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 the approach of projectional coderivatives and directional normal cones is illustrated through an example. Under the framework of parametric systems, we analyse linear constraint systems, linear complementarity problems and affine variational inequalities. For linear constraint systems with a polyhedral setting, we show that by the generalized Mordukhovich criterion it enjoys the Lipschitz-like property relative to its domain automatically. Besides, we derive the corresponding graphical modulus. For linear complementarity problems with a Q0-matrix, we investigate the sufficient and necessary condition for the Lipschitz-like property relative to its convex domain. For affine variational inequality, a generalized critical face condition is obtained to characterize the Lipschitz-like property relative to a polyhedral convex set under a constraint qualification. By exploiting the structure of linear constraint systems, we investigate the Lipschitz-like property of such systems with an explicit set constraint under full perturbations (including the matrix perturbation) and derive some sufficient and necessary conditions. Some other approaches like outer-subdifferentials and error bounds are also taken into scope to characterize such property. The criterion is later applied on a linear portfolio selection optimization problem.