Topics in optimization : solving second-order conic systems with finite precision; Calculus of generalized subdifferentials for nonsmooth functions

Abstract

This work consists of three parts. The first one is devoted to the work done jointly with my supervisor Prof. Felipe Cucker from the City University of Hong Kong and Prof. Javier Pena from the Carnegie-Mellon University on the finite-precision analysis of an interior-point method for solving second-order conic systems. The second part concerns results related to calculus of exhausters, which were obtained jointly or with advice of Prof. Vladimir Fedorovich Demyanov from St.-Petersburg State University. The third chapter is devoted to the calculus of generalized differentials and contains authors's original results on this subject. In Chapter 1, an interior-point method to decide feasibility problems of second- order conic systems is described and analyzed. A main feature of this algorithm is that arithmetic operations are performed with finite precision. Bounds for both the number of arithmetic operations and the finest precision required are exhibited. This work has given rise to publication [21]. Chapter 2 is devoted to the study of exhausters and some related problems. In Section 2.2 we introduce the notions of upper and lower exhausters and give some historic background, then in Sections 2.3 and 2.4 we discuss optimality conditions in terms of exhausters. The optimality conditions in terms of proper exhausters were stated by Demyanov [23], and the optimality conditions in terms of adjoint exhausters were obtained by Roshchina in [105]. In Section 2.5 we address the problem of constructing exhausters, and show how to construct an exhauster of an arbitrary locally Lipschitz function. This was originally published by Roshchina in [100]. Section 2.6 is based on the work [103] and devoted to the problem of minimality of exhausters. In Sections 2.7 and 2.8 the problems of reducing exhausters and converting lower exhausters to upper ones and vice versa are discussed. These two sections are based on the work [104]. In Chapter 3 relationships between exhausters and generalized subdifferentials are discussed. In Section 3.2 we introduce the relationship between exhausters, Frechet and Gateaux subdifferentials and provide some calculus rules based on this relationship. In Section 3.3 we study the relationships between exhausters and the Mordukhovich subdifferential, following the lines of [101]. Section 3.4 is devoted to a study of Mordukhovich subdifferential of a minimum of approximate convex functions. This study was motivated by the results obtained in calculus of exhausters, and is based on [102]. The thesis has two appendices. Appendix A contains some technical details for the finite-precision analysis of Chapter 1, and Appendix B contains some classical facts from Nonsmooth Analysis.