The use of Grossone in Mathematical Programming and Operations Research

TL;DR

This paper explores Sergeyev's grossone methodology for handling infinite and infinitesimal quantities, demonstrating its applications in Operations Research and Mathematical Programming, including improvements in simplex method procedures and penalty function definitions.

Contribution

It introduces the novel application of grossone in optimization algorithms, enhancing numerical precision and methodological robustness in linear and nonlinear programming.

Findings

Grossone improves anti-cycling in simplex method.

Grossone enables exact differentiable penalty functions.

Potential for more precise infinite quantity calculations.

Abstract

The concepts of infinity and infinitesimal in mathematics date back to anciens Greek and have always attracted great attention. Very recently, a new methodology has been proposed by Sergeyev for performing calculations with infinite and infinitesimal quantities, by introducing an infinite unit of measure expressed by the numeral grossone. An important characteristic of this novel approach is its attention to numerical aspects. In this paper we will present some possible applications and use of grossone in Operations Research and Mathematical Programming. In particular, we will show how the use of grossone can be beneficial in anti--cycling procedure for the well-known simplex method for solving Linear Programming Problems and in defining exact differentiable Penalty Functions in Nonlinear Programming.