= Dennis-Moré condition
{c}
{title2=$\|(B_k-H_*)s_k\|/\|s_k\|\to0$}
For a convergent <quasi-Newton method> sequence $x_k\to x_*$ with $\nabla f(x_*)=0$, nonsingular $H_*=\nabla^2f(x_*)$, and continuous Hessian nearby, the displayed condition implies q-<superlinear convergence>. Here $B_ks_k=-\nabla f(x_k)$ and $s_k=x_{k+1}-x_k$. The condition requires Hessian accuracy only along the actual steps, rather than convergence of the full matrix approximations. It is associated with <John E. Dennis> and <Jorge Moré>.
Indeed, the first-order expansion of the <gradient> gives $\nabla f(x_{k+1})=-(B_k-H_*)s_k+o(\|s_k\|)=o(\|s_k\|)$. Nonsingularity of $H_*$ gives $\|x-x_*\|\leq C\|\nabla f(x)\|$ locally. Consequently $\|e_{k+1}\|\leq\varepsilon_k(\|e_k\|+\|e_{k+1}\|)$ with $\varepsilon_k\to0$. Rearrangement proves $\|e_{k+1}\|/\|e_k\|\to0$. Finite termination is understood separately from ratios with zero denominators.
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