Globally convergent conjugate gradient algorithms without the Lipschitz condition for nonconvex optimization

It is well known that under the Wolfe–Powell inexact line search, the global convergence of the nonlinear conjugate gradient method always requires the Lipschitz continuous condition for nonconvex functions. In this paper, we find that the Lipschitz continuous condition is unnecessary for proving the global convergence for particular algorithms if its searching direction has the well-known sufficient descent property and the trust region feature. Thus, the global convergence of the family conjugate gradient algorithms proposed by Yuan et al. (Numer. Algorithms, 84(2020)) is established without the Lipschitz condition for nonconvex functions since they have these two properties. Furthermore, a new algorithm without a line search technique is presented, and its global convergence is also analysed under suitable assumptions. The numerical results show that the performance of the proposed algorithm is competitive with some particular problems.