New modification of conjugate gradient method and its hybridization to solve large-scale unconstrained optimization problems - Mouiyad Mahmoud Bani Yousef
In our daily life, many situations can be expressed mathematically, such as in the fields of engineering, economics, medicine, business, and other sciences. The problems that arise from these situations can be solved using optimization. There are several methods for solving unconstrained optimization problems. One of the most commonly used is the onjugate Gradient (CG) method, which is known for its high efficiency in solving large-scale problems, due to its low memory requirement and global convergence properties. Besides, some methods such as the Fletcher-Reeves (FR), Polak-Ribiere Polyak (PRP), Wei-Yao-Liu (WYL), and Aini-Rivaie-Mustafa (ARM) do not perform well in terms of the number of iteration and Central Processing Unit (CPU) time. To overcome these drawbacks, the modification of (CG) methods has been proposed to improve these methods. In this study, two modifications of the CG coefficient are proposed for solving unconstrained optimization problems under exact line search. The new modifications are nam d as Mouiyad, Mustafa, Rivaie (MMR) and hybrid of MMR-PRP method based on well-known CG method. The basic idea of the hybrid method is to combine the method of MMR with the PRP method to produce a new algorithm that inherits the convergence properties of both methods with better numerical performance. For the numerical test, 35 standard optimization test functions, and three random initial guesses are used, ranging from the point that is nearest to the solutions, to points further away. The test measures the performance of the solvers in terms of iteration number and CPU time. All of the computations are performed on MATLAB R2015. The computational results are plotted using Sigma Plot 10 program. The performance of the new CG method of MMR is compared with PRP, FR, WYL, and ARM methods while the proposed hybrid (MMR-PRP) method is compared with Hybrid methods of Dai- Yuan Zero (HDYZ), Dai and Yuan (LS-CD), Hu and Storey (HUS) respectively. Theoretical proof shows that the proposed methods fulfill sufficient descent condition and possess global convergence properties. Numerical results show the proposed MMR and hybrid MMR-PRP perform better than other CG methods. MMR method has successfully solved 100 % of the entire test problems under exact line search compared to FR, WYL, PRP, and ARM with 91.2%, 97.9%, 97.9% and 98.4% respectively. Additionally, MMR-PRP hybrid CG method has successfully solved 100% of the test problems under exact line search compared to the existing hybrid methods of HDYZ, LS-CD, and HUS. An application in data fitting is also included to prove the applicability of the new approaches in the real-life problems. The proposed MMR and hybrid MMR-PRP methods have shown great efficiency in solving unconstrained optimization test problems and the real-life problems. Moreover, both approaches possess sufficient descent and global convergence property as demonstrated by the theoretical and numerical proofs and thus can be used as alternatives for solving unconstrained optimization problems.