Stable Well-posedness,Generalized Tilt-stability Of A Local Minimizer And Metric Regulanrity With Respect To An Admissible Function

In this thesis, we mainly consider the following four issues:first, extending both the Ekeland variational principle and the Borwein-Preiss smooth variational principle to a more general topological space; second, introducing the notions of the stable well-posedness and the tilt-stable minimizer with respect to the corresponding admissible function for a lower semicontinuous function and establishing the relationships between them; third, establishing the sufficient conditions for a(?)-regular function to be well-posed and stably well-posed after the notion of (?)-regular functions being introduced; fourth, establishing relevant theories of two generalized metric subregularities (metric subregularities with respect to (?) and to (?)r for a multifunction after the needed notions being introduced.We introduce the research background in Chapter1, and then recall some funda-mental tools and some basic concepts and properties of variational analysis and nons-mooth analysis in Chapter2.Chapter3is devoted to establishing the variational principle on a topological space. To the best of our knowledge, all the existing variational principles were established in the framework of a complete metric space or a Banach space. In this chapter, we main-ly consider variational principles in the framework of more general topological spaces without any metrics. We introduce the notion of the Cantor compatibility for a topo-logical space X and gauge-type functions on X. In terms of the Cantor-compatibility, we provide strong variational principles on a topological space by gauge-type function-s, which further extend both the Ekeland variational principle and the Borwein-Preiss smooth variational principle to the topological space case.Chapter4mainly deals with the stable well-posedness. In this chapter, adopting an admissible function (?):R+â†'R+, we introduce and study the stable well-posedness and the (?)-tilt-stable local minimizer for a proper lower semicontinuous extended real-valued function f on a Banach space and the metric regularity of the subdifferential mapping (?)cf with respect to an admissible function. We prove that f has the stable well-posedness at χ with respect to (?) if (?)cf is strongly metrically regular at (χ,0) with respect to (?)’and that the converse also holds under the convexity assumption of f. We also prove that f has the stable well-posedness at χ with respect to (?) if and only if x is a (?’)-1-tilt-stable local minimizer of f. Moreover, our results improve and extend the main results on the tilt-stability and the Holder tilt-stability in the literature in the special cases of (?)(t)=t and (?)(t)=tq.Chapter5first considers the relationships between a (?)-regular function, a strongly ()?-paraconvex function and the hypomonotonicity of the Clarke sudifferential mapping of the corresponding function, and then is dedicated to providing sufficient conditions for a (?)-subregular function to be well-posed and stably well-posed at χ in terms of met-ric subregularity/regularity of the Clarke subdifferential of the concerned (?)-subregular function. Better results are presented in this chapter compared with Chapter4; more-over, the results established in this chapter also improve some existing results in the literature.Chapter6mainly concerns the issue of the metric subregulaities with respect to an admissible function for a multifunction F which is more general than the subdifferential mapping considered in Chapter4and5. Sufficient conditions for F to be metrically subregular with respect to (?) and to (?)r are provided in terms of coderivatives of the concerned multifunction, respectively. Furthermore, sharper results are obtained in the Asplund space case. In the special cases that (?)(t)=t and (?)t(t)=tq, our results reduce to the main results established by Zheng and Ng [SIAM, J. Optim.,20(2010), pp.2119-2136] and Li and Mordukhovich [SIAM, J. Optim.,22(2012), pp.1655-1684], respectively. In particular, our results further improve and extend Ioffe’s classic result on the metric subregularity for an inequality.