An improved differential evolution algorithm with a restart technique to solve systems of nonlinear equations

Jeerayut Wetweerapong, Pikul Puphasuk


In this research, an improved differential evolution algorithm with a restart technique (DE-R) is designed for solutions of systems of nonlinear equations which often occurs in solving complex computational problems involving variables of nonlinear models. DE-R adds a new strategy for mutation operation and a restart technique to prevent premature convergence and stagnation during the evolutionary search to the basic DE algorithm. The proposed method is evaluated on various real world and synthetic problems and compared with the recently developed methods in the literature. Experiment results show that DE-R can successfully solve all the test problems with fast convergence speed and give high quality solutions. It also outperforms the compared methods.


Systems of nonlinear equations; global optimization; differential evolution algorithm; restart technique

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Boussa¨ıd, I., Lepagnot, J., & Siarry, P. (2013). A survey on optimization metaheuristics. Information sciences, 237, 82–117.

Siddique, N., & Adeli, H. (2015). Nature inspired computing: An overview and some future directions. Cogn Comput, 7, 706–714.

Nanda, S. J., & Panda, G. (2014). A survey on nature inspired metaheuristic algorithms for partitional clustering. Swarm and Evolutionary Computation, 16, 1—18.

Jos´e-Garc´ıa, A., & G´omez-Flores, W. (2016). Automatic clustering using nature-inspired metaheuristics: A survey. Applied Soft Com- puting, 41, 192—213.

Hamm, L., Brorsen, B. W., & Hagan, M. T. (2007). Comparison of Stochastic Global Optimization Methods to Estimate Neural Network Weights. Neural Process Lett, 26, 145—158.

Piotrowski, A. P. (2014). Differential evolution algorithms applied to neural network training suffer from stagnation. Applied Soft Computing, 21, 382—406.

Raja, M. A. Z., Umar, M., Sabir, Z., Khan, J. A., & Baleanu, D. (2018). A new stochastic computing paradigm for the dynamics of nonlinear singular heat conduction model of the human head. Eur. Phys. J. Plus, 133(364), DOI 10.1140/epjp/i2018-12153-4.

Sabir, Z., Manzar, M. A., Raja, M. A. Z., Sheraz, M., & Wazwaz, A. M. (2018). Neuroheuristics for nonlinear singular Thomas- Fermi systems. Applied Soft Computing, 65, 152-169.

Raja, M. A. Z., Shah, Z., Manzar, M. A., Ahmad, I., Awais, M.,& Baleanu, D. (2018). A new stochastic computing paradigm for nonlinear Painlev´e II systems in applications of random matrix theory. Eur. Phys. J. Plus, 133(254), DOI 10.1140/epjp/i2018-12080-4.

Raja, M. A. Z., Zameer, A., Kiani, A. K., Shehzad, A., & Khan, A. R. (2018). Natureinspired computational intelligence integration with Nelder-Mead method to solve nonlinear benchmark models. Neural Comput & Applic, 29, 1169-1193.

Ahmad, I., Zahid, H., Ahmad, F., Raja, M. A. Z., & Baleanu, D. (2019). Design of computational intelligent procedure for thermal analysis of porous fin model. Chinese Journal of Physics, 59, 641-655.

Broyden, C. G. (1965). A class of methods for solving nonlinear simultaneous equations. Mathematics of computation, 19(92), 577–593.

Mart´ınez, J. M. (1994). Algorithms for solving nonlinear systems of equations. In : E. Spedicato, ed. Algorithms for Continuous Optimization-The state of the art. Kluwer Academic Publishers, London, 81–108.

Kelley, C. T. (1995). Iterative methods for solving linear and nonlinear equations. SIAM, Philadelphia.

Dennis, J. E., & Schnabel, R. B. (1996). Numerical methods for unconstrained optimization and nonlinear equations. SIAM, Philadelphia.

Ortega, J. M., & Rheinboldt, W. C. (2000). Iterative solution of nonlinear equations in several variables. SIAM, Philadelphia.

Kelley, C. T. (2003). Solving nonlinear equations with Newton’s method. SIAM, Philadelphia.

Karr, C. L., Weck, B., & Freeman, L.M. (1998). Solutions to systems of nonlinear equations via a genetic algorithm. Eng. Appl. Artif. Intell., 11(3), 369–375.

Grosan, C., & Abraham, A. (2008). A new approach for solving nonlinear equations systems. IEEE Transactions on Systems Man and Cybernetics-Part A: Systems and Humans, 38(3), 698–714.

Hirsch M. J., Pardalos, P. M., & Resende, M. G. C. (2009). Solving systems of nonlinear equations with continuous GRASP. Nonlinear Analysis: Real World Applications, 10, 2000– 2006.

Jaberipour, M., Khorram, E., & Karimi, B. (2011). Particle swarm algorithm for solving systems of nonlinear equations. Computers and Mathematics with Applications, 62(2), 566–576.

Pourjafari, E., & Mojallali, H. (2012). Solving nonlinear equations systems with a new approach based on invasive weed optimization algorithm and clustering. Swarm and Evolutionary Computation, 4, 33–43.

