| Summary: | The unconstrained optimization problem has been dealing with different methods to be
solved recently. The most common solution is the conjugate gradient (CG) method due
to its convergence speed, simplicity, low memory requirements, and its capability to
solve large-scale problems. There are many modifications of the CG method with global
convergence properties, however, some of them possess a high processing time when
applying in real problems. Therefore, it is proposed new modifications that have global
convergence properties and efficient compared to other CG methods. Besides, they can
be used in practical applications. In this thesis, two improved methods of Polak- Ribi ere
Polyak are proposed that are Dawahdeh, Mamat, and Rivaie (DMAR) method and
Mahmoud, Mamat, and Rivaie (MMAR) method. The sufficient descent condition, as
well as the global convergence of the proposed methods, are established under strong
Wolfe-Powell (SWP) line search. Furthermore, the performance of these methods is
tested using 33 standard benchmark test problems. Numerical results are analysed using
the performance profile based on the number of iterations and the processing time. All
the algorithms are coded in the Matlab subroutine and the finding is plotted using Sigma
Plot software. The proposed methods are compared with the existing CG methods of
Polak-Ribiere-Polyak plus (PRP+) , Wei, Yao, and Liu (WYL), new Polak-Ribiere
Polyak (NPRP), and Aini, Rivaie, and Mustafa (ARM). The tests cover problems of a
small scale to a large scale whereas the initial points are chosen around the solution
point from the nearest to that furthest. The new CG methods are tested for applicability
by using them to solve a function constructed from real data. DMAR and MMAR
methods shown to have sufficient descent condition and possess global convergence
properties. The result shows both methods have successfully solved 100 % the entire
test problem under SWP line search compared to the methods ofPRP+, WYL, NPRP,
and ARM with 68.50%, 99.10%, 98.80%, and 85.30% respectively. In addition, DMAR
and MMAR methods have the least processing time. Based on the numerical results,
they are superior to the other tested CG methods. Also, the proposed methods are proved
to be applicable in a real case. The proposed CG methods possess global convergence
properties and they are shown to be very efficient and robust compared to the other CG
methods. Hence, the objectives achieved indicate that the proposed CG methods can be
used as an alternative for solving large-scale unconstrained optimization problems.
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