Stability of inclusions: Characterizations via suitable Lipschitz functions and algorithms

Optimization

Volumn 55, Issue 5-6, 2006, Pages 627-660

  Klatte, Diethard ^a   Kummer, Bernd ^b  

^a UNIVERSITY OF ZURICH   (Switzerland)

^b HUMBOLDT UNIVERSITY   (Germany)

Author keywords

(Nonsmooth) Newton method; Ekeland's principle; Generalized equation; Kojima function; Lipschitz stability; Modified successive approximation

Indexed keywords

EID: 33750893609     PISSN: 02331934     EISSN: 10294945     Source Type: Journal    
DOI: 10.1080/02331930600819787     Document Type: Article

References (68)

1. Aubin, J.-P. and Ekeland, I., 1984, Applied Nonlinear Analysis (New York: Wiley).
2. Auslender, A., 1984, Stability in mathematical programming with nondifferentiable data. SIAM Journal on Control and Optimization, 22, 29-41.
3. Bank, B., Guddat, J., Klatte, D., Kummer, B. and Tammer, K., 1982, Non-linear Parametric Optimization (Berlin: Akademie-Verlag; also Basel-Boston: Birkhäuser, 1983).
4. Bonnans, J.F. and Shapiro, A., 2000, Perturbation Analysis of Optimization Problems (New York: Springer).
5. Burke, J.V., 1991, Calmness and exact penalization. SIAM Journal on Control and Optimization, 29, 493-497.
6. Clarke, F.H., 1976, On the inverse function theorem. Pacific Journal of Mathematics, 64, 97-102.
7. Clarke, F.H., 1983, Optimization and Nonsmooth Analysis (New York: Wiley).
8. Cominetti, R., 1990, Metric regularity, tangent sets and second-order optimality conditions. Applied Mathematics and Optimization, 21, 265-287.
9. Dempe, S., 2002, Foundations of Bilevel Programming (Dordrecht-Boston- London: Kluwer Academic Publ.).
10. Dolecki, S. and Rolewicz, S., 1979, Exact penalties for local minima. SIAM Journal on Control and Optimizationd, 17, 596-606.
11. Dontchev, A., 1996, Local convergence of the Newton method for generalized equations. Comptes Rendus de l'Académie des Sciences de Paris, 332, 327-331.
12. Dontchev, A. and Rockafellar, R.T., 1996, Characterizations of strong regularity for variational inequalities over polyhedral convex sets. SIAM Journal on Optimization, 6, 1087-1105.
13. Dontchev, A. and Rockafellar, R.T., 1998, Characterizations of Lipschitz stability in nonlinear programming. In: A.V. Fiacco (Ed.) Mathematical Programming with Data Perturbations (New York: Marcel Dekker), pp. 65-82.
14. Dontchev, A. and Rockafellar, R.T., 2004, Regularity and conditioning of solution mappings in variational analysis. Set-valued Analysis, 12, 79-109.
15. Ekeland, I., 1974, On the variational principle. Journal of Mathematical Analysis and Applications, 47, 324-353.
16. Facchinei, F. and Kanzow, C., 1997, A nonsmooth inexact Newton method for the solution of large-scale nonlinear complementarity problems. Mathematical Programming B, 76(3), 493-512.
17. Fischer, A., 1997, Solutions of monotone complementarity problems with locally Lipschitzian functions. Mathematical Programming B, 76(3), 513-532.
18. Facchinei, F. and Pang, J.-S., 2003, Finite-dimensional Variational Inequalities and Complementary Problems, Vol. I and Vol. II (New York: Springer).
19. n functions. Optimization, 49, 425-446.
20. Graves, L.M., 1950, Some mapping theorems. Duke Mathematical Journal, 17, 11-114.
21. Griewank, A., 1987, The local convergence of Broyden-like methods on Lipschitzian problems in Hilbert spaces. SIAM Journal on Numerical Analysis, 24, 684-705.
22. 1,1-data. Applied Mathematics and Optimization, 11, 43-56.
23. Henrion, R. and Outrata, J., 2001, A subdifferential condition for calmness of multifunctions. Journal of Mathematical Analysis and Applications, 258, 110-130.
24. Ioffe, A.D., 1979, Regular points of Lipschitz functions. Transactions of the American Mathematical Society, 251, 61-69.
25. Ioffe, A.D., 2000, Metric regularity and subdifferential calculus. Russian Mathematical Surveys, 55, 501-558.
26. n, I: Morse Theory, Chebychev Approximation (Frankfurt a.M.-Bern-New York: Peter Lang Verlag).
27. n, II: Transversality, Flows, Parametric Aspects (Frankfurt a.M.-Bern-New York: Peter Lang Verlag).
28. King, A. and Rockafellar, R.T., 1992, Sensitivity analysis for nonsmooth generalized equations. Mathematical Programming, 55, 341-364.
29. Klatte, D. and Kummer, B., 1999, Generalized Kojima functions and Lipschitz stability of critical points. Computational Optimization and Applications, 13, 61-85.
30. Klatte, D. and Kummer, B., 2002, Constrained minima and Lipschitzian penalties in metric spaces. SIAM Journal on Optimization, 13(2), 619-633.
31. Klatte, D. and Kummer, B., 2002, Nonsmooth Equations in Optimization - Regularity, Calculus, Methods and Applications, Nonconvex Optimization and its Applications (Dordrecht-Boston-London: Kluwer, Academic Publ.).
32. Klatte, D. and Kummer, B., 2005, Strong Lipschitz stability of stationary solutions for nonlinear programs and variational inequalities. SIAM Journal on Optimization, 16, 96-119.
33. Kojima, M., 1980, Strongly stable stationary solutions in nonlinear programs. In: S.M. Robinson (Ed.) Analysis and Computation of Fixed Points (New York: Academic Press), pp. 93-138.
