Continuity of Usual Operations and Variational Convergences

Set-Valued Analysis

Volumn 11, Issue 3, 2003, Pages 225-256

  Penot, Jean Paul ^a   Zǎlinescu, Constantin ^b  

^a Lab de Mathematiques Appliquees   (France)

^b ALEXANDRU IOAN CUZA UNIVERSITY OF IASI   (Romania)

Author keywords

Asymptotic functions; Bounded Hausdorff convergence; Bounded hemi convergence; Convergence; Epiconvergence; Hamilton Jacobi equations; Mosco convergence; Variational convergence

Indexed keywords

EID: 1642500111     PISSN: 09276947     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1024432532388     Document Type: Article

References (55)

1. Attouch, H.: Variational Convergence for Functions and Operators, Pitman, Boston, 1984.
2. Attouch, H., Azé, D. and Beer, G.: On some inverse stability problems for the epigraphical sum, Nonlinear Anal. 16 (1991), 241254.
3. Attouch, H., Lucchetti, R. and Wets, R. J.-B.: The topology of the ρ-Hausdorff distance, Ann. Mar. Pura Appl. (4) 160 (1991), 303-320.
4. Attouch, H., Moudafi, A. and Riahi, H.: Quantitative stability analysis for maximal monotone operators and semi-groups of contractions. Nonlinear Anal. 21 (1993), 697-723.
5. Attouch, H., Ndoutoume, J. and Théra, M.: Epigraphical convergence of functions and convergence of their derivatives in Banach spaces, Sém. Anal. Convexe 20 (1990), Exp. No. 9, 45 pp.
6. Attouch, H. and Théra, M.: Convergences en analyse multivoque et variationnelle, Matapli 36 (1993), 22-39.
7. Attouch, H. and Wets, R. J.-B.: Isometries for the Legendre-Fenchel transform, Trans. Amer Math. Soc. 296 (1986), 33-60.
8. Attouch, H. and Wets, R. J.-B.: Epigraphical analysis, Ann. Inst. H. Poincaré Anal. Non Linéaire 6 (suppl.) (1989), 73-100.
9. Attouch, H. and Wets, R. J.-B.: Quantitative stability of variational systems: I. The epigraphical distance, Trans. Amer Math. Soc. 328 (1991), 695-729.
10. Attouch, H. and Wets, R. J.-B.: Quantitative stability of variational systems. II. A framework for nonlinear conditioning, SIAM J. Optim. 3 (1993), 359-381.
11. Attouch, H. and Wets, R. J.-B.: Quantitative stability of variational systems. III: ε-approximate solutions, Math. Programming 61A (1993), 197-214.
12. Aubin, J.-P. and Frankowska, H.: Set-Valued Analysis, Birkhäuser, Basel, 1990.
13. Azé, D. and Penot, J.-P.: Recent quantitative results about the convergence of convex sets and functions, In: Functional analysis and approximation (Bagni di Lucca, 1988), Pitagora, Bologna, 1989, pp. 90110.
14. Azé, D. and Penot, J.-P: Operations on convergent families of sets and functions, Optimization 21 (1990), 521-534.
15. Azé, D. and Penot, J.-P.: Qualitative results about the convergence of convex sets and convex functions, In: Optimization and nonlinear analysis (Haifa, 1990), Longman Sci. Tech., Harlow, 1992, 1-24.
16. Azé, D. and Rahmouni, A.: On primal dual stability in convex optimization, J. Convex Anal. 3 (1996), 309-329.
17. Beer, G.: Conjugate convex functions and the epi-distance topology, Proc. Amer Math. Soc. 108 (1990), 117-126.
18. Beer, G.: Topologies on Closed and Convex Sets, Kluwer Academic Publishers, Dordrecht, 1993.
19. Beer, G. and Lucchetti, R.: Convex optimization and the epi-distance topology, Trans. Amer Math. Soc. 327 (1991), 795-813.
20. Beer, G. and Lucchetti, R.: The epi-distance topology: continuity and stability results with applications to convex optimization problems, Math. Oper Res. 17 (1992), 715-726.
21. Castaing, C. and Valadier, M.: Convex Analysis and Measurable Multifunctions, Lecture Notes in Math. 580, Springer, Berlin, 1977.
22. Contesse, L. and Penot, J.-P.: Continuity of the Fenchel correspondence and continuity of polarities, J. Math. Anal. Appl. 156 (1991), 305-328.
23. Eberhard, A. and Wenczel, R.: Epi-distance convergence of parametrised sums of convex functions in non-reflexive spaces, J. Convex Anal. 7 (2000), 47-71.
24. Kusraev, A. G. and Kutateladze, S. S.: Subdifferentials: Theory and Applications, Kluwer Academic Publishers, Dordrecht, 1995.
