It is well known that the training of Least Squares Support Vector Machines (LSSVM) is carried out via the solution of a Karush-Kuhn-Tucker (KKT) linear system. Such an approach provides improved computational efficiency in the training stage when compared to standard Support Vector Machine (SVM) in which training involves quadratic program solution. However, a disadvantage of LSSVM refers to the lack of sparsity in the optimal solution of Lagrange multipliers, limiting the use of such a model in the context of training on large datasets. In order to induce the obtaining of sparse solutions, a new methodology was developed that employs LASSO regularization in the LSSVM primal problem, with a solution obtained via the Alternating Directions Method of Multipliers (ADMM) accelerated via Nesterov. The experimental results demonstrate that the proposed model presented reduced processing time in the training stage than the standard LSSVM without loss of predictive performance.