Characterizations of error bounds for lower semicontinuous functions on metric spaces

Dominique Azé; Jean-Noël Corvellec

ESAIM: Control, Optimisation and Calculus of Variations (2004)

  • Volume: 10, Issue: 3, page 409-425
  • ISSN: 1292-8119

Abstract

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Refining the variational method introduced in Azé et al. [Nonlinear Anal. 49 (2002) 643-670], we give characterizations of the existence of so-called global and local error bounds, for lower semicontinuous functions defined on complete metric spaces. We thus provide a systematic and synthetic approach to the subject, emphasizing the special case of convex functions defined on arbitrary Banach spaces (refining the abstract part of Azé and Corvellec [SIAM J. Optim. 12 (2002) 913-927], and the characterization of the local metric regularity of closed-graph multifunctions between complete metric spaces.

How to cite

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Azé, Dominique, and Corvellec, Jean-Noël. "Characterizations of error bounds for lower semicontinuous functions on metric spaces." ESAIM: Control, Optimisation and Calculus of Variations 10.3 (2004): 409-425. <http://eudml.org/doc/246088>.

@article{Azé2004,
abstract = {Refining the variational method introduced in Azé et al. [Nonlinear Anal. 49 (2002) 643-670], we give characterizations of the existence of so-called global and local error bounds, for lower semicontinuous functions defined on complete metric spaces. We thus provide a systematic and synthetic approach to the subject, emphasizing the special case of convex functions defined on arbitrary Banach spaces (refining the abstract part of Azé and Corvellec [SIAM J. Optim. 12 (2002) 913-927], and the characterization of the local metric regularity of closed-graph multifunctions between complete metric spaces.},
author = {Azé, Dominique, Corvellec, Jean-Noël},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {error bounds; strong slope; variational principle; metric regularity},
language = {eng},
number = {3},
pages = {409-425},
publisher = {EDP-Sciences},
title = {Characterizations of error bounds for lower semicontinuous functions on metric spaces},
url = {http://eudml.org/doc/246088},
volume = {10},
year = {2004},
}

TY - JOUR
AU - Azé, Dominique
AU - Corvellec, Jean-Noël
TI - Characterizations of error bounds for lower semicontinuous functions on metric spaces
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2004
PB - EDP-Sciences
VL - 10
IS - 3
SP - 409
EP - 425
AB - Refining the variational method introduced in Azé et al. [Nonlinear Anal. 49 (2002) 643-670], we give characterizations of the existence of so-called global and local error bounds, for lower semicontinuous functions defined on complete metric spaces. We thus provide a systematic and synthetic approach to the subject, emphasizing the special case of convex functions defined on arbitrary Banach spaces (refining the abstract part of Azé and Corvellec [SIAM J. Optim. 12 (2002) 913-927], and the characterization of the local metric regularity of closed-graph multifunctions between complete metric spaces.
LA - eng
KW - error bounds; strong slope; variational principle; metric regularity
UR - http://eudml.org/doc/246088
ER -

