The use of grossone in elastic net regularization and sparse support vector machines

Abstract

New algorithms for the numerical solution of optimization problems involving the $$l_{0}$$ pseudo-norm are proposed. They are designed to use a recently proposed computational methodology that is able to deal numerically with finite, infinite and infinitesimal numbers. This new methodology introduces an infinite unit of measure expressed by the numeral $$\textcircled {1}$$ (grossone) and indicating the number of elements of the set $${\text {I}}\!{\text {N}}$$ , of natural numbers. We show how the numerical system built upon $$\textcircled {1}$$ and the proposed approximation of the $$l_0$$ pseudo-norm in terms of $$\textcircled {1}$$ can be successfully used in the solution of elastic net regularization problems and sparse support vector machines classification problems.