Title: A new dispersive flies optimisation algorithm for the sum of three cubes

Authors: Boian Lazov; Tsvetan Vetsov

Addresses: Department of Mathematics, University of Architecture, Civil Engineering and Geodesy, 1164 Sofia, Bulgaria ' Department of Theoretical Physics, Faculty of Physics, Sofia University, Sofia 1164, Bulgaria

Abstract: By first solving the equation x³ + y³ + z³ = k with fixed k for z and then considering the distance to the nearest integer function of the result, we turn the sum of three cubes problem into an optimisation one. To our knowledge, this is a novel approach. We then present a modification of the dispersive flies optimisation (DFO) algorithm and apply it to this function in the case with k = 2. We have two goals: to show the viability of using optimisation when searching for integer solutions and to measure how efficient our modified DFO is. We have significantly improved the performance of DFO for very large and discrete search spaces by adding new mechanisms to increase the exploration behaviour of the flies. As a comparison we also use two implementations of simulated annealing. The efficiency of the algorithms is measured by their running times. We model the data by assuming two underlying probability distributions - exponential and log-normal, and calculate relevant numerical characteristics for them. Finally, we evaluate the statistical distinguishability of our methods with respect to some standard parametric and non-parametric statistical tests.

Keywords: dispersive flies optimisation; DFO; particle swarm optimisation; PSO; Diophantine equations; sum of three cubes; simulated annealing.

DOI: 10.1504/IJICA.2023.131357

International Journal of Innovative Computing and Applications, 2023 Vol.14 No.3, pp.167 - 190

Received: 15 Feb 2022
Received in revised form: 04 Jun 2022
Accepted: 29 Jul 2022
Published online: 07 Jun 2023 *