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

In this paper, we propose a variant FR (VFR) formula ${\beta }_k^{\mathit{VFR}}$ 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 ${\beta }_k^{\mathit{VFR}}$ to ${\beta }_k^{*}$ such that $|{\beta }_k^{*}|⩽{\beta }_k^{\mathit{FR}}$ 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.