Oliveira, H. A., & Petraglia, A. (2013). Solving nonlinear systems of functional equations with fuzzy adaptive simulated annealing. Applied Soft Computing, 13, 4349–4357.

Raja, M. A. Z., Kiani, A. K., Shehzad, A., & Zameer, A. (2016). Memetic computing through bio-inspired heuristics integration with sequential quadratic programming for nonlinear systems arising in different physical model. SpringerPlus, 5:2063, DOI 10.1186/s40064-016-3750-8.

Zhang, X., Wan, Q., & Fan, Y. (2019). Applying modified cuckoo search algorithm for solving systems of nonlinear equations. Neural Comput & Applic, 31, 553–576

Storn, R., & Price, K. (1995). Differential evolution—a simple and efficient adaptive scheme for global optimization over continuous spaces. Technical Report TR-95-012, ICSI, Berkeley.

Storn, R., & Price, K. (1997). Differential evolution: A simple and efficient heuristic for global optimization over continuous spaces. J. Glob Optim, 11(4), 341–359.

Storn, R. (2008). Differential evolution research-Trends and open questions. In : U. K. Chakraborty, ed. Advances in Differential Evolution. Springer, Berlin, 1–31.

Neri, F., & Tirronen, V. (2010). Recent advances in differential evolution: a survey and experimental analysis. Artif Intell Rev, 33, 61–106.

Das, S., & Suganthan, P. N. (2011). Differential evolution: a survey of the state-of-the-art. IEEE Trans Evol Comput, 15(1), 4–31.

Lampinen, J., & Zelinka, I. (2000). On stagnation of the differential evolution algorithm. Proceedings of the 6th international Mendel conference on soft computing, 76–83.

G¨amperle, R., M¨uller, S. D., & Koumoutsakos, P. (2002). A parameter study for differential evolution. Proceedings of the conference in neural networks and applications (NNA), fuzzy sets and fuzzy systems (FSFS) and evo- lutionary computation (EC), WSEAS, 293–298.

Fan, H. Y., & Lampinen, J. (2003). A trigonometric mutation operation to differential evolution. J Glob Optim, 27(1), 105–129.

Das, S., Konar, A., & Chakraborty, U. K. (2005). Two improved differential evolution schemes for faster global search. ACM-SIGEVO Proceedings of genetic and evolutionary computation conference, 991–998.

Kaelo, P., & Ali, M. M. (2007). Differential evolution algorithms using hybrid mutation. Comput Optim Appl, 37, 231–246.

Das, S., Abraham, A., Chakraborty, U. K., & Konar, A. (2009). Differential evolution with a neighborhood-based mutation operator. IEEE Trans Evol Comput, 13(3), 526–553.

Neri, F., & Tirronen, V. (2009). Scale factor local search in differential evolution. Memet Comput J, 1(2), 153–171.

Qin, A. K., & Suganthan, P. N. (2005). Self-adaptive differential evolution algorithm for numerical optimization. Proceedings of the IEEE congress on evolutionary computation, 1785–1791.

Salman, A., Engelbrecht, A. P., & Omran, M. G. (2007). Empirical analysis of self-adaptive differential evolution. Eur J Oper Res, 183(2), 785–804.

Soliman, O. S., & Bui, L. T. (2008). A selfadaptive strategy for controlling parameters in differential evolution. Proceedings of the IEEE congress on evolutionary computation, 2837–2842.

Yang, Z., Tang, K., & Yao, X. (2008). Selfadaptive differential evolution with neighborhood search. Proceedings of the world congress on computational intelligence, 1110–1116.

Qin, A. K., Huang, V. L., & Suganthan, P. N. (2009). Differential evolution algorithm with strategy adaptation for global numerical optimization. IEEE Trans Evol Comput, 13(2), 398–417.

Zaharie, D. (2002). Critical values for control parameters of differential evolution algorithm. Proceedings of the 8th international Mendel conference on soft computing, 62–67.

Zaharie, D. (2003). Control of population diversity and adaptation in differential evolution algorithms. Proceedings of the 9th inter- national Mendel conference on soft computing, 41–46.

Verschelde, J., Verlinden, P., & Cools, R. (1994). Homotopies exploiting newton polytopes for solving sparse polynomial systems. SIAM J. Numer. Anal., 31, 915–930.

Morgan, A., & Shapiro, V. (1987). Boxbisection for solving second-degree systems and the problem of clustering. ACM Transaction on Mathematical Software, 13, 152–167.

Pramanik, S. (2002). Kinematic synthesis of a six-member mechanism for automotive steering. ASME Journal of Mechanical Design, 124, 642–645.

Meintjes, K., & Morgan, A. (1990). Chemical equilibrium systems as numerical test problems. ACM Transaction on Mathematical Software, 16, 143–151.

More, J., Garbow, B., & Hillstrom, K. (1981). Testing unconstrained optimization software. ACM Transaction on Mathematical Software, 7, 17–41.

Bongartz, I., Conn, A., Gould, N., & Toint, Ph. (1995). CUTE: constrained and unconstrained testing environment. ACM Transactions on Mathematical Software, 21, 123–160.



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