Hostname: page-component-8448b6f56d-m8qmq Total loading time: 0 Render date: 2024-04-23T09:48:54.425Z Has data issue: false hasContentIssue false
  • English
  • Français

ŁOJASIEWICZ-TYPE INEQUALITIES AND GLOBAL ERROR BOUNDS FOR NONSMOOTH DEFINABLE FUNCTIONS IN O-MINIMAL STRUCTURES

Part of: Existence theories

Published online by Cambridge University Press:  10 August 2015

DŨNG PHI HOÀNG
DŨNG PHI HOÀNG*
Affiliation:
Laboratory of Applied Mathematics and Computing, Posts and Telecommunications Institute of Technology (PTIT), Vietnam Department of Scientific Fundamentals, PTIT, Km10 Nguyen Trai Road, Ha Dong District, Hanoi, Vietnam email dunghp@ptit.edu.vn
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we give some Łojasiewicz-type inequalities for continuous definable functions in an o-minimal structure. We also give a necessary and sufficient condition for the existence of a global error bound and the relationship between the Palais–Smale condition and this global error bound. Moreover, we give a Łojasiewicz nonsmooth gradient inequality at infinity near the fibre for continuous definable functions in an o-minimal structure.

Keywords

Łojasiewicz inequalitieserror boundso-minimal structuresnonsmooth analysis

MSC classification

Primary: 32B20: Semi-analytic sets and subanalytic sets
Secondary: 90C31: Sensitivity, stability, parametric optimization 49J52: Nonsmooth analysis
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 93 , Issue 1 , February 2016 , pp. 99 - 112
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

