A new hybrid conjugate gradient method close to the memoryless BFGS quasi-Newton method and its application in image restoration and machine learning

A new hybrid conjugate gradient algorithm for solving the unconstrained optimization problem was presented. The algorithm could be considered as a modification of the memoryless Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton method. Based on a normalized gradient difference, we introduced a new combining conjugate gradient direction close to the direction of the memoryless BFGS quasi-Newton direction. It was shown that the search direction satisfied the sufficient descent property independent of the line search. For general nonlinear functions, the global convergence of the algorithm was proved under standard assumptions. Numerical experiments indicated a potential performance of the new algorithm, especially for solving the large-scale problems. In addition, the proposed method was used in practical application problems for image restoration and machine learning.