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Nonsmooth sequential analysis in Asplund spaces

Authors: Boris S. Mordukhovich and Yongheng Shao
Journal: Trans. Amer. Math. Soc. 348 (1996), 1235-1280
MSC (1991): Primary 49J52; Secondary 46B20, 58C20
MathSciNet review: 1333396
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Abstract: We develop a generalized differentiation theory for nonsmooth functions and sets with nonsmooth boundaries defined in Asplund spaces. This broad subclass of Banach spaces provides a convenient framework for many important applications to optimization, sensitivity, variational inequalities, etc. Our basic normal and subdifferential constructions are related to sequential weak-star limits of Fréchet normals and subdifferentials. Using a variational approach, we establish a rich calculus for these nonconvex limiting objects which turn out to be minimal among other set-valued differential constructions with natural properties. The results obtained provide new developments in infinite dimensional nonsmooth analysis and have useful applications to optimization and the geometry of Banach spaces.

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  • 1. E. Asplund, Fréchet differentiability of convex functions, Acta Math. 121 (1968), 31--47. MR 37:6754
  • 2. J.-P. Aubin and H. Frankowska, Set-valued analysis, Birkhäuser, Boston, Mass., 1990. MR 91d:49001
  • 3. J. M. Borwein, Epi-Lipschitz-like sets in Banach spaces: theorems and examples, Nonlinear Anal. 11 (1987), 1207--1217. MR 89b:90025
  • 4. J. M. Borwein and S. P. Fitzpatrick, Weak-star sequential compactness and bornological limit derivatives, J. of Convex Analysis (to appear).
  • 5. J. M. Borwein and A. D. Ioffe, Proximal analysis in smooth spaces, Set-Valued Anal. (to appear).
  • 6. J. M. Borwein and D. Preiss, A smooth variational principle with applications to subdifferentiability and differentiability of convex functions, Trans. Amer. Math. Soc. 303 (1987), 517--527. MR 88k:49013
  • 7. J. M. Borwein and H. M. Strojwas, Tangential approximations, Nonlinear Anal. 9 (1985), 1347--1366. MR 87i:90320
  • 8. J. M. Borwein and H. M. Strojwas, Proximal analysis and boundaries of closed sets in Banach space, Part I: Theory, Can. J. Math. 38 (1986),431--452; Part II: Applications, Can. J. Math. 39(1987), 428--472. MR 87h:90258; MR 88f:46034
  • 9. J. M. Borwein and Q. J. Zhu, Viscosity solutions and viscosity subderivatives in smooth Banach spaces with applications to metric regularity, SIAM J. Control Optim. (to appear).
  • 10. F. H. Clarke, Optimization and nonsmooth analysis, Wiley, 1983. MR 85m:49002
  • 11. F. H. Clarke, Methods of dynamic and nonsmooth optimization, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 57, Soc. Indust. Appl. Math., Philadelphia, Pa., 1989. MR 91j:49001
  • 12. F. H. Clarke, An indirect method in the calculus of variations, Trans. Amer. Math. Soc. 336 (1993), 655--573. MR 93f:49002
  • 13. F. H. Clarke, R. J. Stern, and P. R. Wolenski, Subgradient criteria for monotonicity and the Lipschitz condition, Can. J. Math. 45 (1993), 1167--1183. MR 94j:49018
  • 14. R. Correa, A. Jofré, and L. Thibault, Subdifferential monotonicity as characterization of convex functions, Numer. Funct. Anal. Optimiz. 15 (1994), 531--535. MR 95d:49029
  • 15. M. G. Crandall and J. P. Lions, Hamilton-Jacobi equations in infinite dimensions. I: Uniqueness of viscosity solutions, J. Funct. Anal. 62 (1985), 379--396. MR 86j:35154
  • 16. R. Deville, G. Godefroy, and V. Zizler, Smoothness and renorming in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Math., vol. 64, Longman, 1993. MR 94d:46012
  • 17. I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 325--358. MR 49:11344
  • 18. M. Fabian, Subdifferentiallity and trustworthiness in the light of a new variational principle of Borwein and Preiss, Acta Univ. Carolinae 30 (1989), 51--56. MR 91c:49024
  • 19. B. E. Ginsburg and A. D. Ioffe, The maximum principle in optimal control of systems governed by semilinear equations, Nonsmooth Analysis and Geometric Methods in Deterministic Optimal Control (B. S. Mordukhovich and H. J. Sussmann, eds.), IMA Volumes in Mathematics and its Applications, vol. 78, Springer, 1996, pp. 81--110.
