A limited memory subspace minimization conjugate gradient algorithm for unconstrained optimization

Subspace minimization conjugate gradient (SMCG) methods are a class of quite efficient iterative methods for unconstrained optimization. The orthogonality is an important property of linear conjugate gradient method. It is however observed that the orthogonality of the gradients in linear conjugate gradient method is often lost, which usually causes slow convergence. Based on SMCG\(\_\)BB (Liu and Liu in J Optim Theory Appl 180(3):879–906, 2019), we combine subspace minimization conjugate gradient method with the limited memory technique and present a limited memory subspace minimization conjugate gradient algorithm for unconstrained optimization. The proposed method includes two types of iterations: SMCG iteration and quasi-Newton (QN) iteration. In the SMCG iteration, the search direction is determined by solving a quadratic approximation problem, in which the important parameter is estimated based on some properties of the objective function at the current iterative point. In the QN iteration, a modified quasi-Newton method in the subspace is proposed to improve the orthogonality. Additionally, a modified strategy for choosing the initial stepsize is exploited. The global convergence of the proposed method is established under weak conditions. Some numerical results indicate that, for the tested functions in the CUTEr library, the proposed method has a great improvement over SMCG\(\_\)BB, and it is comparable to the latest limited memory conjugate gradient software package CG\(\_\)DESCENT (6.8) (Hager and Zhang in SIAM J Optim 23(4):2150–2168, 2013) and is also superior to the famous limited memory BFGS (L-BFGS) method.