References

Absil, P. A. and Kurdyka, K., ‘On the stable equilibrium points of gradient systems’, Systems Control Lett. 55 (2006), 573–577.CrossRefGoogle Scholar
Absil, P. A., Mahony, R. and Andrews, B., ‘Convergence of the iterates of descent methods or analytic cost functions’, SIAM J. Optim. 16 (2005), 531–547.CrossRefGoogle Scholar
Auslander, A. and Crouzeix, J.-P., ‘Global regularity theorem’, Math. Oper. Res. 13 (1988), 243–253.CrossRefGoogle Scholar
Bolte, J., Daniilidis, A. and Lewis, A. S., ‘The Łojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems’, SIAM J. Optim. 17(4) (2007), 1205–1223.CrossRefGoogle Scholar
Bolte, J., Daniilidis, A., Ley, O. and Mazet, L., ‘Characterizations of Łojasiewicz inequalities and applications’, Trans. Amer. Math. Soc. 362(6) (2010), 3319–3363.CrossRefGoogle Scholar
Clarke, F. H., Optimization and Nonsmooth Analysis (John Wiley & Sons, New York, 1983).Google Scholar
Corvellec, J. N. and Montreanu, V. V., ‘Nonlinear error bounds for lower semi-continuous functions on metric spaces’, Math. Program. 114 (2008), 291–319.CrossRefGoogle Scholar
Coste, M., An Introduction to O-minimal Geometry (Instituti Editoriali e poligrafici internazionali, Università di Pisa, Pisa, 1999).Google Scholar
Dinh, S. T., Hà, H. V. and Nguyen, T. T., ‘Łojasiewicz inequality for polynomial functions on non-compact domains’, Int. J. Math. 23 (2012), 1250033, doi:10.1142/S0129167X12500334.Google Scholar
van den Dries, L., Tame Topology and O-minimal Structures (Cambridge University Press, Cambridge, 1998).CrossRefGoogle Scholar
van den Dries, L. and Miller, C., ‘Geometric categories and o-minimal structures’, Duke Math. J. 84 (1996), 497–540.CrossRefGoogle Scholar
Ekeland, I., ‘Nonconvex minimization problems’, Bull. Amer. Math. Soc. (N.S.) 1 (1974), 443–474.CrossRefGoogle Scholar
Hà, H. V., ‘Global Hölderian error bound for non-degenerate polynomials’, SIAM J. Optim. 23(2) (2013), 917–933.Google Scholar
Hà, H. V. and Nguyen, H. D., ‘Łojasiewicz inequality at infinity for polynomial in two real variables’, Math. Z. 266 (2010), 243–264.Google Scholar
Hoffman, A. J., ‘On approximate solutions of linear inequalities’, J. Res. Natl. Bur. Stand. 49 (1952), 263–265.CrossRefGoogle Scholar
Hörmander, L., ‘On the division of distributions by polynomials’, Ark. Mat. 3(53) (1958), 555–568.CrossRefGoogle Scholar
Ioffe, A. D., ‘Metric regularity and subdifferential calculus’, Uspekhi Mat. Nauk. 55 (2000), 103–162 (in Russian); English transl. Russian Math. Surveys 55 (2000), 501–588.Google Scholar
Ioffe, A. D., ‘An invitation to tame optimization’, SIAM J. Optim. 19(4) (2009), 1894–1917.CrossRefGoogle Scholar
Klatte, D. and Li, A., ‘Asymptotic constraint qualifications and global error bounds for convex inequalities’, Math. Program. 84 (1999), 137–140.CrossRefGoogle Scholar
Kurdyka, K., ‘On gradients of functions definable in o-minimal structures’, Ann. Inst. Fourier (Grenoble) 48 (1998), 769–783.CrossRefGoogle Scholar
Li, G., ‘On the asymptotic well behaved functions and global error bound for convex polynomials’, SIAM J. Optim. 20(4) (2010), 1923–1943.CrossRefGoogle Scholar
Li, G., ‘Global error bounds for piecewise convex polynomials’, Math. Program. 137(1–2) (2013), 37–64.CrossRefGoogle Scholar
Li, G., Mordukhovich, B. S. and Pham, T. S., ‘New error bounds for polynomial systems with applications to Hölderian stability in optimization and spectral theory of tensors’, Math. Program. (2014), doi:10.1007/s10107-014-0806-9.Google Scholar
Li, G. and Ng, K. F., ‘Error bounds of generalized D-gap functions for nonsmooth and nonmonotone variational inequality problems’, SIAM. J. Optim. 20(2) (2009), 667–690.CrossRefGoogle Scholar
Li, G., Tang, C. and Wei, Z. X., ‘Error bound results for generalized D-gap functions of nonsmooth variational inequality problems’, Int. J. Comput. Appl. Math. 233(11) (2010), 2795–2806.CrossRefGoogle Scholar
Loi, T. L., ‘Łojasiewicz inequalities for sets definable in the structure ℝexp’, Ann. Inst. Fourier (Grenoble) 45 (1995), 951–957.CrossRefGoogle Scholar
Łojasiewicz, S., ‘Sur le problème de la division’, Studia Math. 18 (1959), 87–136.CrossRefGoogle Scholar
Łojasiewicz, S., Ensembles Semi-Analytiques (Publications Mathématiques Institut des Hautes Études Scientifiques, Bures-sur-Yvette, France, 1964).Google Scholar
Luo, X. D. and Luo, Z. Q., ‘Extensions of Hoffman’s error bound to polynomial systems’, SIAM J. Optim. 4 (1994), 383–392.CrossRefGoogle Scholar
Luo, Z. Q., ‘New error bounds and their applications to convergence analysis of iterative algorithms’, Math. Program. 88(2, Ser. B) (2000), 341–355.CrossRefGoogle Scholar
Luo, Z. Q. and Sturm, J. F., ‘Error bound for quadratic systems’, in: High Perfomance Optimization (eds. Frenk, H., Roos, K., Terlaky, T. and Zhang, S.) (Kluwer, Dordrecht, The Netherlands, 2000), 383–404.CrossRefGoogle Scholar
Mangasarian, O. L., ‘A condition number for differentiable convex inequalities’, Math. Oper. Res. 10 (1985), 175–179.CrossRefGoogle Scholar
Robinson, S., ‘An application of error bounds for convex programming in a linear space’, SIAM J. Control Optim. 13 (1975), 271–273.CrossRefGoogle Scholar
Rockafellar, R. T. and Wets, R., Variational Analysis, Grundlehren der Mathematischen Wissenschaften, 317 (Springer, New York, 1998).CrossRefGoogle Scholar
Simon, L., ‘Asymptotics for a class of non-linear evolution equations, with applications to geometric problems’, Ann. of Math. (2) 118 (1983), 525–571.CrossRefGoogle Scholar
Yang, W. H., ‘Error bounds for convex polynomials’, SIAM J. Optim. 19 (2009), 1633–1647.CrossRefGoogle Scholar