Implicit Multifunction Theorems

Set-Valued Analysis

Volumn 7, Issue 3, 1999, Pages 209-238

  Ledyaev, Yuri S ^a   Zhu, Qiji J ^a  

^a WESTERN MICHIGAN UNIVERSITY   (United States)

Author keywords

Coderivatives; Implicit function theorem; Metric regularity; Multidirectional mean value inequality; Nonsmooth analysis; Open mapping theorem; Solvability; Stability; Subdifferentials

Indexed keywords

EID: 0041544886     PISSN: 09276947     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1008775413250     Document Type: Article

References (67)

1. Aubin, J.-P.: Contingent derivatives of set-valued maps and existence of solutions to nonlinear inclusions and differential inclusions, In: Mathematical Analysis and Applications, Part A, Adv. in Math. Suppl. Stud. 7a, Academic Press, New York, 1981, pp. 159-229.
2. Aubin, J.-P. and Frankowska, H.: On inverse function theorems for set-valued maps, J. Math. Pures Appl. 66 (1987), 71-89.
3. Aubin, J.-P. and Frankowska, H.: Set-Valued Analysis, Birkhäuser, Boston, 1990.
4. Azé, D., Chou, C. C. and Penot, J.-P.: Subtraction theorems and approximate openness for multifunctions: topological and infinitesimal viewpoints, J. Math. Anal. Appl. 221 (1998), 33-58.
5. Borwein, J. M.: Stability and regular points of inequality systems, J. Optim. Theory Appl. 48 (1986), 9-52.
6. Borwein, J. M. and Ioffe, A. D.: Proximal analysis in smooth spaces, Set-Valued Anal. 4 (1996), 1-24.
7. Borwein, J. M. and Preiss, D.: A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions, Trans. Amer. Math. Soc. 303 (1987), 517-527.
8. Borwein, J. M. and Zhuang, D. M.: Verifiable necessary and sufficient conditions for regularity of set-valued and single-valued maps, J. Math. Anal. Appl. 134 (1988), 441-459.
9. Borwein, J. M. and Zhu, Q. J.: Viscosity solutions and viscosity subderivatives in smooth Banach spaces with applications to metric regularity, SIAM J. Control Optim. 34 (1996), 1586-1591.
10. Borwein, J. M. and Zhu, Q. J.: A survey of subdifferential calculus with applications, to appear in Nonlinear Anal.
11. Clarke, F. H.: A new approach to Lagrange multipliers, Math. Oper. Res. 1 (1976), 165-174.
12. Clarke, F. H.: On the inverse function theorem, Pacific J. Math. 64 (1976), 97-102.
13. Clarke, F. H.: Optimization and Nonsmooth Analysis, Wiley, New York, 1983; reprinted as Vol. 5 of Classics in Appl. Math., SIAM, Philadelphia, 1990.
14. Clarke, F. H.: Optimization and Nonsmooth Analysis, Wiley, New York, 1983; reprinted as Vol. 5 of Classics in Appl. Math., SIAM, Philadelphia, 1990.
15. Clarke, F. H.: Methods of Dynamic and Nonsmooth Optimization, CBMS-NSF Regional Conf. Ser. in Appl. Math. 57, SIAM, Philadelphia, 1989.
16. Clarke, F. H.: Solving equations by the decrease principle, In: Advances in Mathematical Sciences: CRM's 25 years (Montreal, PQ, 1994), 29-39, CRM Proc. Lecture Notes 11, Amer. Math. Soc., Providence, RI, 1997.
17. Clarke, F. H. and Ledyaev, Yu. S.: Mean value inequalities, Proc. Amer. Math. Soc. 122 (1994), 1075-1083.
18. Clarke, F. H., Ledyaev, Yu. S., Stern, R. J. and Wolenski, P. R.: Nonsmooth Analysis and Control Theory, Grad. Texts in Math. 178, Springer-Verlag, New York, 1998.
