Transversality, Regularity and Error Bounds in Variational Analysis and Optimisation

Abstract

Transversality properties of collections of sets, regularity properties of set-valued mappings and error bounds of extended-real-valued functions lie at the core of variational analysis and optimisation because of their importance for stability analysis, constraint qualifications, qualification conditions in coderivative and subdifferential calculus, and convergence analysis of computational algorithms. The thesis is devoted to the investigation of several research questions related to the aforementioned properties and their applications in several optimisation problems. The main tools of the analysis are standard techniques of modern variational analysis (see [2,10,11,13]).Quantitative analysis of transversality properties of collections of sets is investigated in the convex and nonconvex, linear and nonlinear settings by employing conventional tools of generalisation differentiation. The quantitative relations between transversality and regularity properties of set-valued mappings as well as nonlinear extensions of the new transversality properties of a set-valued mapping to a set in the range space are discussed. These results are presented in recent publications [1,[3][4][5][6]14].We study theoretical and applied aspects of a new property called 'linear semitransversality of collections of set-valued mappings' in metric spaces [9]. The property can be seen as a generalisation of the conventional semitransversality of collections of sets and the negation of the corresponding stationarity which is a weaker property than the extremality of collections of set-valued mappings [12]. Necessary and sufficient conditions in terms of primal and dual objects and quantitative connections with the semiregularity of set-valued mappings are formulated. The results are applied to optimality conditions for a multiobjective optimisation problem with geometric constraints.