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

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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