A comparative study of gradient-based and meta-heuristic optimization methods using Griewank benchmark function/ Um estudo comparativo de métodos de otimização baseados em gradientes e meta-heurísticos usando a função de benchmark do Griewank

Carlos Eduardo Rambalducci Dalla, Wellington Betencurte da Silva, Júlio Cesar Sampaio Dutra, Marcelo José Colaço


Optimization methods are frequently applied to solve real-world problems such, engineering design, computer science, and computational chemistry. This paper aims to compare gradient-based algorithms and the meta-heuristic particle swarm optimization to minimize the multidimensional benchmark Griewank function, a multimodal function with widespread local minima. Several approaches of gradient-based methods such as steepest descent, conjugate gradient with Fletcher-Reeves and Polak-Ribiere formulations, and quasi-Newton Davidon-Fletcher-Powell approach were compared. The results presented showed that the meta-heuristic method is recommended for function with this behavior because is no needed prior information of the search space. The performance comparison includes computation time and convergence of global and local optimum.


optimization, gradient based, particle swarm, Griewank function.

Full Text:



Antoniou Andreas and Lu Wu-Sheng. Applications of Unconstrained Optimization, pages 231–263. Springer US, Boston, MA, 2007.

M. Locatelli. A note on the griewank test function. Journal of Global Optimization, 25(2):169– 174, 2003.

Yan Huang, Jian ping Li, and Peng Wang. Unusual phenomenon of optimizing the griewank function with the increase of dimension. Frontiers of Information Technology & Electronic Engineering, 20(10):1344–1360, October 2019.

Jasbir S. Arora. Numerical methods for unconstrained optimum design. In Introduction to Optimum Design, pages 411–441. Elsevier, 2012.

Xin-She Yang. Optimization algorithms. In Introduction to Algorithms for Data Mining and Machine Learning, pages 45–65. Elsevier, 2019.

M.R. Hestenes and E. Stiefel. Methods of conjugate gradients for solving linear systems. Journal of Research of the National Bureau of Standards, 49(6):409, December 1952.

Lukas Exl, Johann Fischbacher, Alexander Kovacs, Harald Oezelt, Markus Gusenbauer, and Thomas Schrefl. Preconditioned nonlinear conjugate gradient method for micromagnetic energy minimization. Computer Physics Communications, 235:179–186, February 2019.

R. Fletcher. Function minimization by conjugate gradients. The Computer Journal, 7(2):149–154, February 1964.

E. Polak and G. Ribiere. Note sur la convergence de méthodes de directions conjuguées. Revue française d'informatique et de recherche opérationnelle. Série rouge, 3(16):35–43, 1969.

R. E. Ricketts. Practical optimization. International Journal for Numerical Methods in Engineering, 18(6):954–954, June 1982.

Saptarshi Sengupta, Sanchita Basak, and RichardPeters. Particle swarm optimization: A sur vey of historical and recent developments with hybridization perspectives. Machine Learning and Knowledge Extraction, 1(1):157–191, October 2018.

DOI: https://doi.org/10.34117/bjdv7n6-102


  • There are currently no refbacks.