; II, idem, 727–744.

    Google Scholar 

  32. A. Eberhard and R. Wenczel, Epi-distance convergence of parametrised sums of convex functions in non-reflexive spaces, J. Convex Anal. 7 (2000), 47–71.

    MATH  MathSciNet  Google Scholar 

  33. M. Fabian, Theory of Fréchet cones, Casopis Pro Pěstivani Mat., 107 (1982), 37–58.

    MathSciNet  MATH  Google Scholar 

  34. A. D. Ioffe, Regular points of Lipschitz functions, Trans. Amer. Math. Soc. 251 (1979), 61–69.

    Article  MATH  MathSciNet  Google Scholar 

  35. T. Kato, Perturbation theory for linear operators, Springer Verlag, Berlin (1966).

    MATH  Google Scholar 

  36. D. Klatte, On quantitative stability for non-isolated minima, Control Cybern. 23 (1994), 183–200.

    MATH  MathSciNet  Google Scholar 

  37. M. A. Krasnoselskii, Positive solutions of operator equations, Noordhoff, Groningen (1964).

    Google Scholar 

  38. J. Lahrache, Stabilité et convergence dans les espaces non réflexifs, Sém. Anal. Convexe 21 (1991), Exp. No. 10, 50 pp.

    Google Scholar 

  39. D. T. Luc, Recession maps and applications, Optimization 27 (1993), 1–15.

    MATH  MathSciNet  Google Scholar 

  40. D. T. Luc, Recessively compact sets: properties and uses, Set-Valued Anal. 10 (2002), 15–35.

    Article  MATH  MathSciNet  Google Scholar 

  41. D. T. Luc and J.-P. Penot, Convergence of asymptotic directions, Trans. Amer. Math. Soc. 353 (2001), 4095–4121.

    Article  MATH  MathSciNet  Google Scholar 

  42. R. Lucchetti and A. Pasquale, A new approach to a hyperspace theory, J. Convex Anal. 1 (1994), 173–193.

    MATH  MathSciNet  Google Scholar 

  43. R. Lucchetti and A. Torre, Classical convergences and topologies, Set-Valued Anal. 2 (1994), 219–241.

    Article  MATH  MathSciNet  Google Scholar 

  44. L. McLinden and R. C. Bergstrom, Preservation of convergence of convex sets and functions in finite dimensions, Trans. Amer. Math. Soc. 268 (1981), 127–142.

    Article  MATH  MathSciNet  Google Scholar 

  45. F. Mignot, Contrôle dans les inéquations variationelles elliptiques, J. Funct. Anal. 22 (1976), 130–185.

    Article  MATH  MathSciNet  Google Scholar 

  46. J.-J. Moreau, Intersection of moving convex sets in a normed space, Math. Scand. 36 (1975), 159–173.

    MATH  MathSciNet  Google Scholar 

  47. U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Adv. Math. 3 (1969), 510–585.

    Article  MATH  MathSciNet  Google Scholar 

  48. S. B. Nadler, Multivalued contraction mappings, Pacific J. Math. 30 (1969), 475–488.

    MATH  MathSciNet  Google Scholar 

  49. T. Pennanen, R. T. Rockafellar, M. Théra, Graphical convergence of sums of monotone mappings, Proc. Amer. Math. Soc. 130 (2002), 2261–2269.

    Article  MATH  MathSciNet  Google Scholar 

  50. J.-P. Penot, On regularity conditions in mathematical programming, Math. Prog. Study 19 (1982), 167–199.

    MATH  MathSciNet  Google Scholar 

  51. J.-P. Penot, Compact nets, filters and relations, J. Math. Anal. Appl., 93 (1983), 400–417.

    Article  MATH  MathSciNet  Google Scholar 

  52. J.-P. Penot, Differentiability of relations and differential stability of perturbed optimization problems, SIAM J. Control Optim. 22 (1984), 529–551.

    Article  MATH  MathSciNet  Google Scholar 

  53. J.-P. Penot, Preservation of persistence and stability under intersections and operations, Part I: Persistence, J. Optim. Theory Appl. 79 (1993), 525–550; Part II: Stability, idem, 551–561.

    Article  MATH  MathSciNet  Google Scholar 

  54. J.-P. Penot, The cosmic Hausdorff topology, the bounded Hausdorff topology and continuity of polarity, Proc. Amer. Math. Soc, 113 (1991), 275–285.

