SUT J. Math. 54 (1), 79-98, (June 2018) DOI: 10.55937/sut/1547386572
KEYWORDS: Unconstrained optimization, memoryless quasi-Newton method, Broyden family, three-term conjugate gradient method, sufficient descent condition, global convergence, 90C30, 90C06
Memoryless quasi-Newton methods are studied for solving large-scale unconstrained optimization problems. Nakayama et al. (2017) proposed a memoryless quasi-Newton method based on the spectral-scaling Broyden family and showed that the method satisfies the sufficient descent condition and converges globally. To relax the conditions on parameters in the method, we apply the modification technique by Kou and Dai (2015) to the method of Nakayama et al., and we give a hybrid method of the three-term conjugate gradient method and the memoryless quasi-Newton method based on the spectral-scaling Broyden family. We show that our method satisfies the sufficient descent condition, and we prove that the method converges globally. Furthermore, we give a concrete choice of parameters for our method. Finally, some numerical results are given.