Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Convergence of asymptotic directions


Authors: Dinh The Luc and Jean-Paul Penot
Journal: Trans. Amer. Math. Soc. 353 (2001), 4095-4121
MSC (1991): Primary 54A20
DOI: https://doi.org/10.1090/S0002-9947-01-02664-2
Published electronically: May 17, 2001
MathSciNet review: 1837222
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study convergence properties of asymptotic directions of unbounded sets in normed spaces. The links between the continuity of a set-valued map and the convergence of asymptotic directions are examined. The results are applied to investigate continuity properties of marginal functions and asymptotic directions of level sets.


References [Enhancements On Off] (What's this?)

  • 1. S. ADLY, D. GOELEVEN and M. THERA, Recession mappings and noncoercive variational inequalities, J. Math. Anal. Appl. 26 (1996), 1573-1603. MR 97e:49005
  • 2. A. AGADI and J.-P. PENOT, New asymptotic cones and usual tangent cones, submitted.
  • 3. A. AGADI and J.-P. PENOT, Asymptotic approximation of sets with ap= plication in mathematical programming, preprint, Univ. of Pau, February 1996.
  • 4. A. AUSLENDER, How to deal with the unboundedness in optimization: Theory and algorithms, Math. Programming serie B 31(1997), 3-19. MR 98h:90051
  • 5. A. AUSLENDER, Noncoercive optimization problems, Math. Operations Research 21 (1996), 769-782. MR 97j:49010
  • 6. J.-P. AUBIN and H. FRANKOWSKA, ``Set-valued Analysis'', Birkhäuser, Basel, 1990.
  • 7. C. BAIOCCHI, G. BUTTAZZO, F. GASTALDI and F. TOMARELLI, General existence theorems for unilateral problems in continum mechanics, Arch. Ration. Mech. Analysis 100 (1988), 149-180. MR 88k:73014
  • 8. C. BERGE, ``Espaces Topologiques'', Dunod, Paris, 1963. MR 21:4401
  • 9. G. CHOQUET, Ensembles et cônes convexes faiblement complets, C. R. Acad. Sci. Paris 254 (1962), 1908-1910. MR 24:A2823
  • 10. L. CONTESSE and J.-P. PENOT, Continuity of the Fenchel correspondence and continuity of polarities, J. Math. Anal. Appl. 156 (1991), 305-328. MR 92i:49025
  • 11. J.-P. CROUZEIX, Pseudomonotone variational inequality problems: existence of solutions, Math. Programming 78 (1997), 305-314. MR 98i:90101
  • 12. A. DANILIIDIS and N. HADJISSAVAS, Coercivity conditions and variational inequalities, preprint, Univ. of the Aegean, October 1997.
  • 13. G. B. DANTZIG, J. FOLKMAN and N. SHAPIRO, On the continuity of the minimum set of continuous function, J. Math. Anal. Appl. 17 (1967), 519-548. MR 34:7241
  • 14. G. DEBREU, ``Theory of value", New Haven and London, Yale University Press, 1975.
  • 15. J.-P. DEDIEU, Cône asymptote d'un ensemble non convexe. Application à l'optimisation, C. R. Acad. Sci. Paris 287 (1977), 501-503. MR 56:16320
  • 16. J.-P. DEDIEU, L'image de la limite supérieure d'une famille d'ensembles est-elle égale à la limite supérieure de la famille des images?, Ann. Fac. Sc. Toulouse 11 (1990), 91-103. MR 93f:46009
  • 17. J. DURDILL, On the geometric characterization of differentiability II, Com. Math. University Carolinae 15 (1974), 727-744.
  • 18. M. FABIAN, Theory of Fréchet cones, Casopis Propestivani Mat. 107 (1982), 37-58. MR 83d:46053
  • 19. W. FENCHEL, ``Convex cones, sets and functions", mimeographed lecture notes, Princeton University, 1951.
  • 20. V. L. KLEE, Asymptotes and projections of convex sets, Math. Scand. 8 (1960), 356-362. MR 28:1534
  • 21. V. L. KLEIN and A.C. THOMPSON, ``Theory of Correspondences", Wiley, New York, 1984. MR 86a:90012
  • 22. D. KLATTE, On the lower semicontinuity of optimal sets in convex parametric optimization, Math. Progr. Study 10 (1979), 104-109. MR 82i:90108
  • 23. I. KOUADA, Upper-semi-continuity and cone-concavity of multi-valued vector functions in a duality theory for vector optimization, Math. Methods Oper. Res. 46 (1997), 169-192. MR 98h:90069
  • 24. B. KUMMER, A note on the continuity of the solution set of special dual optimization problems, Math. Progr. Study. 10 (1979), 110-114. MR 80f:90105
  • 25. J.-P. EVANS and F. J. GOULD, Stability in nonlinear programming, Operation Research 18 (1970), 107-118. MR 41:9573
