Adaptive three-term PRP algorithms without gradient Lipschitz continuity condition for nonconvex functions

At present, many conjugate gradient methods with global convergence have been proposed in unconstrained optimization, such as MPRP algorithm proposed by Zhang et al. (IMA J. Numer. Anal. 26(4):629-640, 2006). Unfortunately, almost all of these methods require gradient Lipschitz continuity condition. As far as we know, how do the current conjugate gradient methods deal with gradient non-Lipschitz continuity problems is basically blank. For gradient non-Lipschitz continuity problems, Algorithm 1 and Algorithm 2 are proposed in this paper based on MPRP algorithm. The proposed algorithms have the following characteristics: (i) Algorithm 1 retains sufficient descent property independent of line search technology in MPRP algorithm; (ii) for nonconvex and gradient non-Lipschitz continuous functions, the global convergence of Algorithm 1 is obtained in combination with the trust region property and the weak Wolfe-Powell line search technique; (iii) based on Algorithm 1, Algorithm 2 is further improved which global convergence can be obtained independently of line search technique; (iv) according to numerical experiments, the proposed algorithms perform competitively with other similar algorithms.

Automated summary

This paper introduces two conjugate gradient methods for solving the gradient non-Lipschitz continuity problem, and points out that these methods perform competitively in numerical experiments. Algorithm 1, combined with the MPRP algorithm and independent of line search technology, achieves global convergence; Algorithm 2 further improves the performance based on Algorithm 1, achieving global convergence independently of line search technique.