  • 20. A. D. Ioffe, Nonsmooth analysis: differential calculus of nondifferentiable mappings, Trans. Amer. Math. Soc. 266 (1981), 1--56. MR 82j:58018
  • 21. A. D. Ioffe, Calculus of Dini subdifferentials and contingent derivatives of set-valued maps, Nonlinear Anal. 8 (1984), 517--539. MR 85k:46049
  • 22. A. D. Ioffe, Approximate subdifferentials and applications. I: The finite dimensional theory, Trans. Amer. Math. Soc. 281 (1984), 389--416. MR 89m:49029
  • 23. A. D. Ioffe, Approximate subdifferentials and applications. II: Functions on locally convex spaces, Mathematika 33 (1986), 111--128. MR 87k:49028
  • 24. A. D. Ioffe, Approximate subdifferential and applications. III: The metric theory, Mathematika 36 (1989), 1--38. MR 90g:49012
  • 25. A. D. Ioffe, Proximal analysis and approximate subdifferentials, J. London Math. Soc. 41 (1990), 175--192. MR 91i:46045
  • 26. A. D. Ioffe, Nonsmooth subdifferentials: their calculus and applications, Proceedings of the First World Congress of Nonlinear Analysts, (V. Lakshmikantham, ed.), W. de Gruyter, Berlin, 1995.
  • 27. A. D. Ioffe and R. T. Rockafellar, The Euler and Weierstrass conditions for nonsmooth variational problems, Calculus of Variations and PDEs (to appear).
  • 28. A. Jofré, D. T. Luc, and M. Théra, $\varepsilon $-Subdifferential and $\varepsilon $-monotonicity, preprint, January 1995.
  • 29. A. Jourani and L. Thibault, A note of Fréchet and approximate subdifferentials of composite functions, Bull. Austral. Math. Soc. 49 (1994), 111--116. MR 95f:49019
  • 30. A. Jourani and L. Thibault, Extensions of subdifferential calculus rules in Banach spaces and applications, Can. J. Math. (to appear).
  • 31. A. Y. Kruger, Properties of generalized differentials, Siberian Math. J. 26 (1985), 822--832. MR 87d:90147
  • 32. A. Y. Kruger, Generalized differentials of nonsmooth functions and necessary conditions for an extremum, Siberian Math. J. 26 (1985), 370--379. MR 86j:49038
  • 33. A. Y. Kruger, A covering theorem for set-valued mappings, Optimization 19 (1988), 763--780. MR 89m:49030
  • 34. A. Y. Kruger and B. S. Mordukhovich, Extremal points and the Euler equation in nonsmooth optimization, Dokl. Akad. Nauk BSSR 24 (1980), 684--687. (Russian) MR 82b:90127
  • 35. A. Y. Kruger and B. S. Mordukhovich, Generalized normals and derivatives, and necessary optimality conditions in nondifferentiable programming, Part I: Depon. VINITI, No. 408-80; Part II: Depon. VINITI, No. 494-80, Moscow, 1980. (Russian)
  • 36. E. B. Leach, A note on inverse function theorem, Proc. Amer. Math. Soc. 12 (1961), 694--697. MR 23:A3442
  • 37. P. D. Loewen, Proximal normal formula in Hilbert spaces, Nonlinear Anal. 11 (1987), 979-995. MR 89b:49017
  • 38. P. D. Loewen, Limits of Fréchet normals in nonsmooth analysis, Optimization and Nonlinear Analysis (A. D. Ioffe, L. Marcus, and S. Reich, eds.), Pitman Research Notes Math. Ser. 244, 1992, pp. 178--188. MR 93h:49028
  • 39. P. D. Loewen, Optimal control via nonsmooth analysis, CRM Proceedings and Lecture Notes, Vol. 2, Amer. Math. Soc., Providence, RI, 1993. MR 94h:49003
  • 40. P. D. Loewen, A mean value theorem for Fréchet subgradients, Nonlinear Anal. 23 (1994), 1365--1381. MR 95h:49023
  • 41. B. S. Mordukhovich, Maximum principle in problems of time optimal control with nonsmooth constraints, J. Appl. Math. Mech. 40 (1976), 960 - 969. MR 58:7284
  • 42. B. S. Mordukhovich, Metric approximations and necessary optimality conditions for general classes of nonsmooth extremal problems, Soviet Math. Dokl. 22 (1980), 526--530. MR 82b:90104
  • 43. B. S. Mordukhovich, Nonsmooth analysis with nonconvex generalized differentials and adjoint mappings, Dokl. Akad. Nauk BSSR 28 (1984), 976--979. MR 86c:49018
  • 44. B. S. Mordukhovich, Approximation methods in problems of optimization and control, ``Nauka", Moscow, 1988. MR 89m:49001
  • 45. B. S. Mordukhovich, Sensitivity analysis in nonsmooth optimization, Theoretical Aspects of Industrial Design (D. A. Field and V. Komkov, eds.), SIAM Proc. Appl. Math., vol. 58, Soc. Indust. Appl. Math., Philadelphia, Pa., 1992, pp. 32--46. MR 93a:49012
  • 46. B. S. Mordukhovich, Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions, Trans. Amer. Math. Soc. 340 (1993), 1--35. MR 94a:49011
