Metaheuristic Algorithms for Mathematical Problem Solving: A Comparative Experimental Study on Nonlinear Systems, Boundary Value Problems, and NP-Hard Combinatorial Optimization

Authors

  • Nadia Ghasemabadi * Department of Computer Engineering, Ayandegan Institute of Higher Education, Tonkabon, Iran. https://orcid.org/0009-0002-0943-1993
  • Zahra Joorbonyan Ayandegan Institute of Higher Education, Gilan University, International Society of Fuzzy Set Extensions and Applications (ISFSEA), Morvarid Intelligent Industrial Systems Research Group, Iran, Research Expansion Alliance (REA) Press. https://orcid.org/0000-0003-3579-830X

https://doi.org/10.48313/maa.v2i3.47

Abstract

This study presents a comprehensive experimental comparison of six metaheuristic algorithms Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Differential Evolution (DE), Grey Wolf Optimizer (GWO), Salp Swarm Algorithm (SSA), and Sine Cosine Algorithm (SCA) applied to three fundamental classes of mathematical problems: 1) systems of nonlinear equations with dimensionality ranging from 10 to 100 variables, 2) parameter estimation in ordinary and Partial Differential Equations (PDE), and 3) NP-hard combinatorial optimization including the Traveling Salesman Problem (TSP) and graph coloring. Experiments were conducted on 23  Congress on Evolutionary Computation (CEC) 2014 benchmark functions in dimensions 10, 30, 50, and 100 under 51 independent runs with 500,000 function evaluations per run; 8 nonlinear systems drawn from the scientific and engineering literature; 5 Ordinary Differential Equation (ODE)/PDE parameter estimation problems with synthetic and semi-synthetic data; and 10 Traveling Salesman Problem Library (TSPLIB) instances alongside 5  Discrete Mathematics and Theoretical Computer Science Center (DIMACS) graph coloring instances. Rigorous statistical analysis was performed using Friedman non-parametric tests, Nemenyi post-hoc comparisons, and Wilcoxon signed-rank pairwise tests with Bonferroni correction. Results demonstrate domain-specific algorithmic superiority: DE achieves the best mean Friedman rank on continuous benchmark functions (mean rank 1.43 at D=30), GWO excels in nonlinear equation solving with a 97.8% success rate, and GA outperforms all competitors on combinatorial problems with a gap-to-optimal of 0.87% across TSP instances. No single algorithm dominates all problem domains, confirming the No Free Lunch theorem in the context of mathematical optimization. Practical guidelines for algorithm selection across mathematical problem types are provided, alongside discussion of scalability, computational complexity, and directions for future hybrid and adaptive approaches.

Keywords:

Metaheuristic algorithms, Mathematical optimization, Nonlinear equations, Differential equations, Combinatorial optimization, Benchmark functions, Convergence analysis, Friedman test, Swarm intelligence, Evolutionary computation

