Characterizations of error bounds for lower semicontinuous functions on metric spaces

Azé, Dominique; Corvellec, Jean-Noël

ESAIM: Control, Optimisation and Calculus of Variations, Tome 10 (2004) no. 3, pp. 409-425.

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.

Classification : 49J52, 90C26, 90C25, 49J53
Mots clés : error bounds, strong slope, variational principle, metric regularity

[1] A. Auslender and J.-P. Crouzeix, Well behaved asymptotical convex functions. Ann. Inst. H. Poincaré, Anal. Non Linéaire 6 (1989) 101-121. | Numdam | MR | Zbl

[2] A. Auslender, R. Cominetti and J.-P. Crouzeix, Convex functions with unbounded level sets. SIAM J. Optim. 3 (1993) 669-687. | MR | Zbl

[3] A. Auslender and M. Teboulle, Asymptotic Cones and Functions in Optimization and Variational Inequalities. Springer Monogr. Math. (2003). | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[7] J.V. Burke and M.C. Ferris, Weak sharp minima in mathematical programming. SIAM J. Control Optim. 31 (1993) 1340-1359. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[10] I. Ekeland, Nonconvex minimization problems. Bull. Amer. Math. Soc. 1 (1979) 443-474. | MR | Zbl

[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. | MR | Zbl

[12] A.J. Hoffman, On approximate solutions of systems of linear inequalities. J. Res. Nat. Bur. Stand. 49 (1952) 263-265. | MR

[13] A. Ioffe, Regular points of Lipschitz functions. Trans. Amer. Math. Soc. 251 (1979) 61-69. | MR | Zbl

[14] A. Ioffe, On the local surjection property. Nonlinear Anal. 11 (1987) 565-592. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[17] M. Lassonde, First order rules for nonsmooth constrained optimization. Nonlinear Anal. 44 (2001) 1031-1056. | MR | Zbl

[18] B. Lemaire, Well-posedness, conditioning and regularization of minimization, inclusion and fixed-point problems. Pliska Stud. Math. Bulgar. 12 (1998) 71-84. | MR | Zbl

[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). | MR | Zbl

[20] O.L. Mangasarian, Error bounds for nondifferentiable convex inequalities under a strong Slater constraint qualification. Math. Program. 83 (1998) 187-194. | MR | Zbl

[21] B.S. Mordukhovich, Metric approximations and necessary optimality conditions for general classes of nonsmooth extremal problems. Soviet Math. Dokl. 22 (1980) 526-530. | Zbl

[22] B.S. Mordukhovich and Y. Shao, Differential characterizations of covering, metric regularity and Lipschitzian properties of multifunctions. Nonlinear Anal. 25 (1995) 1401-1428. | MR | Zbl

[23] K.F. Ng and X.Y. Zheng, Error bounds for lower semicontinuous functions in normed spaces. SIAM J. Optim. 12 (2001) 1-17. | MR | Zbl

[24] S. Simons, Subdifferentials of convex functions. Contemp. Math. 204 (1997) 217-246. | MR | Zbl

[25] M. Studniarski and D.E. Ward, Weak sharp minima: characterizations and sufficient conditions. SIAM J. Control Optim. 38 (1999) 219-236. | MR | Zbl

[26] Z. Wu and J. Ye, On error bounds for lower semicontinuous functions. Math. Program. 92 (2002) 301-314. | MR | Zbl

[27] Z. Wu and J. Ye, First-order and second-order conditions for error bounds. Preprint (2002). | MR | Zbl

[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] C. Zălinescu, Convex Analysis in General Vector Spaces. World Scientific Publ. Co., River Edge, NJ (2002). | MR | Zbl