Hybrid Conjugate Gradient Method with Exact Line Search for Grey Wolf Optimization Algorithm

Abstract

Optimization is a process of obtaining optimal solution in solving objective function.  Conjugate Gradient (CG) method is known for solving unconstrained optimization problems due to its simplicity, low memory storage and global convergence properties. Over the years, researchers improved classical CG method and introduced hybrid, three term and modified CG method. The hybrid conjugate gradient (HCG) method is the combination of two or more CG coefficients where the rapid computational efficiency of one method is paired with the robust global convergence of another. These CG methods have been implemented in various applications such as data fitting, robotic motion control and Grey Wolf Optimization (GWO) algorithm. GWO is a metaheuristic optimization algorithm that has proven effective  in solving multiple optimization problems. However, GWO is said to have poor population diversity, slow convergence in later stages and prone to get stuck in local optimums where it proves the imbalance of exploration and exploitation search. Thus, this research aims to find the best HCG method   under exact line search that can be implemented in GWO to improve search process through the hybrid GWO-CG algorithm. The performance of five HCG methods, Linda-Aini-Mustafa-Rivaie (LAMR)  coefficient hybrid with Kamil-Mustafa-Aini-Rivaie (KMAR), Norrlaili-Rivaie-Mustafa-Ismail (NRMI), Hestenes–Stiefel (HS), Polak–Ribière–Polyak (PRP) and Liu–Storey (LS) coefficients respectively are tested using 15 standard test functions with different initial points and variables ranging from 2 to 10,000. The numerical results are computed based on number of iteration (NOI) and CPU time. The results are plotted using performance profile to evaluate its efficiency and robustness. Numerically, LAMR-PRP outperforms other methods by solving all test functions with least NOI and CPU time. Lastly, LAMR-PRP is implemented in GWO and resulting effective implementation as it performs better that original GWO.