 Solution

Gradient descent uses only the current gradient and a suitable step size; its additional storage is order $d$. It is attractive for very large $d$ or when low per-iteration cost matters most. For a smooth strongly convex function, standard step choices give linear convergence, with a rate sensitive to the condition number. Mere convexity gives a global-minimum interpretation but does not alone imply a linear rate; for nonconvex objectives the usual guarantee concerns stationary points.

BFGS uses successive gradients to update an approximation to inverse curvature, incorporating information that can correct poor scaling. Full BFGS uses order $d^2$ storage and update work. With suitable line searches and local smoothness and positive-curvature assumptions near a nondegenerate minimum, it has superlinear convergence. This often favours it in moderate dimensions when gradients are not extremely cheap relative to the extra work.

Neither gradient descent nor BFGS requires an explicitly evaluated Hessian matrix. Thus an expensive Hessian matrix is compatible with both, rather than a reason to reject BFGS. If an accurate Hessian matrix and its factorization are exceptionally cheap, Newton's method in optimization becomes an additional reasonable option. Convexity helps identify a stationary point as a global minimum; without it, both methods need suitable descent and curvature safeguards. The BFGS update learns curvature from the Quasi-Newton secant equation.^[ref]