References

  1. [1] Newton, I. (1711). De analysi per aequationes numero terminorum infinitas. Royal Society London. https://www.newtonproject.ox.ac.uk/view/texts/normalized/NATP00204
  2. [2] Raphson, J. (1702). Analysis aequationum universalis. Typis TB Prostant Venales Apud A. & I. Churchill. https://openlibrary.org/books/OL52949866M/Analysis_aequationum_universalis
  3. [3] Hestenes, M. R., & Stiefel, E. (1952). Methods of conjugate gradients for solving linear systems. Journal of research of the national bureau of standards, 49(6), 409–436. https://www.stat.uchicago.edu/~lekheng/courses/302/classics/hestenes-stiefel.pdf
  4. [4] Dantzig, G. B. (1947). Maximization of a linear function of variables subject to linear inequalities. Activity analysis of production and allocation, 13(1), 339–347. https://www.sciepub.com/reference/82259
  5. [5] Karmarkar, N. (1984). A new polynomial-time algorithm for linear programming. Proceedings of the sixteenth annual ACM symposium on theory of computing (pp. 302-311). Association for Computing Machinery (ACM). https://doi.org/10.1145/800057.808695
  6. [6] Broyden, C. G. (1965). A class of methods for solving nonlinear simultaneous equations. Mathematics of computation, 19(92), 577–593. http://dx.doi.org/10.1090/S0025-5718-1965-0198670-6
  7. [7] Holland, J. H. (1992). Adaptation in natural and artificial systems: An introductory analysis with applications to biology, control, and artificial intelligence. MIT Press. https://mitpress.mit.edu/9780262082136/adaptation-in-natural-and-artificial-systems/
  8. [8] Goldberg, D. E. (1989). Genetic algorithms in search, optimization, and machine learning Addison. Reading. https://www2.fiit.stuba.sk/~kvasnicka/Free books/Goldberg_Genetic_Algorithms_in_Search.pdf
  9. [9] Kennedy, J., & Eberhart, R. (1995). Particle swarm optimization. Proceedings of icnn’95-international conference on neural networks (pp. 1942–1948). IEEE. https://www.cs.tufts.edu/comp/150GA/homeworks/hw3/_reading6 1995 particle swarming.pdf
  10. [10] Storn, R., & Price, K. (1997). Differential evolution-a simple and efficient heuristic for global optimization over continuous spaces. Journal of global optimization, 11(4), 341–359. https://doi.org/10.1023/A:1008202821328
  11. [11] Mirjalili, S., Mirjalili, S. M., & Lewis, A. (2014). Grey wolf optimizer. Advances in engineering software, 69, 46–61. https://doi.org/10.1016/j.advengsoft.2013.12.007
  12. [12] Mirjalili, S., Gandomi, A. H., Mirjalili, S. Z., Saremi, S., Faris, H., & Mirjalili, S. M. (2017). Salp swarm algorithm: A bio-inspired optimizer for engineering design problems. Advances in engineering software, 114, 163–191. https://doi.org/10.1016/j.advengsoft.2017.07.002
  13. [13] Mirjalili, S. (2016). SCA: A sine cosine algorithm for solving optimization problems. Knowledge-based systems, 96, 120–133. https://doi.org/10.1016/j.knosys.2015.12.022
  14. [14] Liang, J. J., Qu, B. Y., Suganthan, P. N., & Hernández-Diaz, A. G. (2013). Problem definitions and evaluation criteria for the CEC 2013 special session on real-parameter optimization. Computational intelligence laboratory, zhengzhou university, zhengzhou, china and nanyang technological university, singapore, technical report, 201212(34), 281–295. https://peerj.com/articles/cs-2671/CEC2013.pdf
  15. [15] Liang, J. J., Qu, B. Y., & Suganthan, P. N. (2013). Problem definitions and evaluation criteria for the CEC 2014 special session and competition on single objective real-parameter numerical optimization. Computational intelligence laboratory, zhengzhou university, zhengzhou china and technical report, nanyang technological university, singapore, 635(2), 2014. http://bee22.com/manual/tf_images/Liang CEC2014.pdf
  16. [16] Awad, N. H., Ali, M. Z., & Suganthan, P. N. (2017). Ensemble sinusoidal differential covariance matrix adaptation with euclidean neighborhood for solving cec2017 benchmark problems. 2017 IEEE congress on evolutionary computation (CEC) (pp. 372–379). IEEE. https://doi.org/10.1109/CEC.2017.7969336