34. 1 equations. Journal of the Operational Research Society of Japan, 29, 352-372.
35. Kruger, A.Y. and Mordukhovich, B.S., 1980, Extremal points and Euler equations in nonsmooth optimization (in Russian). Doklady Akademii Nauk BSSR, 24, 684-687.
36. Kruger, A.Y., 2002, Strict (ε,δ)-subdifferentials and extremality conditions. Optimization, 51, 539-554.
37. Kummer, B., 1984, Generalized equations: solvability and regularity. Mathematical Programming Study, 21, 199-212.
38. 1,1 optimization. Journal of Optimization Theory and Applications, 70, 559-580.
39. 1,1-optimization. Journal of Mathematical Analysis and Applications, 158, 35-46.
40. Kummer, B., 1992, Newton's method based on generalized derivatives for nonsmooth functions: convergence analysis. In: W. Oettli and D. Pallaschke (Eds) Advances in Optimization (Berlin: Springer), pp. 171-194.
41. Kummer, B., 1998, Lipschitzian and pseudo-Lipschitzian inverse functions and applications to nonlinear programming. In: A.V. Fiacco (Ed.) Mathematical Programming with Data Perturbations (New York: Marcel Dekker), pp. 201-222.
42. Kummer, B., 1999, Metric regularity: characterizations, nonsmooth variations and successive approximation. Optimization, 46, pp. 247-281.
43. Kummer, B., 2000, Inverse functions of pseudo regular mappings and regularity conditions. Mathematical Programming, Series B, 88, 313-339.
44. Levy, A.B., 2001, Solution sensitivity from general principles. SIAM Journal on Optimization, 40, 1-38.
45. Levy, A.B., 2001, Lipschitzian multifunctions and a Lipschitzian inverse mapping theorem. Mathematics of Operations Research, 26, 105-118.
46. Levy, A.B., Poliquin, R.A. and Rockafellar, R.T., 2000, Stability of locally optimal solutions. SIAM Journal on Optimization, 10, 580-604.
47. Levy, A.B. and Rockafellar, R.T., 1996, Variational conditions and the proto-differentiation of partial subgradient mappings. Nonlinear Analysis: Theory, Methods and Applications, 26, 1951-1964.
48. Lyusternik, L., 1934, Conditional extrema of functions. Matematicheskii Sbornik, 41, 390-401.
49. Mifflin, R., 1977, Semismooth and semiconvex functions in constrained optimization. SIAM Journal on Control and Optimization, 15, 957-972.
50. Mordukhovich, B.S., 1988, Approximation Methods in Problems of Optimization and Control, in Russian (Moscow: Nauka).
51. Mordukhovich, B.S., 1993, Complete characterization of openness, metric regularity and Lipschitzian properties of multifunctions. Transactions of the American Mathematical Society, 340, 1-35.
52. Mordukhovich, B.S., 1994, Stability theory for parametric generalized equations and variational inequalities via nonsmooth analysis. Transactions of the American Mathematical Society, 343, 609-657.
53. Mordukhovich, B.S. and Shao, Y., 1998, Mixed coderivatives of set-valued mappings in variational analysis. Journal of Applied Analysis, 4, 269-294.
54. Outrata, J., Kočvara, M. and Zowe, J., 1998, Nonsmooth Approach to Optimization Problems with Equilibrium Constraints (Dordrecht-Boston-London: Kluwer Academic Publ.).
55. Pallaschke, D. and Rolewicz, S., 1997, Foundations of Mathematical Optimization (Dordrecht-Boston-London: Kluwer Academic Publ.).
56. Poliquin, R.A. and Rockafellar, R.T., 1994, Proto-derivative formulas for basic subgradient mappings in mathematical programming. Set-valued Analysis, 2, 275-290.
57. Poliquin, R.A. and Rockafellar, R.T., 1995, Prox-regular functions in variational analysis. Transactions of the American Mathematical Society, 348, 1805-1838.
58. Ralph, D. and Scholtes, S., 1997, Sensivitity analysis of composite piecewise smooth equations. Mathematical Programming, Series B, 76, 593-612.
59. Robinson, S.M., 1979, Generalized equations and their solutions, Part I: basic theory. Mathematical Programming Study, 10, 128-141.
60. Robinson, S.M., 1980, Strongly regular generalized equations. Mathematics of Operations Research, 5, 43-62.
61. Robinson, S.M., 1982, Generalized equations and their solutions. Part II: applications to nonlinear programming. Mathematical Programming Study, 19, 200-221.
62. Robinson, S.M., 2003, Variational conditions with smooth constraints: structure and analysis. Mathematical Programming, 97, 245-265.
63. Rockafellar, R.T., 1988, First and second order epi-differentiability in nonlinear programming. Transactions of the American Mathematical Society, 207, 75-108.
64. Rockafellar, R.T. and Wets, R.J.-B., 1998, Variational Analysis (Berlin: Springer).
65. Sun, D. and Qi, L., 1999, On NCP functions. Computational Optimization and Applications, 13, 201-220.
66. Thibault, L., 1980, Subdifferentials of compactly Lipschitz vector-valued functions. Annali di Matematica Pura ed Applicata, 4, 157-192.
67. Thibault, L., 1982, On generalized differentials and subdifferentials of Lipschitz vector-valued functions. Nonlinear Analysis: Theory, Methods and Applications, 6, 1037-1053.
68. Zeidler, E., 1976, Vorlesungen über nichtlineare Funktionalanalysis I - Fixpunktsätze- (Leipzig: Teubner Verlagsgesellschaft).