Finding multiple roots of a box-constrained system of nonlinear equations with a biased random-key genetic algorithm
- 246 Downloads
Several numerical methods for solving nonlinear systems of equations assume that derivative information is available. Furthermore, these approaches usually do not consider the problem of finding all solutions to a nonlinear system. Rather, most methods output a single solution. In this paper, we address the problem of finding all roots of a system of equations. Our method makes use of a biased random-key genetic algorithm (BRKGA). Given a nonlinear system, we construct a corresponding optimization problem, which we solve multiple times, making use of a BRKGA, with areas of repulsion around roots that have already been found. The heuristic makes no use of derivative information. We illustrate the approach on seven nonlinear equations systems with multiple roots from the literature.
KeywordsNonlinear systems of equations Global optimization Continuous optimization Heuristic Stochastic algorithm Nonlinear programming BRKGA
The research of R.M.A Silva was partially supported by the Brazilian National Council for Scientific and Technological Development (CNPq), the Foundation for Support of Research of the State of Minas Gerais, Brazil (FAPEMIG), Coordination for the Improvement of Higher Education Personnel, Brazil (CAPES), Foundation for the Support of Development of the Federal University of Pernambuco, Brazil (FADE), the Office for Research and Graduate Studies of the Federal University of Pernambuco (PROPESQ), and the Foundation for Support of Science and Technology of the State of Pernambuco (FACEPE).
- 1.Aguiar e Oliveira, H., Jr., Ingber, L., Petraglia, A., Petraglia, M.R., Machado, M.A.S.: Nonlinear equation solving. In: Stochastic Global Optimization and Its Applications with Fuzzy Adaptive Simulated Annealing. Intelligent Systems Reference Library, vol. 35, pp. 169–187. Springer, Berlin Heidelberg (2012) ISBN 978-3-642-27478-7. doi: 10.1007/978-3-642-27479-4_10
- 12.Lester, I.: Adaptive simulated annealing (asa): lessons learned. Control Cybern. 25, 33–54 (1996)Google Scholar
- 17.Merlet, J.P.: The COPRIN examples page. http://www-sop.inria.fr/coprin/logiciels/ALIAS/Benches/benches.html (2006)
- 18.Oliveira Jr, H.: Fuzzy control of stochastic global optimization algorithms and VSFR. Naval Res. Mag. 16, 103–113 (2003)Google Scholar
- 20.R Development Core Team. R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria (2011). URL: http://www.R-project.org/ ISBN 3-900051-07-0
- 21.Reklaitis, G., Ragsdell, K.: Engineering Optimization. Wiley, New York (1983)Google Scholar
- 22.Royden, H.L.: Real Analysis, 3rd edn. Macmillan Publishing, New York (1988)Google Scholar
- 23.Spears, W.M., DeJong K.A.: On the virtues of parameterized uniform crossover. In: Proceedings of the Fourth International Conference on Genetic Algorithms, pp. 230–236 (1991)Google Scholar