19. Craven, B. D.: Implicit function theorems and Lagrange multipliers, Numer. Funct. Anal. Optim. 2 (1980), 473-486.
20. Demyanov, V. F.: Implicit function theorems in nonsmooth analysis, Vestnik S.-Peterburg. Univ. Mat. Mekh. Astronom. 149(3) (1994), 17-22 (Russian).
21. Devdariani, E. N. and Ledyaev, Yu. S.: Optimality conditions for implicit controlled systems, Dynam. Sys. J. Math. Sci. 80 (1996), 1519-1532.
22. Deville, R., Godefroy, G. and Zizler, V.: A smooth variational principle with applications to Hamilton-Jacobi equations in infinite dimensions, J. Funct. Anal. 111 (1993), 197-212.
23. Dien, P. H. and Yen, N. D.: On implicit functions theorems for set-valued maps and their applications to mathematical programming under inclusion constraints, Appl. Math. Optim. 24 (1991), 35-54.
24. Dontchev, A. L. and Hager, W. W.: Implicit functions, Lipschitz maps, and stability in optimization, Math. Oper. Res. 19 (1994), 753-768.
25. Dontchev, A. L. and Hager, W. W.: An inverse mapping theorem for set-valued maps, Proc. Amer. Math. Soc. 121 (1994), 481-489.
26. Ekeland, I.: On the variational principle, J. Math. Anal. Appl. 47 (1974), 324-353.
27. Frankowska, H.: An open mapping principle for set-valued maps, J. Math. Anal. Appl. 127 (1987), 172-180.
28. Frankowska, H.: Some inverse mapping theorems, Ann. Inst. H. Poincaré. Anal. Non Linéaire 7 (1990), 183-234.
29. Ioffe, A. D.: Regular points of Lipschitz functions, Trans. Amer. Math. Soc. 251 (1979), 61-69.
30. Ioffe, A. D.: Nonsmooth analysis: Differential calculus of nondifferentiable mappings, Trans. Amer. Math. Soc. 266 (1981), 1-56.
31. Ioffe, A. D.: Calculus of Dini subdifferentials of functions and contingent derivatives of set-valued maps, Nonlinear Anal. 8 (1984), 517-539.
32. Ioffe, A. D.: Global surjection and global inverse mapping theorems in Banach spaces, Ann. New York Acad. Sci. 491 (1987), 181-188.
33. Ioffe, A. D.: On the local surjection property, Nonlinear Anal. 11 (1987), 565-592.
34. Ioffe, A. D.: Approximate subdifferentials and applications 3: the metric theory, Mathematika 36 (1989), 1-38.
35. Ioffe, A. D.: Fuzzy principles and characterization of trustworthiness, Set-Valued Anal. 6 (1998), 265-276.
36. Ioffe, A. D. and Penot, J. P.: Subdifferentials of performance functions and calculus of coderivatives of set-valued mappings, Serdica Math. J. 22 (1996), 257-282.
37. Lassonde, M.: First-order rules for nonsmooth constrained optimization, Preprint.
38. Ledyaev, Yu. S.: Theorems on an implicitly given set-valued mapping, Dokl. Akad. Nauk SSSR 276 (1984), 543-546 (Russian).
39. Ledyaev, Yu. S.: Extremal problems in differential game theory, Dr. Sci. Dissertation, Steklov Institute of Mathematics, Moscow, 1990.
40. Levy, A. B.: Implicit multifunction theorems for the sensitivity analysis of variatinal conditions, Math. Programming 74 (1996), 333-350.
41. Levy, A. B. and Rockafellar, R. T.: Sensitivity of solutions to generalized equations, Trans. Amer. Math. Soc. 345 (1994), 661-671.
42. Mordukhovich, B. S.: Metric approximations and necessary optimality conditions for general classes of nonsmooth extremal problems, Soviet Math. Dokl. 22 (1980), 526-530.
43. Mordukhovich, B. S.: Approximation Methods in Problems of Optimization and Control, Nauka, Moscow, 1988 (Russian; English transl., Wiley-Interscience).
