Discrete particle swarm optimization in the numerical solution of a system of linear diophantine equations

Abstract

This article proposes the use of a discrete version of the well known Particle Swarm Optimization, DPSO, a metaheuristic optimization algorithm for numerically solving a system of linear Diophantine equations. Likewise, the transformation of this type of problem (i.e. solving a system of equations) into an optimization one is also shown. The current algorithm is able to find all the integer roots in a given search domain, at least for the examples shown. Simple problems are used to show its efficacy. Moreover, aspects related to the processing time, as well as to the effect of increasing the population and the search space, are discussed. It was found that the strategy shown herein represents a good approach when dealing with systems that have more unknowns than equations, or when it becomes of considerable size, since a big search domain is required.