  17. [17] Das, S., & Suganthan, P. N. (2010). Differential evolution: A survey of the state-of-the-art. IEEE transactions on evolutionary computation, 15(1), 4–31. https://doi.org/10.1109/TEVC.2010.2059031
  18. [18] Tanabe, R., & Fukunaga, A. (2013). Success-history based parameter adaptation for differential evolution. 2013 IEEE congress on evolutionary computation (pp. 71–78). IEEE. https://doi.org/10.1109/CEC.2013.6557555
  19. [19] Mirjalili, S., & Lewis, A. (2016). The whale optimization algorithm. Advances in engineering software, 95, 51–67. https://doi.org/10.1016/j.advengsoft.2016.01.008
  20. [20] Mohamed, A. W., & Suganthan, P. N. (2018). Real-parameter unconstrained optimization based on enhanced fitness-adaptive differential evolution algorithm with novel mutation. Soft computing, 22(10), 3215–3235. https://doi.org/10.1007/s00500-017-2777-2
  21. [21] Clerc, M., & Kennedy, J. (2002). The particle swarm-explosion, stability, and convergence in a multidimensional complex space. IEEE transactions on evolutionary computation, 6(1), 58–73. https://doi.org/10.1109/4235.985692
  22. [22] Liang, J. J., Qin, A. K., Suganthan, P. N., & Baskar, S. (2006). Comprehensive learning particle swarm optimizer for global optimization of multimodal functions. IEEE transactions on evolutionary computation, 10(3), 281–295. https://doi.org/10.1109/TEVC.2005.857610
  23. [23] Faris, H., Aljarah, I., Al Betar, M. A., & Mirjalili, S. (2018). Grey wolf optimizer: A review of recent variants and applications. Neural computing and applications, 30(2), 413–435. https://doi.org/10.1007/s00521-017-3272-5
  24. [24] 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. https://doi.org/10.1109/TSMCA.2008.918599
  25. [25] Mo, Y., Liu, H., & Wang, Q. (2009). Conjugate direction particle swarm optimization solving systems of nonlinear equations. Computers & mathematics with applications, 57(11–12), 1877–1882. https://doi.org/10.1016/j.camwa.2008.10.005
  26. [26] 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. https://doi.org/10.1016/j.swevo.2011.12.001
  27. [27] Hirsch, M. J., Meneses, C. N., Pardalos, P. M., & Resende, M. G. C. (2007). Global optimization by continuous grasp. Optimization letters, 1(2), 201–212. https://doi.org/10.1007/s11590-006-0021-6
  28. [28] Nadimi Shahraki, M. H., Taghian, S., Mirjalili, S., Abualigah, L., Abd Elaziz, M., & Oliva, D. (2021). EWOA-OPF: Effective whale optimization algorithm to solve optimal power flow problem. Electronics, 10(23), 2975. https://doi.org/10.3390/electronics10232975
  29. [29] Babaei, M. (2013). A general approach to approximate solutions of nonlinear differential equations using particle swarm optimization. Applied soft computing, 13(7), 3354–3365. https://doi.org/10.1016/j.asoc.2013.02.005
  30. [30] Ahmadianfar, I., Heidari, A. A., Gandomi, A. H., Chu, X., & Chen, H. (2021). RUN beyond the metaphor: An efficient optimization algorithm based on Runge Kutta method. Expert systems with applications, 181, 115079. https://doi.org/10.1016/j.eswa.2021.115079
  31. [31] Mendes, P., & Kell, D. (1998). Non-linear optimization of biochemical pathways: Applications to metabolic engineering and parameter estimation. Bioinformatics (oxford, england), 14(10), 869–883. https://doi.org/10.1093/bioinformatics/14.10.869
  32. [32] Rodriguez Fernandez, M., Egea, J. A., & Banga, J. R. (2006). Novel metaheuristic for parameter estimation in nonlinear dynamic biological systems. BMC bioinformatics, 7(1), 483. https://doi.org/10.1186/1471-2105-7-483
  33. [33] Godio, A., Pace, F., & Vergnano, A. (2020). SEIR modeling of the Italian epidemic of SARS-CoV-2 using computational swarm intelligence. International journal of environmental research and public health, 17(10), 3535. https://doi.org/10.3390/ijerph17103535
  34. [34] Dorigo, M., & Gambardella, L. M. (2002). Ant colony system: A cooperative learning approach to the traveling salesman problem. IEEE transactions on evolutionary computation, 1(1), 53–66. https://doi.org/10.1109/4235.585892
  35. [35] Helsgaun, K. (2000). An effective implementation of the Lin-Kernighan traveling salesman heuristic. European journal of operational research, 126(1), 106–130. https://doi.org/10.1016/S0377-2217(99)00284-2