44. Mordukhovich, B. S.: Complete charachterization of openness, metric regularity, and Lipschitzian properties of multifunctions, Trans. Amer. Math. Soc. 340 (1993), 1-35.
45. Mordukhovich, B. S.: Generalized differential calculus for nonsmooth and set-valued mappings, J. Math. Anal. Appl. 183 (1994), 250-288.
46. Mordukhovich, B. S.: Lipschitzian stability of constraint systems and generalized equations, Nonlinear Anal. 22 (1994), 173-206.
47. Mordukhovich, B. S.: Sensitivity analysis for constraint and variational systems by means of set-valued differentiation, Optimization 31 (1994), 13-46.
48. Mordukhovich, B. S. and Shao, Y.: Stability of set-valued mappings in infinite dimensions: point criteria and applications, SIAM J. Control Optim. 35 (1997), 285-314.
49. Mordukhovich, B. S. and Shao, Y.: Differential characterizations of covering, metric regularity and Lipschitzian properties of multifunctions between Banach spaces, Nonlinear Anal. 25 (1995), 1401-1428.
50. Mordukhovich, B. S. and Shao, Y.: Nonsmooth sequential analysis in Asplund spaces, Trans. Amer. Math. Soc. 348 (1996), 1235-1280.
51. Penot, J. P.: On the regularity conditions in mathematical programming, Math. Programming Study 19 (1982), 167-199.
52. Penot, J. P.: Metric regularity, openness and Lipschitzian behavior of multifunctions, Nonlinear Anal. 13 (1989), 629-643.
53. Penot, J. P.: Inverse function theorems for mappings and multimappings, SEA Bull. Math. 19 (1995), 1-16.
54. Phelps, R. R.: Convex Functions, Monotone Operators and Differentiability, Lecture Notes in Math. 1364, Springer-Verlag, New York, 1988, 2nd edn, 1993.
55. Pshenichnyǐ, B. N.: Necessary Conditions for an Extremum, Marcel Dekker, New York, 1971.
56. Pshenichnyǐ, B. N.: Theorems on implicit functions for multivalued mappings, Dokl. Akad. Nauk SSSR 291 (1986), 1063-1067 (Russian).
57. Robinson, S. M.: Stability theory for systems of inequalities, Part II: differentiable nonlinear systems, SIAM J. Numer. Anal. 13 (1976), 497-513.
58. Robinson, S. M.: Regularity and stability of convex multivalued functions, Math. Oper. Res. 1 (1976), 130-145.
59. Robinson, S. M.: An implicit-function theorem for a class of nonsmooth functions, Math. Oper. Res. 16 (1991), 292-309.
60. Rockafellar, R. T.: Lipschitz properties of multifunctions, Nonlinear Anal. 9 (1985), 867-885.
61. Sussmann, H. J.: Multidifferential calculus: chain rule, open mapping and transversal intersection theorems, to appear in William Hager (ed.), Proc. IFIP Conf. "Optimal Control: Theory Algorithms, and Applications", University of Florida, Gainesville, 1997.
62. Thibault, L.: On subdifferentials of optimal value functions, SIAM J. Control Optim. 29 (1991), 1019-1036.
63. Ursescu, C.: Multifunctions with closed convex graphs, Czech. Math. J. 25 (1975), 438-441.
64. Warga, J.: Fat homeomorphisms and unbounded derivate containers, J. Math. Anal. Appl. 81 (1981), 545-560.
65. 1 approximations, Nonlinear Anal. 12 (1988), 593-597.
66. Zhu, Q. J.: Clarke-Ledyaev mean value inequalities in smooth Banach spaces, Nonlinear Anal. 32 (1998), 315-324.
67. Zhu, Q. J.: The equivalence of several basic theorems for subdifferentials, Set-Valued Anal. 6 (1998), 171-185.