Worst case complexity of direct search under convexity

• Autor(a) principal: Dodangeh, Mahdi
• Data de Publicação: 2016
• Outros Autores: Vicente, Luís Nunes
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
• Texto Completo: http://hdl.handle.net/10316/45246; https://doi.org/10.1007/s10107-014-0847-0

In this paper we prove that the broad class of direct-search methods of directional type, based on imposing sufficient decrease to accept new iterates, exhibits the same worst case complexity bound and global rate of the gradient method for the unconstrained minimization of a convex and smooth function. More precisely, it will be shown that the number of iterations needed to reduce the norm of the gradient of the objective function below a certain threshold is at most proportional to the inverse of the threshold. It will be also shown that the absolute error in the function values decay at a sublinear rate proportional to the inverse of the iteration counter. In addition, we prove that the sequence of absolute errors of function values and iterates converges r-linearly in the strongly convex case.