Regularity and conditioning of solution mappings in variational analysis

Set-Valued Analysis

Volumn 12, Issue 1-2, 2004, Pages 79-109

  Dontchev, A L ^a   Rockafellar, R T ^b  

^a Mathematical Reviews   (United States)

^b UNIVERSITY OF WASHINGTON   (United States)

Author keywords

Calmness; Conditioning; Distance to ill posedness; Graphical derivatives; Inverse mapping theorems; Lipschitz properties; Metric regularity; Radius of regularity; Solution mappings; Strong regularity; Strong subregularity; Subregularity

Indexed keywords

EID: 3943050689     PISSN: 09276947     EISSN: None     Source Type: Journal    
DOI: 10.1023/b:svan.0000023394.19482.30     Document Type: Article

References (39)

1. Bonnans, J. F. and Shapiro, A.: Perturbation Analysis of Optimization Problems, Springer Ser. Oper. Res., Springer, New York, 2000.
2. Canovas, M. J., López, M. A., Parra, J. and Toledo, F. J.: Distance to ill-posedness and consistency value of linear semi-infinite inequality systems, Preprint, 2003.
3. Clarke, F. W.: A new approach to Lagrange multipliers, Math. Oper. Res. 1 (1976), 165-174.
4. Cheung, D. and Cucker, F.: Probabilistic analysis of condition numbers for linear programming, J. Optim. Theory Appl. 114 (2002), 55-67.
5. Dedieu, J.-P.: Approximate solutions of numerical problems, condition number analysis and condition number theorem, In: The Mathematics of Numerical Analysis (Park City, UT, 1995), Lectures in Appl. Math. 32, Amer. Math. Soc., Providence, RI, 1996, pp. 263-283.
6. Demmel, J.: The condition number and the distance to the nearest ill-posed problem, Numer. Math. 51 (1987), 251-289.
7. Dmitruk, A. V., Milyutin, A. A. and Osmolovskiǐ, N. P.: The Lyusternik theorem and the theory of extremum, Uspekhi Mat. Nauk 35 (1980) 11-46.
8. Dontchev, A. L.: Characterizations of Lipschitz stability in optimization, In: Recent Developments in Well-Posed Variational Problems, Math. Appl. 331, Kluwer, Dordrecht, 1995, pp. 95-115.
9. Dontchev, A. L. and Rockafellar, R. T.: Characterizations of strong regularity for variational inequalities over polyhedral convex sets, SIAM J. Optim. 6 (1996), 1087-1105.
10. Dontchev, A. L. and Rockafellar, R. T.: Ample parameterization of variational inclusions, SIAM J. Optim. 12 (2001), 170-187.
11. Dontchev, A. L., Lewis, A. S. and Rockafellar, R. T.: The radius of metric regularity, Trans. Amer. Math. Soc. 355 (2003), 493-517.
12. Eckart, C. and Young, G.: The approximation of one matrix by another of lower rank, Psychometrika 1 (1936), 211-218.
13. Epelman, M. and Freund, R. M.: A new condition measure, preconditioners, and relations between different measures of conditioning for conic linear systems, SIAM J. Optim. 12 (2002), 627-655.
14. Facchinei, F. and Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, Vols. I and II, Springer Ser. Oper. Res., Springer, New York, 2003.
15. Freund, R. M. and Vera, J. R.: Some characterizations and properties of the 'distance to ill-posedness' and the condition measure of a conic linear system, Math. Progr. Ser. A 86 (1999), 225-260.
16. Graves, L. M.: Some mapping theorems, Duke Math. J. 17 (1950), 111-114.
17. Henrion, R. and Outrata, J.: A subdifferential condition for calmness of multifunctions, J. Math. Anal. Appl. 258 (2001), 110-130.
18. Ioffe, A. D.: Metric regularity and subdifferential calculus, Uspekhi Mat. Nauk 55 (2000), 103-162; translation in Russian Math. Surveys 55 (2000), 501-558.
19. Ioffe, A. D.: Metric regularity and subdifferential calculus, Uspekhi Mat. Nauk 55 (2000), 103-162; translation in Russian Math. Surveys 55 (2000), 501-558.
20. Ioffe, A. D.: On stability estimates for the regularity of maps, Preprint, 2002.
21. King, A. J. and Rockafellar, R. T.: Sensitivity analysis for nonsmooth generalized equations, Math. Progr. Ser. A 55(2) (1992), 193-212.
22. Klatte, D. and Kummer, B.: Nonsmooth Equations in Optimization. Regularity, Calculus, Methods and Applications, Nonconvex Optimization and its Applications 60, Kluwer Academic Publishers, Dordrecht, 2002.
23. Levy, A. B.: Implicit multifunction theorems for the sensitivity of variational conditions, Math. Progr. Ser. A 74(3) (1996), 333-350.
24. Luo, Z.-Q. and Pang, J.-S.: Error Bounds in Mathematical Programming (1998 Kowloon conference volume), Math. Progr. Ser. B 88(2) (2000).
25. Lyusternik, L. A.: On the conditional extrema of functionals, Math. Sbornik 41 (1934), 390-401.
26. Mangasarian, O. L.: A condition number for differentiable convex inequalities, Math. Oper. Res. 10 (1985), 175-179.
27. Mordukhovich, B. S.: Coderivative analysis of variational systems, J. Global Optim., to appear.
28. Nayakkankuppam, M. H. and Overton, M. L.: Conditioning of semidefinite programs, Math. Progr. Ser. A 85 (1999), 525-540.
29. Peña, J.: Conditioning of convex programs from a primal-dual perspective, Math. Oper. Res. 26 (2001), 206-220.
30. Peña, J.: Block-structured distance to infeasibility, Preprint.
31. Renegar, J.: Linear programming, complexity theory and elementary functional analysis, Math. Progr. Ser. A 70 (1995), 279-351.
32. Robinson, S. M.: Strongly regular generalized equations, Math. Oper. Res. 5 (1980), 43-62.
33. Robinson, S. M.: Some continuity properties of polyhedral multifunctions, Math. Progr. Study 14 (1981), 206-214.
34. Robinson, S. M.: An implicit-function theorem for a class of nonsmooth functions, Math. Oper. Res. 16 (1991), 292-308.
35. Rockafellar, R. T.: Proto-differentiability of set-valued mappings and its applications in optimization, In: Analyse non linéaire (Perpignan, 1987); Ann. Inst. H. Poincaré Anal. Non Linéaire 6 (1989), suppl., 449-482.
36. Rockafellar, R. T. and Wets, R. J.-B.: Variational Analysis, Springer, Berlin, 1997.
37. Shapiro, A., Homem-de-Mello, T. and Kim, J.: Conditioning of convex piecewise linear stochastic programs, Math. Progr. Ser. A 94 (2002), 1-19.
38. Wright M. H.: Ill-conditioning and computational error in interior methods for nonlinear programming, SIAM J. Optim. 9 (1999), 84-111.
39. Zolezzi, T.: On the distance theorem in quadratic optimization, J. Convex Anal. 9 (2002), 693-700.