Métodos de regularização cúbica com aproximações preguiçosas da hessiana

• Ano de defesa: 2025
• Autor(a) principal: Gehlen Filho, Vilmar
• Orientador(a): Gonçalves, Max Leandro Nobre
• Banca de defesa: Gonçalves, Max Leandro Nobre, Melo, Jefferson Divino Gonçalves de, Grapiglia, Geovani Nunes, Santos, Luiz Rafael dos
• Tipo de documento: Dissertação
• Tipo de acesso: Acesso aberto
• dARK ID: ark:/38995/001300000g5d4
• Idioma: eng
• Instituição de defesa: Universidade Federal de Goiás
• Programa de Pós-Graduação: Programa de Pós-graduação em Matemática (IME)
• Departamento: Instituto de Matemática e Estatística - IME (RMG)
• País: Brasil
• Palavras-chave em Português: Método de regularização cúbica; Análise de complexidade; Hessiana preguiçosa; Otimização não-convexa
• Palavras-chave em Inglês: Cubic regularization method; Complexity analysis; Lazy hessian; Non- convex optimization
• Área do conhecimento CNPq: CIENCIAS EXATAS E DA TERRA::MATEMATICA
• Link de acesso: http://repositorio.bc.ufg.br/tede/handle/tede/14032

In this work, we present variants of the Cubic Regularization Newton's (CRN) method incorporating lazy Hessian approximations for solving general non-convex optimization problems (0-3). We propose two approaches: the first (Algorithm 1) employs the exact gradient while reusing the same Hessian approximation for a block of \( m \) iterations, whereas the second (Algorithm 2) extends this idea by additionally allowing the use of inexact gradients. Implementations of methods, where information about derivatives are computed through finite difference strategies, are presented. One interesting feature of our algorithms is that the regularization parameter and the accuracy of the derivative approximations (when they are updated) are jointly adjusted using a nonmonotone line search criterion. We establish first-order complexity results for both methods. Specifically, for a given precision $\epsilon$, it is shown that the Algorithm~1 and Algorithm~2 require at most {$\mathcal{O}\left( m^{1/2} \epsilon^{-3/2}\right)$} outer iterations to generate an $\epsilon-$approximate critical point for aforementioned problem. When the derivatives are computed by finite difference approaches, we show that Algorithm~1 (resp. Algorithm~2) needs at most {$\mathcal{O}\left((n+m)m^{-1/2}\epsilon^{-3/2}+(n+m)\right)$} (resp. {$\mathcal{O}\left((n^2+mn)m^{-1/2}\epsilon^{-3/2}+(n^2+mn)\right)$}) gradient and function (resp. function) evaluations to generate an $\epsilon$-approximate critical point, where $n$ is the dimension of the domain of the objective function.