A new nonliner conjugate gradient coefficient via hybrid of fletcher-reeves and hestenes-stiefel methods for unconstrained optimization problems - Saleh Nazzal al-Suliman
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 Fletcher 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.