New family of nonlinear conjugate gradient methods for solving unconstrained optimization problems and its application - Norrlaili Shapiee
Conjugate gradient (CG) method is one of the most effective algorithms for solving unconstrained optimization problem. Some CG methods cycle infinitely without reaching the solution point and do not possess global convergence properties. Even if the method reaches a solution point, the process often results in high number of iterations. The new developments in CG method are sometimes difficult to implement. Furthermore, some particular conditions lead to failure hence making it unable to retain the global optimum solution. Nowadays CG method is less used in real life problems. To overcome such problems, two new CG methods for solving unconstrained problems have been proposed. They are denoted as NRM] (Norrlaili, Rivaie, Mustafa and Ismail) and NRM1+. The global convergence is established by using exact line search. These new CG methods are compared with some classical CG methods such as RMJL (Rivaie, Mustafa, Ismail and Leong) method, HS (Hestenes and Steifel) method as well as the FR (Fletcher and Reeves) method. The CG methods are tested with twenty-five standard optimization test problems by using MatlabR20 II b subroutine programming. For every problem, four initial points are used; ranging from the point that is nearest to the solution point, to one that is furthest away from it. The numerical results are based on the number of iterations and CPU time. The efficiency of the CG algorithms is analyzed by using performance profile. In addition, the new CG methods are used to solve determine the function between year and the index of road deaths. Theoretical proofs show that these new CG methods fulfill sufficient descent conditions and possess global convergence properties. From the results, FR method only solves 89.75% of the problems. The NRMI and NRMI+ methods successfully solve 100%. The HS method solves 95.5% while the RMIL method solves 97% of the problems. These new methods have been shown to outperform the FR, HS and RMIL methods. They are also successful to determine the function between year and the index of road deaths. Hence, conclude that the new CG methods are successful for testing the standard optimization test problems, index of road deaths problems and also possess global convergence properties.