  • 47. B. S. Mordukhovich, Lipschitzian stability of constraint systems and generalized equations, Nonlinear Anal. 22 (1994), 173--206. MR 94m:49041
  • 48. B. S. Mordukhovich, Stability theory for parametric generalized equations and variational inequalities via nonsmooth analysis, Trans. Amer. Math. Soc. 343 (1994), 609--658. MR 94h:49031
  • 49. B. S. Mordukhovich, Generalized differential calculus for nonsmooth and set-valued mappings, J. Math. Anal. Appl. 183 (1994), 250--288. MR 95i:49029
  • 50. B. S. Mordukhovich and Y. Shao, Differential characterizations of covering, metric regularity, and Lipschitzian properties of multifunctions between Banach spaces, Nonlinear Anal. 25 (1995), 1401--1424. CMP 96:02
  • 51. B. S. Mordukhovich and Y. Shao, Extremal characterizations of Asplund spaces, Proc. Amer. Math. Soc. 124 (1996), 197--205.
  • 52. B. S. Mordukhovich and Y. Shao, On nonconvex subdifferential calculus in Banach spaces, J. of Convex Analysis 2 (1995), 211--228.
  • 53. B. S. Mordukhovich and Y. Shao, Stability of set-valued mappings in infinite dimensions: point criteria and applications, SIAM J. Control Optim. (to appear).
  • 54. J.-P. Penot, A mean value theorem with small subdifferentials, preprint, February 1995.
  • 55. R. R. Phelps, Convex functions, monotone operators and differentiability, 2nd edition, Lecture Notes in Mathematics, vol. 1364, Springer, 1993. MR 94f:46055
  • 56. R. Poliquin, Subgradient monotonicity and convex functions, Nonlinear Anal. 14 (1990), 305--317. MR 91b:90155
  • 57. D. Preiss, Differentiability of Lipschitz functions on Banach spaces, J. Func. Anal. 91 (1990), 312--345. MR 91g:46051
  • 58. R. T. Rockafellar, Generalized directional derivatives and subgradients of nonconvex functions, Can. J. Math. 32 (1980), 257--280. MR 81f:49006
  • 59. R. T. Rockafellar, Proximal subgradients, marginal values, and augmented Lagrangians in nonconvex optimization, Math. Oper. Res. 6 (1981), 424--436. MR 83m:90088
  • 60. R. T. Rockafellar, Extensions of subgradient calculus with applications to optimization, Nonlinear Anal. 9 (1985), 665--698. MR 87a:90148
  • 61. R. T. Rockafellar, Lipschitzian properties of multifunctions, Nonlinear Anal. 9 (1985), 867--885. MR 87a:90149
  • 62. R. T. Rockafellar, Maximal monotone relations and the second derivatives of nonsmooth functions, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), 167--184. MR 87c:49021
  • 63. R. T. Rockafellar and R. J-B Wets, Variational analysis, Springer (to appear).
  • 64. L. Thibault, Subdifferentials of compactly Lipschitzian vector-valued functions, Ann. Mat. Pura Appl. 125 (1980), 157--192. MR 92h:58006
  • 65. L. Thibault, On subdifferentials of optimal value functions, SIAM J. Control Optim. 29 (1991), 1019--1036. MR 92e:49027
  • 66. L. Thibault, A note on the Zagrodny mean value theorem, preprint, July 1994.
  • 67. J. S. Treiman, Clarke's gradients and epsilon-subgradients in Banach spaces, Trans. Amer. Math. Soc. 294 (1986), 65--78. MR 87d:90188
  • 68. D. E. Ward and J. M. Borwein, Nonconvex calculus in finite dimensions, SIAM J. Control Optim. 25 (1987), 1312--1340. MR 88m:58011
  • 69. J. Warga, Fat homeomorphisms and unbounded derivate containers, J. Math. Anal. Appl. 81 (1981), 545--560; errata, ibid 81 (1981), 582--583. MR 83f:58007; MR 84b:58016
  • 70. D. Zagrodny, Approximate mean value theorem for upper subderivatives, Nonlinear Anal. 12 (1988), 1413--1428. MR 89k:58034

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Additional Information

Boris S. Mordukhovich
Affiliation: Department of Mathematics, Wayne State University, Detroit, Michigan 48202

Yongheng Shao
Affiliation: Department of Mathematics, Wayne State University, Detroit, Michigan 48202

Keywords: Nonsmooth analysis, generalized differentiation, nonconvex calculus, Asplund spaces, variational principles, Fr\'{e}chet normals and subdifferentials, sequential limits
Received by editor(s): June 8, 1994
Received by editor(s) in revised form: April 3, 1995
Additional Notes: This research was partially supported by the National Science Foundation under grants DMS–9206989 and DMS-9404128
Article copyright: © Copyright 1996 American Mathematical Society

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