A variant spectral-type FR conjugate gradient method and its global convergence

In this paper, we propose a variant FR (VFR) formula βkVFR and corresponding spectral-type conjugate gradient method (SVFR) such that the direction generated is always a descent direction for the objective function. We also extend βkVFR to βk∗ such that |βk∗|⩽βkFR and have similar conclusions. Under appropriate conditions, we prove that the proposed method is globally convergent under not only Wolfe line search but also Armijo-type line search. Numerical experiments show the SVFR method performs well.