Complete characterization of openness, Metric regularity, And lipschitzian properties of multifunctions

Transactions of the American Mathematical Society

Volumn 340, Issue 1, 1993, Pages 1-35

  Mordukhovich, Boris ^a  

^a WAYNE STATE UNIVERSITY   (United States)

Author keywords

Lipschitzian behavior; Metric regularity; Multifunctions; Nonsmooth analysis; Open mapping principle; Optimization

Indexed keywords

EID: 84966250815     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0002-9947-1993-1156300-4     Document Type: Article

References (63)

1. J.-P. Aubin, Lipschitz behavior of solutions to convex minimization problems, Math. Oper. Res. 9 (1984), 87-111.
2. J.-P. Aubin and I. Ekeland, Applied nonlinear analysis, Wiley, 1984.
3. J.-P. Aubin and H. Frankowska, Set-valued analysis, Birkhauser, Boston, Mass., 1990.
4. J. M. Borwein, Stability and regular points of inequality systems, J. Optim. Theory Appl. 48 (1986), 9-52.
5. J. M. Borwein and D. M. Zhuang, Verifiable necessary and sufficient conditions for regularity of set-valued and single-valued maps, J. Math. Anal. Appl. 134 (1988), 441-459.
6. J. V. Burke, An exact penalization viewpoint of constrained optimization, SIAM J. Control Optim. 29 (1991), 968-998.
7. F. H. Clarke, Generalized gradients and applications, Trans. Amer. Math. Soc. 205 (1975), 247-262.
8. F. H. Clarke, Optimization and nonsmooth analysis, Wiley, 1983.
9. F. H. Clarke, Methods of dynamic and nonsmooth optimization, Soc. Indust. Appl. Math., Philadelphia, Pa., 1989.
10. R. Cominetti, Metric regularity, tangent sets, and second-order optimality conditions, Appl. Math. Optim. 21 (1990), 265-287.
11. A. V. Dmitruk, A. A. Miljutin, and N. P. Osmolovskii, Ljusternik's theorem and the theory of extrema, Russian Math. Surveys 35 (1980), 11-51.
12. A. L. Dontchev and W. W. Hager, Lipschitzian stability in nonlinear control and optimization, SIAM J. Control Optim. 31 (1993), 569-603.
13. I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324-358.
14. A. V. Fiacco, Introduction to sensitivity analysis in mathematical programming, Academic Press, 1983.
15. H. Frankowska, An open mapping principle for set-valued maps, J. Math. Anal. Appl. 127 (1987), 172-180.
16. H. Frankowska, Some inverse mapping theorems, Ann. Inst. H. Poincare Anal. Non Lineaire 7 (1990), 183-234.
17. L. M. Graves, Some mapping theorems, Duke Math. J. 17 (1950), 111-114.
18. H. Halkin, Interior mapping theorem with set-valued derivatives, J. Analyse Math. 30 (1976), 200-207.
19. J.-B. Hiriart-Urruty, A short proof of the variational principle for approximate solutions of a minimization problem, Amer. Math. Monthly 90 (1983), 206-207.
20. A. D. Ioffe, Regular points of Lipschitz functions, Trans. Amer. Math. Soc. 251 (1979), 61-69.
21. A. D. Ioffe, Nonsmooth analysis: differential calculus of nondifferentiable mappings, Trans. Amer. Math. Soc. 266 (1981), 1-56.
22. A. D. Ioffe, Approximate subdifferentials and applications. I: The finite dimensional theory, Trans. Amer. Math. Soc. 281 (1984), 389-416.
23. A. D. Ioffe, On the local surjection property, Nonlinear Anal. 11 (1987), 565-592.
24. A. D. Ioffe, Approximate subdifferentials and applications. Ill: The metric theory, Mathematika 36 (1989), 1-38.
25. A. D. Ioffe and V. M. Tihomirov, Theory of extremal problems, North-Holland, Amsterdam, 1979.
26. A. Jourani and L. Thibault, Approximate subdifferentials and metric regularity: the finite dimensional case, Math. Programming 47 (1990), 203-218.
27. A. J. King and R. T. Rockafellar, Sensitivity analysis for nonsmooth generalized equations, Math. Programming 55 (1992), 193-212.
28. A. Kruger, Properties of generalized differentials, Siberian Math. J. 26 (1985), 822-832.
29. A. Kruger, A covering theorem for set-valued mappings, Optimization 19 (1988), 763-780.
30. A. Kruger and B. Mordukhovich, Generalized normals and derivatives, and necessary conditions for extrema in nondifferentiable programming. I, Depon. VINITI No. 408-80, Moscow, (Russian)
31. A. Kruger and B. Mordukhovich, Generalizes normals and derivatives, and necessary conditions for extrema in nondifferentiable programming. II, Depon. VINITI No. 494-80, Moscow, 1980. (Russian)
