Algoritmos primais-duais de ponto fixo aplicados ao problema Ridge Regression

Abstract

In this work we propose algorithms for solving a fixed-point general primal-dual formulation applied to the Ridge Regression problem. We study the primal formulation for regularized least squares problems, especially L2-norm, named Ridge Regression and then describe convex duality for that class of problems. Our strategy was to consider together primal and dual formulations and minimize the duality gap between them. We established the primal-dual fixed point algorithm, named SRP and a reformulation for this method, the main contribution of the thesis, which was more efficient and robust, called acc-SRP method or accelerated version of the SRP method. The theoretical study of the algorithms was done through the analysis of the spectral properties of the associated iteration matrices. We proved the linear convergence of algorithms and some numerical examples comparing two variants for each algorithm proposed were presented. We also showed that our best method, acc-SRP, has excellent numerical performance for solving very ill-conditioned problems, when compared to the conjugate gradient method, which makes it computationally more attractive.