A new hybrid conjugate gradient algorithm based on the Newton direction to solve unconstrained optimization problems

Abstract

In this paper, we propose a new hybrid conjugate gradient method to solve unconstrained optimization problems. This new method is defined as a convex combination of DY and DL conjugate gradient methods. The special feature is that our search direction respects Newton’s direction, but without the need to store or calculate the second derivative (the Hessian matrix), due to the use of the secant equation that allows us to remove the troublesome part required by the Newton method. Our search direction not only satisfies the descent property, but also the sufficient descent condition through the use of the strong Wolfe line search, the global convergence is proved. The numerical comparison shows the efficiency of the new algorithm, as it outperforms both the DY and DL algorithms.