32. A. Kruger and B. Mordukhovich, Extremal points and the Euler equation in nonsmooth optimization, Dokl. Akad. Nauk BSSR 24 (1980), 684-687. (Russian)
33. 1 -optimization, J. Math. Anal. Appl. 158 (1991), 35-46.
34. L. A. Ljustemik, Conditional extrema of functionals, Math. USSR-Sb. 41 (1934), 390-401.
35. G. J. Minty, Monotone (nonlinear) operations in Hilbert space, Duke Math. J. 29 (1962), 341-346.
36. B. Mordukhovich, The maximum principle in the problem of time-optimal control with nonsmooth constraints, J. Appl. Math. Mech. 40 (1976), 960-969.
37. B. Mordukhovich, Metric approximations and necessary optimality conditions for general classes of nonsmooth extremal problems, Soviet Math. Dokl. 22 (1980), 526-530.
38. B. Mordukhovich, Nonsmooth analysis with nonconvex generalized differentials and adjoint mappings, Dokl. Akad. Nauk BSSR 28 (1984), 976-979. (Russian)
39. B. Mordukhovich, Approximation methods in problems of optimization and control, “Nauka”, Moscow, 1988. (Russian; English translation to appear in Wiley-Interscience.)
40. B. Mordukhovich, Sensitivity analysis in nonsmooth optimization, Theoretical Aspects of Industrial De sign (D. A. Field and V. Komkov, eds.), SIAM Proc. Appl. Math., vol. 58, SIAM, Philadelphia, Pa., 1992, pp. 32-46.
41. B. Mordukhovich, On variational analysis of differential inclusions, Optimization and Nonlinear Analysis (A. Ioffe, L. Marcus, and S. Reich, eds.), Pitman Res. Notes Math. Ser. 244, 1992, pp. 199-213.
42. B. Mordukhovich, Lipschitzian stability of constraint systems and generalized equations, Nonlinear Anal, (to appear).
43. B. Mordukhovich, Generalized differential calculus for nonsmooth and set-valued mappings, J. Math. Anal. Appl. (to appear).
44. J. J. Moreau, Fonctionelles convexes, College de France, Paris, 1966.
45. J.-P. Penot, Metric regularity, openness and Lipschitzian behavior of multifunctions, Nonlinear Anal. 13 (1989), 629-643.
46. B. H. Pourciau, Analysis and optimization of Lipschitz continuous mappings, J. Optim. Theory Appl. 22 (1977), 433-452.
47. S. M. Robinson, Regularity and stability of convex multivalued functions, Math. Oper. Res. 1 (1976), 130-145.
48. S. M. Robinson, Stability theory for systems of inequalities. II: Differentiable nonlinear systems, SIAM J. Numer. Anal. 13 (1976), 497-513.
49. S. M. Robinson, Generalized equations and their solutions. I: Basic theory, Math. Programming Stud. 10 (1979), 128-141.
50. R. T. Rockafellar, Convex analysis, Princeton Univ. Press, Princeton, N.J., 1970.
51. R. T. Rockafellar, The theory of subgradients and its applications to problems of optimization: convex and nonconvex functions, Heldermann-Verlag, Berlin, 1981.
52. R. T. Rockafellar, Proximal subgradients, marginal functions, and augmented Lagrangians in nonconvex optimization, Math. Oper. Res. 6 (1981), 427-437.
53. R. T. Rockafellar, Extension of subgradient calculus with applications to optimization, Nonlinear Anal. 9 (1985), 665-698.
54. R. T. Rockafellar, Lipschitzian properties of multifunctions, Nonlinear Anal. 9 (1985), 867-885.
55. R. T. Rockafellar, Maximal monotone relations and the second derivatives of nonsmooth functions, Ann. Inst. H. Poincare Anal. Non Lineaire 2 (1985), 167-184.
56. R. T. Rockafellar, Proto-differentiability of set-valued mappings and its applications in optimization, Ann. Inst. H. Poincare Anal. Non Lineaire 6 (1989), 449-482.
57. R. T. Rockafellar, Dualization of subgradient conditions for optimality, Nonlinear Anal. 20 (1993), 627-646.
58. A. Shapiro and J. F. Bonnans, Sensitivity analysis of parametrized programs under cone constraints, SIAM J. Control Optim. 30 (1992), 1409-1422.
59. L. Thibault, Subdifferentials of compactly Lipschitzian functions, Ann. Mat. Pura Appl. (4) 125 (1980), 157-192.
60. L. Thibault, On subdifferentials of optimal value functions, SIAM J. Control Optim. 29 (1991), 1019-1036.
61. J. S. Treiman, Clarke's gradients and epsilon-subgradients in Banach spaces, Trans. Amer. Math. Soc. 294 (1986), 65-78.
62. C. Ursescu, Multifunctions with closed convex graph, Czechoslovak Math. J. 25 (1975), 438-441.
63. J. Warga, Fat homeomorphisms and unbounded derivate containers, J. Math. Anal. Appl. 81 (1981), 545-560.