New family of conjugate gradient methods with sufficient descent condition and global convergence for unconstrained optimizations - Ibrahim Jusoh
Conjugate gradient methods are a family of significance methods for solving of large-scale unconstrained optimization problems. This is due to both the simplicity of its algorithm and low memory requirement. A lot of efforts have been done to improve those methods since 1964 when the work of Fletcher and Reeve had opened the way to nonlinear conjugate gradient methods. In this research two new simple modifications of conjugate gradient coefficient 13k have been proposed. Both algorithms satisfy sufficient descent conditions and global convergence for exact line search and strong Wolfe line search. The convergence rate is super linear and its search directions fulfill the angle conditions. Based on the fact that a proof of global convergence for an algorithm does not ensure that it is an efficient method, then the new 13k is tested with twenty eight standard optimization test problems using MATLAB version 7.10.0 (R 2010a) subroutine programming and compared with five well- known conjugate gradient methods, which are Fletcher and Reeves (FR), Polak-Ribiere-Polyak (PRP), Hestenes and Steifel (HS), Wei-Yao-Liu (WYL) and Dai and Yuan (DY). Numerical results based on number of iterations and CPU time are analyzed and presented using performance profile of Dolan and Moore. For every test function four initial points are selected, some are close to the solution and some are further away. It is found out that both new formulas perform better than the other formulas for exact line search. However IMRI performs better than the other formulas for strong Wolfe line search.