First- and Second-Order Conditions for Stability Properties and Error Bounds for Generalized Equations, and Applications

Many real-world problems can be modeled by generalized equations. The solution of the latter can be a challenging task, and typically requires the use of some efficient numerical procedures, whose convergence analysis often relies on stability properties of a solution in question, and on a suitable over-estimate for the distance of a given point to the solution set of the problem, called error bound. With this thesis, we aim at a unified approach to first- and second-order conditions for stability properties and error bounds for generalized equations. To this end, we study existing and develop new concepts for generalized first-order derivatives of set-valued mappings, and use them to formulate criteria for Lipschitzian stability properties and Lipschitzian error bound conditions. These criteria can all be regarded as the property that a suitable generalized least singular value of a generalized derivative is nonzero. By considering generalized least singular values as an extended real-valued function that depends on arguments of an underlying mapping, we will be able to obtain second-order conditions arising from generalized derivatives of this function to guarantee non-Lipschitzian stability properties and non-Lipschitzian error bound conditions. This allows us to extend the territory covered by some seminal monographs dealing with stability properties and error bounds for generalized equations under first-order conditions. Furthermore, we discuss some specializations of our findings, and work out relations to existing results. Finally, we also investigate correlations between stability properties and error bounds with respect to different problem-formulations of one and the same generalized equation.