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Error bound for conic inequality in Hilbert spaces. (English) Zbl 1474.49052

Summary: We consider error bound issue for conic inequalities in Hilbert spaces. In terms of proximal subdifferentials of vector-valued functions, we provide sufficient conditions for the existence of a local error bound for a conic inequality. In the Hilbert space case, our result improves and extends some existing results on local error bounds.

MSC:

49K40 Sensitivity, stability, well-posedness
49J52 Nonsmooth analysis
90C31 Sensitivity, stability, parametric optimization
90C48 Programming in abstract spaces
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[1] Hoffman, A. J., On approximate solutions of systems of linear inequalities, Journal of Research of the National Bureau of Standards, 49, 263-265, (1952)
[2] Azé, D.; Corvellec, J. N., Characterizations of error bounds for lower semicontinuous functions on metric spaces, ESAIM. Control, Optimisation and Calculus of Variations, 10, 3, 409-425, (2004) · Zbl 1085.49019
[3] Ioffe, A. D., Metric regularity and subdifferential calculus, Russian Mathematical Surveys, 55, 3, 501-558, (2000) · Zbl 0979.49017
[4] Lewis, A. S.; Pang, J. S.; Crouzeix, J. P.; Martinez-Legaz, J. E.; Volle, M., Error bounds for convex inequality systems, Generalized Convexity, Generalized Monotonicity: Recent Results. Generalized Convexity, Generalized Monotonicity: Recent Results, Nonconvex Optimization and Its Applications, 27, 75-110, (1998), Dordrecht, The Netherlands: Kluwer Academic Publishers, Dordrecht, The Netherlands
[5] Ng, K. F.; Zheng, X. Y., Error bounds for lower semicontinuous functions in normed spaces, SIAM Journal on Optimization, 12, 1, 1-17, (2001) · Zbl 1040.90041
[6] van Ngai, H.; Théra, M., Error bounds for systems of lower semicontinuous functions in Asplund spaces, Mathematical Programming, 116, 1-2, 397-427, (2009) · Zbl 1215.49028
[7] Pang, J. S., Error bounds in mathematical programming, Mathematical Programming B, 79, 1–3, 299-332, (1997) · Zbl 0887.90165
[8] Wu, Z.; Ye, J. J., On error bounds for lower semicontinuous functions, Mathematical Programming, 92, 2, 301-314, (2002) · Zbl 1041.90053
[9] Zalinescu, C., Weak sharp minima, well-behaving functions and global error bounds for convex inequalities in Banach spaces, Proceedings of the 12th Baikal International Conference on Optimization Methods and Their Applications
[10] Ioffe, A. D., Regular points of Lipschitz functions, Transactions of the American Mathematical Society, 251, 61-69, (1979) · Zbl 0427.58008
[11] Zheng, X. Y.; Ng, K. F., Metric subregularity and calmness for nonconvex generalized equations in Banach spaces, SIAM Journal on Optimization, 20, 5, 2119-2136, (2010) · Zbl 1229.90220
[12] Zheng, X. Y.; Ng, K. F., Error bound for conic inequality · Zbl 1335.90099
[13] Clarke, F. H., Optimization and Nonsmooth Analysis, (1983), New York, NY, USA: John Wiley & Sons, New York, NY, USA · Zbl 0582.49001
[14] Clarke, F. H.; Ledyaev, Yu. S.; Stern, R. J.; Wolenski, P. R., Nonsmooth Analysis and Control Theory, (1998), New York, NY, USA: Springer, New York, NY, USA · Zbl 1047.49500
[15] Mordukhovich, B. S., Variational Analysis and Generalized Differentiation. I/II, (2006), Berlin, Germany: Springer, Berlin, Germany
[16] Schirotzek, W., Nonsmooth Analysis, (2007), Berlin, Germany: Springer, Berlin, Germany · Zbl 1120.49001
[17] Clarke, F. H.; Ledyaev, Yu. S.; Wolenski, P. R., Proximal analysis and minimization principles, Journal of Mathematical Analysis and Applications, 196, 2, 722-735, (1995) · Zbl 0865.49015
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