    Article  MATH  MathSciNet  Google Scholar 

  55. J.-P. Penot, Topologies and convergences on the space of convex functions, Nonlinear Anal. 18 (1992), 905–916.

    Article  MATH  MathSciNet  Google Scholar 

  56. J.-P. Penot, On the convergence of subdifferentials of convex functions, Nonlinear Anal. 21 (1993), 87–101.

    Article  MATH  MathSciNet  Google Scholar 

  57. J.-P. Penot, Conditioning convex and nonconvex problems, J. Optim. Theory Appl. 90 (1996), 535–554.

    Article  MATH  MathSciNet  Google Scholar 

  58. J.-P. Penot, Metric estimates for the calculus of multimappings, Set-Valued Anal. 5 (1997), 291–308.

    Article  MATH  MathSciNet  Google Scholar 

  59. J.-P. Penot, What is quasiconvex analysis? Optimization 47 (2000), 35–110.

    MATH  MathSciNet  Google Scholar 

  60. J.-P. Penot, A metric approach to asymptotic analysis, Bull. Sci. Maths,.

    Google Scholar 

  61. J.-P. Penot and C. Zălinescu, Approximation of functions and sets, in Approximation, Optimization and Mathematical Economics, M. Lassonde ed., Physica-Verlag, Heidelberg, (2001), 255–274.

    Google Scholar 

  62. J.-P. Penot and C. Zălinescu, Continuity of usual operations and variational convergences, Set-Valued Anal. 11 (2003), 225–256.

    Article  MATH  MathSciNet  Google Scholar 

  63. J.-P. Penot and C. Zălinescu, Persistence and stability of solutions to Hamilton-Jacobi equations, preprint, Univ. of Pau, June 2000.

    Google Scholar 

  64. J.-P. Penot and C. Zălinescu, Fenchel-Legendre transform and variational convergences, preprint, 2003.

    Google Scholar 

  65. H. Rådström, An embedding theorem for spaces of convex sets, Proc. Amer. Math. Soc. 3 (1952), 165–169.

    Article  MATH  MathSciNet  Google Scholar 

  66. S. M. Robinson, Stability theory for systems of inequalities, Part I: linear systems, SIAM J. Numer. Anal., 12 (1975), 754–769.

    Article  MATH  MathSciNet  Google Scholar 

  67. S. M. Robinson, Regularity and stability for convex multivalued functions, Math. Oper. Res. 1 (1976), 130–143.

    Article  MATH  MathSciNet  Google Scholar 

  68. R. T. Rockafellar and R. J.-B. Wets, Cosmic convergence, in: Optimization and Nonlinear Analysis, A. Ioffe et al. eds., Pitman Notes 244, Longman, Harlow, 1992, 249–272.

    Google Scholar 

  69. R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Springer, Berlin, 1997.

    Google Scholar 

  70. Y. Sonntag and C. Zălinescu, Set convergences. An attempt of classification, Trans. Amer. Math. Soc. 340 (1993), 199–226.

    Article  MATH  MathSciNet  Google Scholar 

  71. Y. Sonntag and C. Zălinescu, Set convergences: a survey and a classification, Set-Valued Analysis 2 (1994), 339–356.

    Article  MATH  MathSciNet  Google Scholar 

  72. T. Strömberg, The operation of infimal convolution, Dissert. Math. 352 (1996), 1–58.

    Google Scholar 

  73. S. Villa, A W-convergence and well-posedness of non convex functions, J. Convex Anal. (2003), to appear.

    Google Scholar 

  74. C. Zălinescu, On convex sets in general position, Linear Algebra Appl. 64 (1985), 191–198.

    Article  MathSciNet  MATH  Google Scholar 

  75. C. Zălinescu, Stability for a class of nonlinear optimization problems and applications, in Nonsmooth Optimization and Related Fields, F.H. Clarke et al. eds., Plenum Press, London and New York (1989), 437–458.

    Google Scholar 

  76. C. Zălinescu, Recession cones and asymptotically compact sets, J. Optim. Theory Appl., 77 (1993), 209–220.

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Laboratoire de Mathématiques appliquéès, Facultè des Sciences, PAU, France

    Jean-Paul Penot

  2. Faculty of Mathematics, University “Al. I.Cuza” Iasi, Iasi, Rumania

    Constantin Zălinescu

Authors
  1. Jean-Paul Penot
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  2. Constantin Zălinescu
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Editor information

Editors and Affiliations

  1. University of Pisa, Italy

    Franco Giannessi

  2. University of Catania, Italy

    Antonino Maugeri

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Penot, JP., Zălinescu, C. (2005). Bounded (Hausdorff) Convergence: Basic Facts and Applications. In: Giannessi, F., Maugeri, A. (eds) Variational Analysis and Applications. Nonconvex Optimization and Its Applications, vol 79. Springer, Boston, MA. https://doi.org/10.1007/0-387-24276-7_49

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