An improved differential evolution algorithm with a restart technique to solve systems of nonlinear equations
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.
[1] Boussa¨ıd, I., Lepagnot, J., & Siarry, P.(2013). A survey on optimization metaheuris- tics. Information sciences, 237, 82–117 .
[2] Siddique, N., & Adeli, H. (2015). Nature in- spired computing: An overview and some fu- ture directions. Cogn Comput, 7, 706–714.
[3] Nanda, S. J., & Panda, G. (2014). A survey on nature inspired metaheuristic algorithms for partitional clustering. Swarm and Evolu- tionary Computation, 16, 1—18.
[4] 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.
[5] Hamm, L., Brorsen, B. W., & Hagan, M. T.(2007). Comparison of Stochastic Global Op- timization Methods to Estimate Neural Net- work Weights. Neural Process Lett, 26, 145— 158.
[6] Piotrowski, A. P. (2014). Differential evo- lution algorithms applied to neural network training suffer from stagnation. Applied Soft Computing, 21, 382—406.
[7] Raja, M. A. Z., Umar, M., Sabir, Z., Khan, J. A., & Baleanu, D. (2018). A new stochastic computing paradigm for the dynamics of non- linear singular heat conduction model of the human head. Eur. Phys. J. Plus, 133(364), DOI 10.1140/epjp/i2018-12153-4.
[8] Sabir, Z., Manzar, M. A., Raja, M. A. Z., Sheraz, M., & Wazwaz, A. M. (2018). Neuro- heuristics for nonlinear singular Thomas- Fermi systems. Applied Soft Computing, 65, 152-169.
[9] Raja, M. A. Z., Shah, Z., Manzar, M. A., Ah- mad, I., Awais, M.,& Baleanu, D. (2018). A new stochastic computing paradigm for non- linear Painlev´e II systems in applications of random matrix theory. Eur. Phys. J. Plus, 133(254), DOI 10.1140/epjp/i2018-12080-4.
[10] Raja, M. A. Z., Zameer, A., Kiani, A. K., Shehzad, A., & Khan, A. R. (2018). Nature- inspired computational intelligence integra- tion with Nelder-Mead method to solve non- linear benchmark models. Neural Comput & Applic, 29, 1169-1193.
[11] Ahmad, I., Zahid, H., Ahmad, F., Raja, M. A. Z., & Baleanu, D. (2019). Design of com- putational intelligent procedure for thermal analysis of porous fin model. Chinese Journal of Physics, 59, 641-655.
[12] Broyden, C. G. (1965). A class of meth- ods for solving nonlinear simultaneous equa- tions. Mathematics of computation, 19(92), 577–593.
[13] Mart´ınez, J. M. (1994). Algorithms for solv- ing nonlinear systems of equations. In : E. Spedicato, ed. Algorithms for Continuous Optimization-The state of the art. Kluwer Academic Publishers, London, 81–108.
[14] Kelley, C. T. (1995). Iterative methods for solving linear and nonlinear equations. SIAM, Philadelphia.
[15] Dennis, J. E., & Schnabel, R. B. (1996). Nu- merical methods for unconstrained optimiza- tion and nonlinear equations. SIAM, Philadel- phia.
[16] Ortega, J. M., & Rheinboldt, W. C. (2000). Iterative solution of nonlinear equations in several variables. SIAM, Philadelphia.
[17] Kelley, C. T. (2003). Solving nonlinear equa- tions with Newton’s method. SIAM, Philadel- phia.
[18] 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.
[19] Grosan, C., & Abraham, A. (2008). A new approach for solving nonlinear equations sys- tems. IEEE Transactions on Systems Man and Cybernetics-Part A: Systems and Hu- mans, 38(3), 698–714.
[20] 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.
[21] 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.
[22] Pourjafari, E., & Mojallali, H. (2012). Solv- ing nonlinear equations systems with a new approach based on invasive weed optimization algorithm and clustering. Swarm and Evolu- tionary Computation, 4, 33–43.
[23] Oliveira, H. A., & Petraglia, A. (2013). Solv- ing nonlinear systems of functional equations with fuzzy adaptive simulated annealing. Ap- plied Soft Computing, 13, 4349–4357.
[24] Raja, M. A. Z., Kiani, A. K., Shehzad, A., & Zameer, A. (2016). Memetic com- puting through bio-inspired heuristics inte- gration with sequential quadratic program- ming for nonlinear systems arising in different physical model. SpringerPlus, 5:2063, DOI 10.1186/s40064-016-3750-8.
