Hybrid methods of Polak-Ribiere-Polyak, Wei-Yao-Liu and Polak-Ribiere-Polyak, Dai-Wen methods for unconstrained optimization problems - Yasir Salih Mohammednour Mhmoud
Conjugate Gradiet (CG) methods one of the most popular methods for solving unconstrained optimization problems due to its simplicity and ability to improve low memory requirement and computational cost. However, the CG method has a weak global convergence, low-performance in terms of number of iterations and the Central Processing Unit (CPU) time. To overcome these problems, a procedure under exact and inexact line search techniques is introduced. Hybrid CG methods of Polak-RibierePolyak, Wei-Yao-Liu (PRP-WYL) and Polak-Ribiere-Polyak, Dai-Wen (PRPDWPRP) under some mild condition are suggested. PRP-WYL and PRP-DWPRP are combined together using exact, inexact line search methods and satisfies the sufficient descent and global convergence properties correspondence with the PRP md DWPRP CG methods to form a hybrid CG method. On the th r hand, the condition imposed on exact line search method resolves to be zero while for inexact line search method the condition would be less than or equal to the square of norm of the gradient function. The importance of these approaches is to reduce the CPU time and number of iterations respectively. Complete computational experiments are carried out to compare PRP-WYL and PRP-DWPRP with other CG methods for solving unconstrained optimization problems based on number of iterations and CPU time. All the methods are tested on one hundred and thirty-seven standard optimization test functions using MATLAB version R2014a subroutine program on 2.40Gz CPU processor, with 4GBRAM memory and Windows XP professional operating system. For each standard !est functions, four initial values are selected using dimensions ranging from two to tenthousand variables. The numerical results are analysed using the performance profile. The numerical results showed that the proposed Hybrid CG methods performed remarkably and effectively on some CG methods in terms of CPU time and number of iterations. The Hybrid CG methods could solve the entire standard test functions with 100% of success compared to Polak-Ribiere-Polyak (PRP) method with 93%, FletcherReeves (FR) with 72%, Dai-Wen (DWPRP) method with 88.4% and Wei-Yao-Liu (WYL) method with 98%. Hybrid CG methods are effective, efficient and reliable in terms of number of iterations and CPU time. Furthermore, the proposed methods possess a global convergence as well as sufficient descent properties and can be an alternative to the CG methods for solving large scale unconstrained optimization problems.