25. Lahrache, J.: Stabilité et convergence dans les espaces non réflexifs, Sém. Anal. Convexe 21 (1991), Exp. No. 10, 50 pp.
26. Luc, D. T. and Penot, J.-P.: Convergence of asymptotic directions, Trans. Amer Math. Soc. 353 (2001), 4095-4121.
27. Lucchetti, R. and Pasquale, A.: A new approach to a hyperspace theory, J. Convex Anal. 1 (1994), 173-193.
28. Lucchetti, R. and Torre, A.: Classical convergences and topologies, Set-Valued Anal. 2 (1994), 219-241.
29. McLinden, L. and Bergstrom, R. C.: Preservation of convergence of convex sets and functions in finite dimensions, Trans. Amer Math. Soc. 268 (1981), 127-142.
30. Michel, Ph.: Problème des inégalités. Application à la programmation et au contrôle optimal, Bull. Soc. Math. France 101 (1973), 413-439.
31. Penot, J.-P.: Regularity conditions in mathematical programming, Math. Programming Study 19 (1982), 167-199.
32. Penot, J.-P.: Compact nets, filters, and relations, J. Math. Anal. Appl. 93 (1983), 400-417.
33. Penot, J.-P.: The cosmic Hausdorff topology, the bounded Hausdorff topology and continuity of polarity, Proc. Amer Math. Soc. 113 (1991), 275-285.
34. Penot, J.-P.: Topologies and convergences on the space of convex functions, Nonlinear Anal. 18 (1992), 905-916.
35. Penot, J.-P.: Preservation of persistence and stability under intersections and operations. I. Persistence, J. Optim. Theory Appl. 79 (1993), 525-550.
36. Penot, J.-P.: Preservation of persistence and stability under intersections and operations. II. Stability, J. Optim. Theory Appl. 79 (1993), 551-561.
37. Penot, J.-P.: On the convergence of subdifferentials of convex functions, Nonlinear Anal. 21 (1993), 87-101.
38. Penot, J.-P.: Miscellaneous incidences of convergence theories in optimization and nonlinear analysis, I: Behavior of solutions, Set-Valued Anal. 2 (1994), 259-274.
39. Penot, J.-P.: Miscellaneous incidences of convergence theories in optimization and nonlinear analysis, II: Applications in nonsmooth analysis, In: Recent Advances in Nonsmooth Optimization, D. Z. Du et al. (eds), World Scientific, Singapore, 1995, pp. 289-321.
40. Penot, J.-P.: Conditioning convex and nonconvex problems, J. Optim. Theory Appl. 90 (1996), 535-554.
41. Penot, J.-P.: Metric estimates for the calculus of multimappings, Set-Valued Anal. 5 (1997), 291-308.
42. Penot, J.-P.: Proximal mappings, J. Approx. Theory 94 (1998), 203-221.
43. Penot, J.-P.: What is quasiconvex analysis? Optimization 47 (2000), 35-110.
44. Penot, J.-P. and Volle, M.: Explicit solutions to Hamilton Jacobi equations under mild continuity and convexity assumptions, J. Nonlinear Convex Anal. 1 (2000), 177-199.
45. Penot, J.-P. and Zǎlinescu, C.: Approximation of functions and sets, In: Approximation, Optimization and Mathematical Economics, M. Lassonde (ed.), Physica-Verlag, Heidelberg, 2001, pp. 255-274.
46. Penot, J.-P. and Zǎlinescu, C.: Persistence and stability of solutions to Hamilton Jacobi equations, Preprint, Univ. of Pau, June 2000.
47. Robinson, S.M.: Regularity and stability for convex multivalued functions, Math. Oper. Res. 1 (1976), 130-143.
48. Rockafellar, R. T. and Wets, R. J.-B.: Cosmic convergence, In: Optimization and Nonlinear Analysis, A. Ioffe et al. (eds), Pitman Notes 244, Longman, Harlow, 1992, pp. 249-272.
49. Rockafellar, R. T. and Wets, R. J.-B.: Variational Analysis, Springer, Berlin, 1997.
50. Sonntag, Y. and Zǎlinescu, C.: Set convergences. An attempt of classification, Trans. Amer. Math. Soc. 340 (1993), 199-226.
51. Stromberg, T.: A study of the operation of infimal convolution, Doctoral Thesis, Lulea Univ. of Technology, 1994.
52. Ursescu, C.: Multifunctions with convex closed graph, Czechoslovak Math. J. 25 (100) (1975), 438-441.
53. Zǎlinescu, C.: 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, pp. 437-458.
54. Zǎlinescu, C.: On convex sets in general position, Linear Algebra Appl. 64 (1985), 191-198.
55. Zǎlinescu, C.: Convex Analysis in General Vector Spaces, World Scientific, Singapore, 2002.