  • 26. D. T. LUC, ``Theory of vector optimization'', Lecture Notes Econ. and Math. Systems 319, Springer-Verlag, Berlin, New York, 1989. MR 92e:90003
  • 27. D. T. LUC, Recession cones and the domination property in vector optimization, Math. Programming 49 (1990), 113-122. MR 91i:46012
  • 28. D. T. LUC, Recession maps and applications, Optimization 27 (1993), 1-15. MR 95e:49021
  • 29. D. T. LUC, Recessively compact sets: uses and properties, Set-Valued Analysis (to appear).
  • 30. D. T. LUC, Existence results for densely pseudomonotone variational inequalities, Journal of Mathematical Analysis and Applications 254 (2001), 291-308.
  • 31. D. T. LUC and P. H. DIEN, Differentiable selection of optimal solutions in parametric linear programming, Proc. Amer. Math. Society 125 (1997), 883-892. MR 97e:90030
  • 32. D. T. LUC and M.THERA, Derivatives with support and applications, Math. Oper. Research 19 (1994), 659-675. MR 95e:49022
  • 33. D. T. LUC and M.VOLLE, Level sets under infimal convolution and level addition, J. Optim. Theory and Applic. 94 (1997), 695-714. MR 98j:90097
  • 34. P.K. MONTEIRO, F.H. PAGE Jr, M.H. WOODERS, Arbitrage, equilibrium and gains from trades: A counterexample, J. Math. Econ. 28 (1997) 481-501. MR 98m:90034
  • 35. F.H. PAGE Jr., M.H. WOODERS, A necessary and sufficient condition for the compactness of individually rational and feasible outcomes and the existence of an equilibrium, Economics Letters 52 (1996), 153-162. MR 97j:90021
  • 36. J.- P. PENOT, Continuity properties of performance functions, in: Optimization, Theory and Algorithms , eds: W.Oettli and J.Stoer, Lecture Notes in Pure and Applied Mathematics 86, Marcel Dekker, New York, 1983, 77-90. MR 85f:49018
  • 37. J.- P. PENOT, Compact nets, filters and relations, J. Math. Anal. Appl. 93 (1983), 400-417. MR 84h:49032
  • 38. 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. MR 94k:49013
  • 39. J.- P. PENOT, The cosmic Hausdorff topology, the bounded Hausdorff topology and continuity of polarity, Proc. Amer. Math. Soc. 113 (1991), 275-285. MR 91k:54012
  • 40. S. M. ROBINSON, Stability theory for systems of inequalities, Part I: linear systems, SIAM J. Numer. Anal. 12 (1975), 754-769. MR 53:14270
  • 41. R. T. ROCKAFELLAR, ``Convex analysis", Princeton University Press, Princeton, New Jersey 1970. MR 43:445
  • 42. R. T. ROCKAFELLAR and R. J.-B. WETS, Cosmic convergence, in: Optimization and Nonlinear Analysis, eds: A. Ioffe, M. Marcus and S. Reich, Pitman Notes 244, Longman, Harlow, 1992, 249-272. MR 93j:49015
  • 43. R. T. ROCKAFELLAR and R. J.-B. WETS, Variational Analysis, Springer Verlag, 1998. MR 98m:49001
  • 44. J. E. SPINGARN, Fixed and variable constraints in sensitivity analysis, SIAM J. Control and Optim. 18 (1980), 297-310. MR 81c:90084
  • 45. R. E. STEINITZ, Bedingt konvergente Reihen und konvexe Systeme I,II,III, J. Math. 143 (1913), 128-175; 144 (1914), 1-40; 146 (1916), 1-52.
  • 46. C. ZALINESCU, Stability for a class of nonlinear optimization, Proceedings of the Summer School ``Nonsmooth optimization and related topics", Erice 1989. F.H. Clarke, V.F. Dem'yanov and F. Giannessi, eds., Plenum Press, New York and London, 1989. MR 91d:90075
  • 47. C. ZALINESCU, Recession cones and asymptotically compact sets, J. Optim. Theory Appl. 77 (1993), 209-220. MR 94d:90057

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 54A20

Retrieve articles in all journals with MSC (1991): 54A20


Additional Information

Dinh The Luc
Affiliation: Département de Mathematiques, Université d’Avignon, Avignon, France; Hanoi Institute of Mathematics, Hanoi, Vietnam
Email: dtluc@univ-avignon.fr

Jean-Paul Penot
Affiliation: Département de Mathématiques, Université de Pau, Pau, France
Email: Jean-Paul.Penot@univ-pau.fr

DOI: https://doi.org/10.1090/S0002-9947-01-02664-2
Keywords: Asymptotic cone, cosmic continuity, marginal function, recession cone, recession function, level set, extreme desirability condition
Received by editor(s): December 27, 1994
Received by editor(s) in revised form: December 27, 1999
Published electronically: May 17, 2001
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society