Relative Lipschitz-like Property of Parametric Systems via Projectional Coderivatives

Abstract

In this thesis, fundamental properties of a newly introduced tool, projectional coderivatives, are illustrated. Some examples of calculation are also presented. When the set we refer to is a smooth manifold, the projectional coderivative can be simplified as a fixed-point expression. Therefore, we extend the generalized Mordukhovich criterion to such a setting. Moreover, chain rules and sum rules are developed for projectional coderivatives. Different levels of constraint qualifications are incorporated to generate upper estimates accordingly and all these upper estimates converge under the setting of smooth manifolds. By applying the sum rule to parametric systems, we obtain the upper estimate of the projectional coderivative of the solution mapping, which is also an implicit mapping, making it possible to analyse the relative Lipschitz-like property via projectional coderivatives. The difference between Foremost, I would like to express my heartfelt gratitude to my supervisor, Prof.Yang Xiaoqi for his continuous support and guidance of my PhD study, as well as his unwavering patience and overwhelming encouragement in working with me for the past five years. Without his constant help, this dissertation would not have been possible. I'm greatly indebted to him. I would like to thank Dr. Meng Kaiwen and Dr. Li Minghua for sharing their insightful ideas and providing detailed and constructive suggestions to improve this dissertation. The enjoyable but also rigid discussions we had have inspired and empowered me on research skills and the logic of presentation. I benefited tremendously from collaborations with them. Many thanks also to Prof. Zhang Kai and Dr. Hu Yaohua for their kind help in exploring research topics in my early PhD life. I'm also grateful to my undergraduate advisor, Dr. Cai Tao, for introducing me to academic research. His optimism, kind patience and constant encouragement have empowered me in pursuit of challenge. I benefited a lot from the scientific and quantitative analytic skills I learned from him. I would like to express my sincerest thanks to my friends and colleagues, Ms.