[25] Zhang, X., Wan, Q., & Fan, Y. (2019). Ap- plying modified cuckoo search algorithm for solving systems of nonlinear equations. Neu- ral Comput & Applic, 31, 553–576.
[26] Storn, R., & Price, K. (1995). Differen- tial evolution—a simple and efficient adap- tive scheme for global optimization over con- tinuous spaces. Technical Report TR-95-012, ICSI, Berkeley.
[27] 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.
[28] Storn, R. (2008). Differential evolution research-Trends and open questions. In : U. K. Chakraborty, ed. Advances in Differential Evolution. Springer, Berlin, 1–31.
[29] Neri, F., & Tirronen, V. (2010). Recent ad- vances in differential evolution: a survey and experimental analysis. Artif Intell Rev, 33, 61–106.
[30] Das, S., & Suganthan, P. N. (2011). Differen- tial evolution: a survey of the state-of-the-art. IEEE Trans Evol Comput, 15(1), 4–31.
[31] Lampinen, J., & Zelinka, I. (2000). On stag- nation of the differential evolution algorithm. Proceedings of the 6th international Mendel conference on soft computing, 76–83.
[32] G¨amperle, R., M¨uller, S. D., & Koumout- sakos, P. (2002). A parameter study for differ- ential 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.
[33] Fan, H. Y., & Lampinen, J. (2003). A trigonometric mutation operation to differen- tial evolution. J Glob Optim, 27(1), 105–129.
[34] Das, S., Konar, A., & Chakraborty, U. K. (2005). Two improved differential evolu- tion schemes for faster global search. ACM- SIGEVO Proceedings of genetic and evolu- tionary computation conference, 991–998.
[35] Kaelo, P., & Ali, M. M. (2007). Differential evolution algorithms using hybrid mutation. Comput Optim Appl, 37, 231–246.
[36] Das, S., Abraham, A., Chakraborty, U. K., & Konar, A. (2009). Differential evolution with a neighborhood-based mutation opera- tor. IEEE Trans Evol Comput, 13(3), 526– 553.
[37] Neri, F., & Tirronen, V. (2009). Scale factor local search in differential evolution. Memet Comput J, 1(2), 153–171.
[38] 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.
[39] Salman, A., Engelbrecht, A. P., & Omran, M. G. (2007). Empirical analysis of self-adaptive differential evolution. Eur J Oper Res, 183(2), 785–804.
[40] Soliman, O. S., & Bui, L. T. (2008). A self- adaptive strategy for controlling parameters in differential evolution. Proceedings of the IEEE congress on evolutionary computation, 2837–2842.
[41] Yang, Z., Tang, K., & Yao, X. (2008). Self- adaptive differential evolution with neighbor- hood search. Proceedings of the world congress on computational intelligence, 1110–1116.
[42] Qin, A. K., Huang, V. L., & Suganthan, P. N.(2009). Differential evolution algorithm with strategy adaptation for global numerical op- timization. IEEE Trans Evol Comput, 13(2), 398–417.
[43] Zaharie, D. (2002). Critical values for con- trol parameters of differential evolution al- gorithm. Proceedings of the 8th international Mendel conference on soft computing, 62–67.
[44] Zaharie, D. (2003). Control of population di- versity and adaptation in differential evolu- tion algorithms. Proceedings of the 9th inter- national Mendel conference on soft comput- ing, 41–46.
[45] Verschelde, J., Verlinden, P., & Cools, R.(1994). Homotopies exploiting newton poly- topes for solving sparse polynomial systems. SIAM J. Numer. Anal. , 31, 915–930.
[46] Morgan, A., & Shapiro, V. (1987). Box- bisection for solving second-degree systems and the problem of clustering. ACM Transac- tion on Mathematical Software, 13, 152–167.
[47] Pramanik, S. (2002). Kinematic synthesis of a six-member mechanism for automotive steering. ASME Journal of Mechanical De- sign, 124, 642–645.
[48] Meintjes, K., & Morgan, A. (1990). Chem- ical equilibrium systems as numerical test problems. ACM Transaction on Mathemati- cal Software, 16, 143–151.
[49] More, J., Garbow, B., & Hillstrom, K.(1981). Testing unconstrained optimization software. ACM Transaction on Mathematical Software, 7, 17–41.
[50] Bongartz, I., Conn, A., Gould, N., & Toint, Ph. (1995). CUTE: constrained and uncon- strained testing environment. ACM Transac- tions on Mathematical Software, 21, 123–160.