Modification of Polak-Ribiere-Polyak (PRP) conjugate gradient coefficient for unconstrained optimization problems - Mohamed Ahmed Hamoda
Conjugate gradient (CG) method is an important class of methods for solving unconstrained optimization problems, especially when the dimensions are large. This thesis presents three modifications of Polak-Ribiere-Polyak (PRP) conjugate gradient methods for solving this type of problems, where the modifications named as HRMl, HRM2, and HRM3. The modifications are iterative methods that modify the PRP method by using exact and inexact line search techniques, wherein inexact employing strong Wolfe-Powell line search._ Theoretical proof has shown that these three modifications possess a guaranteed global convergence on general non-convex objective functions. They are also in line with the angle condition, suggesting that these coefficients always converge faster than steepest descent methods. All of these new Ih (HRMl, HRM2, and HRM3) are tested based on thirty-four standard optimization functions using MATLAB R2014a (8.3.0.532) subroutine programming. Based on the numerical results, a comparative analysis is conducted assessing the new methods against other CG methods, which included the FR, PRP, MPRP, DPRP, WYL, and AMRl. The chosen test functions consist of small and large-scale problems. For every problem, four different arbitrary initial points x E R" are used, based on the one that is nearest to the solution point. The numerical results based on the number of iterations and CPU times are analyzed using the performance profile developed by Dolan and More. Numerical results show that the new methods are efficient, robust, and outperforms the FR, PRP, NPRP, MPRP, DPRP, WYL and AMRl methods. While achieving superior performance, the new methods still retain its simplicity, possess sufficient descent condition and global convergence properties. It is also shown that the numerical results are in line with the theoretical proof.