| Summary: | The prominent methods for solving optimization problems are the Newton type methods
but these methods have some shortcomings which includes computation and storage of
the Jacobian matrix in every iteration. Numerous research and modifications have been
done recently to improve the efficiency of these methods. The nonlinear conjugate
gradient (CG) methods are among the best scheme proposed for solving optimization
problems, due to their low memory requirement and good convergence properties.
However, recent modifications of nonlinear CG methods are very complex and unable
to solve large-scale unconstrained optimization problems. Therefore, this study
proposed a new modification of nonlinear CG coefficient called Saleh and Mamat (SM),
based on combination of the famous methods of Hestenes-Stiefel (HS), and FletcherÂ
Reeves (FR) under Strong Wolfe-Powell (SWP) inexact line search. Numerical
computational have been carried out using fifteen standard optimization benchmark
problems. This is done illustrate the efficiency of the proposed method when compare
to the well-known nonlinear CG methods of FR, HS, Conjugate Descent (CD), and
Rivaie, Mustafa, Ismail, Leong (RMIL). The comparison is based on number of
iterations and central processing unit (CPU) time. The method is coded in MATLAB
version (R2018a) subroutine programming. Four initial points with different dimension
were selected for each problem starting from the points close to the solution to points
far away. Numerical results show that the new nonlinear CG method is able to solve all
the test problems with 100 % success compared to the well-known methods of HS,
Conjugate Descent (CD), FR and RMIL (Rivaie, Mustafa, Ismail, Leong) with 42.16
%, 83.95 %, 85.44 % and 71.27 % respectively. Also, the sufficient descent ondition
and the global convergence properties of the proposed method has been proved. Th
best and most efficient method is the method that has the ability to reach the solution
point in the least number of iterations and shortest CPU time. Therefore, the proposed
method considered is efficient, robust and reliable in terms of the number of iteration
and CPU time. Thus, the method is good alternative for solving large-scale
unconstrained optimization problems.
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