References

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  1. [1] A. Auslender and J.-P. Crouzeix, Well behaved asymptotical convex functions. Ann. Inst. H. Poincaré, Anal. Non Linéaire 6 (1989) 101-121. Zbl0675.90070MR1019110
  2. [2] A. Auslender, R. Cominetti and J.-P. Crouzeix, Convex functions with unbounded level sets. SIAM J. Optim. 3 (1993) 669-687. Zbl0808.90102MR1244046
  3. [3] A. Auslender and M. Teboulle, Asymptotic Cones and Functions in Optimization and Variational Inequalities. Springer Monogr. Math. (2003). Zbl1017.49001MR1931309
  4. [4] D. Azé and J.-N. Corvellec, On the sensitivity analysis of Hoffman constants for systems of linear inequalities. SIAM J. Optim. 12 (2002) 913-927. Zbl1014.65046MR1922502
  5. [5] D. Azé, J.-N. Corvellec and R.E. Lucchetti, Variational pairs and applications to stability in nonsmooth analysis. Nonlinear Anal. 49 (2002) 643-670. Zbl1035.49014MR1894302
  6. [6] D. Azé and J.-B. Hiriart-Urruty, Optimal Hoffman-type estimates in eigenvalue and semidefinite inequality constraints. J. Global Optim. 24 (2002) 133-147. Zbl1047.90066MR1934024
  7. [7] J.V. Burke and M.C. Ferris, Weak sharp minima in mathematical programming. SIAM J. Control Optim. 31 (1993) 1340-1359. Zbl0791.90040MR1234006
  8. [8] O. Cornejo, A. Jourani and C. Zălinescu, Conditioning and upper-Lipschitz inverse subdifferentials in nonsmooth optimization problems. J. Optim. Theory Appl. 95 (1997) 127-148. Zbl0890.90164MR1477353
  9. [9] E. De Giorgi, A. Marino and M. Tosques, Problemi di evoluzione in spazi metrici e curve di massima pendenza (Evolution problems in metric spaces and curves of maximal slope). Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 68 (1980) 180-187. Zbl0465.47041MR636814
  10. [10] I. Ekeland, Nonconvex minimization problems. Bull. Amer. Math. Soc. 1 (1979) 443-474. Zbl0441.49011MR526967
  11. [11] M. Fabian, Subdifferentiability and trustworthiness in the light of the new variational principle of Borwein and Preiss. Acta Univ. Carolin. 30 (1989) 51-56. Zbl0714.49022MR1046445
  12. [12] A.J. Hoffman, On approximate solutions of systems of linear inequalities. J. Res. Nat. Bur. Stand. 49 (1952) 263-265. MR51275
  13. [13] A. Ioffe, Regular points of Lipschitz functions. Trans. Amer. Math. Soc. 251 (1979) 61-69. Zbl0427.58008MR531969
  14. [14] A. Ioffe, On the local surjection property. Nonlinear Anal. 11 (1987) 565-592. Zbl0642.49010MR886649
  15. [15] A. Ioffe, Variational methods in local and global non-smooth analysis, in Nonlinear Analysis, Differential Equations and Control, Montréal, 1998, F.H. Clarke and R.J. Stern Eds., Kluwer, Dordrecht, NATO Sc. Ser., C 528 (1999) 447-502. Zbl0940.49018MR1695012
  16. [16] A. Ioffe, Towards metric theory of metric regularity, in Approximation, Optimization and Mathematical Economics, Guadeloupe, 1999, M. Lassonde Ed., Physica-Verlag, Heidelberg (2001) 165-176. Zbl0986.54039MR1842886
  17. [17] M. Lassonde, First order rules for nonsmooth constrained optimization. Nonlinear Anal. 44 (2001) 1031-1056. Zbl1033.49030MR1830858
  18. [18] B. Lemaire, Well-posedness, conditioning and regularization of minimization, inclusion and fixed-point problems. Pliska Stud. Math. Bulgar. 12 (1998) 71-84. Zbl0947.65072MR1686513
  19. [19] A.S. Lewis and J.S. Pang, Error bounds for convex inequality systems, in Generalized Convexity, Generalized Monotonicity: Recent Results, Marseille, 1996, J.-P. Crouzeix et al. Eds., Kluwer, Dordrecht, Nonconvex Optim. Appl. 27 (1998). Zbl0953.90048MR1646951
  20. [20] O.L. Mangasarian, Error bounds for nondifferentiable convex inequalities under a strong Slater constraint qualification. Math. Program. 83 (1998) 187-194. Zbl0920.90119MR1647841
  21. [21] B.S. Mordukhovich, Metric approximations and necessary optimality conditions for general classes of nonsmooth extremal problems. Soviet Math. Dokl. 22 (1980) 526-530. Zbl0491.49011
  22. [22] B.S. Mordukhovich and Y. Shao, Differential characterizations of covering, metric regularity and Lipschitzian properties of multifunctions. Nonlinear Anal. 25 (1995) 1401-1428. Zbl0863.47030MR1355730
  23. [23] K.F. Ng and X.Y. Zheng, Error bounds for lower semicontinuous functions in normed spaces. SIAM J. Optim. 12 (2001) 1-17. Zbl1040.90041MR1870584
  24. [24] S. Simons, Subdifferentials of convex functions. Contemp. Math. 204 (1997) 217-246. Zbl0877.49009MR1443005
  25. [25] M. Studniarski and D.E. Ward, Weak sharp minima: characterizations and sufficient conditions. SIAM J. Control Optim. 38 (1999) 219-236. Zbl0946.49011MR1740599
  26. [26] Z. Wu and J. Ye, On error bounds for lower semicontinuous functions. Math. Program. 92 (2002) 301-314. Zbl1041.90053MR1901263
  27. [27] Z. Wu and J. Ye, First-order and second-order conditions for error bounds. Preprint (2002). Zbl1061.49017MR2085934
  28. [28] C. Zălinescu, Weak sharp minima, well behaving functions and global error bounds for convex inequalities in Banach spaces, in Optimization Methods and their Applications, V. Bulatov and V. Baturin Eds., Irkutsk, Baikal (2001) 272-284. 
  29. [29] C. Zălinescu, Convex Analysis in General Vector Spaces. World Scientific Publ. Co., River Edge, NJ (2002). Zbl1023.46003MR1921556

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