On Several Types of Basic Constraint Qualifications via Coderivatives for Generalized Equations
- 103 Downloads
In this paper, we study several types of basic constraint qualifications in terms of Clarke/Fréchet coderivatives for generalized equations. Several necessary and/or sufficient conditions are given to ensure these constraint qualifications. It is proved that basic constraint qualification and strong basic constraint qualification for convex generalized equations can be obtained by these constraint qualifications, and the existing results on constraint qualifications for the inequality system can be deduced from the given conditions in this paper. The main work of this paper is an extension of the study on constraint qualifications from inequality systems to generalized equations.
KeywordsBasic constraint qualification Normal cone Coderivative Generalized equation End set
Mathematics Subject Classification90C31 90C25 49J52 46B20
The authors wish to thank the referees for many valuable comments which help us to improve the original presentation of this paper. The research of the first author was supported by the National Natural Science Foundations of P. R. China (Grants 11771384 and 11461080) and the Fok Ying-Tung Education Foundation (Grant 151101). The second author was partially supported by the Grant MOST 105-2115-M-039-002-MY3.
- 6.Zălinescu, C.: Weak sharp minima, well-behaving functions and global error bounds for convex inequalities in Banach spaces, in Proceedings of 12th Baikal International Conference on Optimization Methods and Their Applied Irkutsk, Russia, pp. 272–284 (2001)Google Scholar
- 8.Lyusternik, L.A.: On conditional extrema of functionals. Math. Sbornik. 41, 390–401 (1934). (in Russian)Google Scholar
- 23.Deutsch, F.: The role of the strong conical hull intersection property in convex optimization and approximation. In: Chui, C.K., Schumaker, L.L. (eds.) Approximation Theory IX, Vol. I: Theoretical Aspects, pp. 105–112. Vanderbilt University Press, Nashville (1998)Google Scholar
- 24.Lewis, A.S., Pang, J.S.: Error bounds for convex inequality systems. In: Crouzeix, J.P. (ed.) Generalized Convexity, Proceedings of the Fifth Sysposium on Generalized Convexity, pp. 75–110. Luminy, Marseille (1997)Google Scholar
- 28.Wei, Z., Yao, J.-C.: On constraint qualifications of a nonconvex inequality. Optim. Lett. (2017). https://doi.org/10.1007/s11590-017-1172-3