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 x (grossone) and indicating the number of elements of the set IN, of natural numbers. We show how the numerical system built upon x and the proposed approximation of the l 0 pseudo-norm in terms of x can be successfully used in the solution of elastic net regularization problems and sparse Support Vector Machines classification problems.

Non-Archimedean zero-sum games

“…It has been proposed by Sergeyev, and [31] contains an exhaustive discussion of the topic. GM has found several applications in optimization theory, such as regularization [14], conjugate gradient methods [15], and especially in lexicographic multi-objective LP. In [8], indeed, a Grossone-version of the Simplex algorithm (the G-Simplex) has been implemented and theoretically studied.…”

“…x is larger than expected. This is due to the fact that here we use the same strategy used in the design of the I-Big-M method [7], i.e., we have added to the problem a set of artificial variables, which have been infinitely penalized (by the term x, as done in the pioneering works [9,14] by De Leone et al). This allowed us to have an initial basis to start from (the one made of only artificial variables).…”

Non-standard analysis revisited: An easy axiomatic presentation oriented towards numerical applications

“…Sergeyev, who has introduced the grossone methodology (Sergeyev, 2017). Since its appearance in 2003, a number of applications have emerged in extremely disparate fields: optimization (De Leone et al, 2020b;Lai et al, 2021a;2021b;De Leone, 2018;Cococcioni and Fiaschi, 2020; ordinary differential equations (Sergeyev et al, 2016;Amodio et al, 2017;Iavernaro et al, 2020) and machine learning (De Leone et al, 2020a;Astorino and Fuduli, 2020), among others. In addition, an implementation of the grossone methodology within Simulink R has been recently introduced (Falcone et al, 2020a;.…”

To the special issue dedicated to the 3rd international conference “Numerical Computations: Theory and Algorithms—NUMTA 2019” June 15–21, 2019, Isola Capo Rizzuto, Italy

“…The first group discusses optimization problems and algorithms including local, global, and multi-criteria optimization (see Franchini et al 2020;Ž ilinskas and Litvinas 2020;Nesterov 2020;Posypkin et al 2020;Shao et al 2020;De Leone et al 2020;Capuano et al 2020;Lančinskas et al 2020;Sergeyev et al 2020;Crisci et al 2020;Cavallaro et al 2020;Astorino and Fuduli 2020;Candelieri et al 2020). The second group of papers (see D'Alotto 2020; Pepelyshev and Zhigljavsky 2020;Falcone et al 2020;Amodio et al 2020;Gangle et al 2020;De Leone et al 2020;Astorino and Fuduli 2020) deals with problems and algorithms using the already mentioned recent computational framework allowing one to work with different infinities and infinitesimals numerically. It can be seen from these articles that, in addition to its already known applications, the new computational methodology can be successfully used in such fields as optimization, ordinary differential equations, game theory, classification, logic, and fractals.…”