| Summary: | Optimization refers to a common procedure applied within the science and engineering
domain to determine variables that produce the best performance values. One of the
most efficient techniques to solve large-scale unconstrained optimization issues is the
conjugate gradient (CG) method, primarily due to its simplicity, low memory
requirements, and global convergence properties. This method embeds n-step to attain
a minimum point, where convergence properties are absent. Several techniques do not
perform well in terms of the number of iteration and Central Processing Unit (CPU)
time. In order to address these shortcomings, this study proposed new CG coefficients,
Pk> which are Tal'at, Mamat, Rivaie (TMR) and Hybrid Tal'at and Mamat (HTM). TMR
and HTM are modifications of Hestenes and Stiefel (HS) to enhance the capabilities of
HS. The convergence properties of both methods were assessed, while numerical
performance was evaluated via exact and inexact line searches. Both the TMR and HTM
methods were examined by using 36 standard optimization test problems, along with
three random initial points, beginning with a point close and a point far away from the
solution point. All the standard optimization test problems were tested from small to
large-scale dimensions, whereas numerical experiments were run via MATLAB
R2015a using a computer with Intel® Core ? i3-M350 (2.27GHz) CPU, 4GB RAM.
Analysis of the findings was discussed based on the number of iteration and CPU time
using a performance profile in graphic form. The TMR was compared with other CG
methods, namely Fletcher and Reeves (FR), HS, and Modified HS (MHS), whereas the
HTM method was compared with Hybrid Hu and Storey (HHUS) and Hybrid Gilbert
and Nocedal (HGN). The theoretical evidence showed that both TMR and HTM
fulfilled sufficient descent condition, apart from exhibiting global convergence
properties. The numerical findings revealed that TMR and HTM methods are better than
the other CG approaches in terms of the number of iteration and CPU time. The TMR
and HTM methods successfully solved all the test problems under exact line search,
followed by HHUS and HGN approaches with a similar percentage at 98.46%.
Meanwhile, MHS, HS, and FR methods solved the test problems at the following
percentages; 97.77%, 91.11 %, and 67.00%, respectively. Both TMR and HTM, which
were tested under inexact line search, resulted in the highest percentage at 98.88% and
99.22%, respectively, followed by HHUS and HGN with success rates at 95.29% and
94.9%. The other methods, namely FR, HS, and MHS, displayed success rates at
92.15%, 76.68%, and 65.09%, respectively. The efficiency exerted by TMR and HTM
methods was assessed in real-life applications to solve issues related to regression
analysis. All the findings showed that both TMR and HTM methods are indeed robust
and superior, apart from showcasing better performance when compared to other
methods, as well as appropriate for practical use.
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