New hybrid conjugate gradient method for unconstrained optimization

Conjugate gradient methods are widely used for unconstrained optimization, especially large scale problems. Most of conjugate gradient methods don’t always generate a descent search direction, so the descent condition is usually assumed in the analyses and implementations. Dai and Yuan (1999) proposed the conjugate gradient method which generates a descent direction at every iteration. Yabe and Sakaiwa (2003) gave a new conjugate gradient method based on the idea of Dai and Yuan, and showed that their method always produces a descent direction and converges globally if the Wolfe conditions are satisfied. On the other hand, expecting the fast convergence of the methods, Dai and Liao (2001) used secant condition of quasi-Newton methods. Yabe and Takano (2003) proposed a new conjugate gradient method following to Dai and Liao. Both Yabe and Sakaiwa method and Yabe and Takano method use the modified secant condition given by Zhang, Deng and Chen (1999), Zhang and Xu (2001). In this paper, we make a combination of Yabe and Sakaiwa method and Yabe and Takano method. Our method always produces a descent search direction and is shown to be globally convergent under some assumptions. Numerical results are reported.