The Use of Infinities and Infinitesimals for Sparse Classification Problems

In this chapter we discuss the use of grossone and the new approach to infinitesimal and infinite proposed by Sergeyev in determining sparse solutions for special classes of optimization problems. In fact, in various optimization and regression problems, and in solving overdetermined systems of linear equations it is often necessary to determine a sparse solution, that is a solution with as many as possible zero components. Expanding on the results in [16], we show how continuously differentiable concave approximations of the l0 pseudo–norm can be constructed using grossone, and discuss the properties of some new approximations. Finally, we will conclude discussing some applications in elastic net regularization and Sparse Support Vector Machines.