Improved Conjugate Gradient Methods For Unconstrained Optimization

Conjugate gradient method is an important method in solving unconstrainedoptimization problems. This method only uses the value of the present gradient functionand the previous search direction, which can avoid the slow convergence disadvantageof the steepest descent method and the computation and storage of matrix needed byNewton method. Due to the features of low memory requirement and simplecomputation, the nonlinear conjugate gradient method is used widely in solving largescale optimization problems. Therefore, it becomes a hot research direction. In thisarticle, we do some research given as follows:1Based on the modified PRP conjugate gradient method proposed by Cheng etal., we propose another modified PRP conjugate gradient method, and prove itssufficient descent property and global convergence property under the standard Armijolinear search condition. The convergence rate of this method is at least R-linearly. Thenumerical results show that this method has better computational performance.2We propose a new method which is always sufficient descent under any linearsearch condition. Under either the standard Armijo linear search condition or themodified Armijo linear search condition, we establish the global convergence of thismethod. Some numerical results are given to show this method is superior to othermethods.3Based on the GN method, we propose a spectral conjugate gradient method.This method can satisfy the sufficient descent condition under any linear searchcondition. Under some appropriate conditions, we establish the global convergence ofthis spectral method, the numerical results show that this spectral GN method isefficient.