Topics in nonlinear conic optimization and applications

• Ano de defesa: 2022
• Autor(a) principal: Mito, Leonardo Makoto
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: eng
• Instituição de defesa: Biblioteca Digitais de Teses e Dissertações da USP
• Palavras-chave em Português: Algorithms; Algoritmos; Condições de otimalidade de primeira ordem; Condições de otimalidade de segunda ordem; Condições de qualificação; Conic optimization; Constraint qualifications; Convergência global; First-order optimality conditions; Global convergence; Otimização cônica; Second-order optimality conditions
• Link de acesso: https://www.teses.usp.br/teses/disponiveis/45/45132/tde-30032022-212754/

This thesis has three main parts: in part one, we develop new sequential optimality conditions for Nonlinear Conic Programming (NCP) problems, which are used to study convergence of algorithms in a simplified and unified way. In part two, we extend the so-called Constant Rank Constraint Qualification (CRCQ) and the Constant Rank of the Subspace Component (CRSC) conditions to the context of NCP over reducible cones by means of new geometric characterizations of them; we use these conditions to prove strong second-order optimality results that improve the classical one obtained under Robinson\'s Constraint Qualification, and we show how CRSC is related to a nonlinear extension of the celebrated facial reduction preprocessing technique. In part three, we present an alternative approach to extending CRCQ and the Constant Positive Linear Dependence (CPLD) conditions to Nonlinear Semidefinite and Second-Order Cone Programming, which has applications in the global convergence theory of a class of numerical methods to first-order stationary points. Then, we combine some of the ideas presented in part two with the CRCQ extension of part three to derive a Weak Constant Rank property that modifies the second-order optimality condition induced by Robinson\'s CQ to a notion that better suits convergence of algorithms.