  36. [36] Larranaga, P., Kuijpers, C. M. H., Murga, R. H., Inza, I., & Dizdarevic, S. (1999). Genetic algorithms for the travelling salesman problem: A review of representations and operators. Artificial intelligence review, 13(2), 129–170. https://doi.org/10.1023/A:1006529012972
  37. [37] Malaguti, E., Monaci, M., & Toth, P. (2008). A metaheuristic approach for the vertex coloring problem. INFORMS journal on computing, 20(2), 302–316. https://doi.org/10.1287/ijoc.1070.0245
  38. [38] Lü, Z., & Hao, J. K. (2010). A memetic algorithm for graph coloring. European journal of operational research, 203(1), 241–250. https://doi.org/10.1016/j.ejor.2009.07.016
  39. [39] Moalic, L., & Gondran, A. (2018). Variations on memetic algorithms for graph coloring problems. Journal of heuristics, 24(1), 1–24. https://doi.org/10.1007/s10732-017-9354-9
  40. [40] Deb, K., Pratap, A., Agarwal, S., & Meyarivan, T. (2002). A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE transactions on evolutionary computation, 6(2), 182–197. https://doi.org/10.1109/4235.996017
  41. [41] Shi, Y., & Eberhart, R. (1998). A modified particle swarm optimizer. Evolutionary computation proceedings (pp. 69-73). IEEE. https://doi.org/10.1109/ICEC.1998.699146
  42. [42] Meintjes, K., & Morgan, A. P. (1990). Chemical equilibrium systems as numerical test problems. ACM transactions on mathematical software (TOMS), 16(2), 143–151. https://doi.org/10.1145/78928.78930
  43. [43] Tsai, L. W., & Morgan, A. P. (1985). Solving the kinematics of the most general six-and five-degree-of-freedom manipulators by continuation methods. Journal of mechanisms, transmissions, and automation in design (ASME), 107, 189-200. https://doi.org/10.1115/1.3258708
  44. [44] Sinha, N., Chakrabarti, R., & Chattopadhyay, P. K. (2003). Evolutionary programming techniques for economic load dispatch. IEEE transactions on evolutionary computation, 7(1), 83–94. https://doi.org/10.1109/TEVC.2002.806788
  45. [45] Abdollahi, A., Ghadimi, A. A., Miveh, M. R., Mohammadi, F., & Jurado, F. (2020). Optimal power flow incorporating FACTS devices and stochastic wind power generation using krill herd algorithm. Electronics, 9(6), 1043. https://doi.org/10.3390/electronics9061043
  46. [46] Reinelt, G. (1991). TSPLIB—A traveling salesman problem library. ORSA journal on computing, 3(4), 376–384. http://dx.doi.org/10.1287/ijoc.3.4.376
  47. [47] Derrac, J., Garcia, S., Molina, D., & Herrera, F. (2011). A practical tutorial on the use of nonparametric statistical tests as a methodology for comparing evolutionary and swarm intelligence algorithms. Swarm and evolutionary computation, 1(1), 3–18. https://doi.org/10.1016/j.swevo.2011.02.002
  48. [48] Friedman, M. (1937). The use of ranks to avoid the assumption of normality implicit in the analysis of variance. Journal of the american statistical association, 32(200), 675–701. https://www.tandfonline.com/doi/pdf/10.1080/01621459.1937.10503522
  49. [49] Nemenyi, P. B. (1963). Distribution-free multiple comparisons. Princeton University. https://www.bibsonomy.org/bibtex/2be58c4cd02deec08771241724168b663/sveng
  50. [50] Zaharie, D. (2003). Control of population diversity and adaptation in differential evolution algorithms. Proceedings of mendel 2003, 9th international conference on soft computing (pp. 41–46). Springer-Verlag. https://staff.fmi.uvt.ro/~daniela.zaharie/mendel03.pdf
  51. [51] Wolpert, D. H., & Macready, W. G. (2002). No free lunch theorems for optimization. IEEE transactions on evolutionary computation, 1(1), 67–82. https://doi.org/10.1109/4235.585893

Published

2026-05-12

Issue

Vol. 2 No. 3 (2025): Metaheuristic Algorithms with Applications

Section

Articles

How to Cite

Ghasemabadi, N., & Joorbonyan, Z. (2026). Metaheuristic Algorithms for Mathematical Problem Solving: A Comparative Experimental Study on Nonlinear Systems, Boundary Value Problems, and NP-Hard Combinatorial Optimization. Metaheuristic Algorithms With Applications, 2(3), 209-228. https://doi.org/10.48313/maa.v2i3.47

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