| Summary: | 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.
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