Stationarity from uniformly bounded quasi-Newton matrices (${\sum }_k\Vert \nabla f(x_k){\Vert }^2<\infty$)

Suppose $f$ is bounded below, $0\prec B_k⪯MI$ for one fixed $M$, $B_k{s}_k=-\nabla f(x_k)$, and $x_{k+1}=x_k+{\alpha }_k{s}_k$ satisfies the Armijo condition with ${\alpha }_k\ge C>0$. ThenA telescoping sum proves the displayed convergence of a nonnegative series and hence $\nabla f(x_k)\to 0$. A uniform positive lower bound on the eigenvalues of $B_k$ is unnecessary for this conclusion. The conclusion is first-order stationarity; without further assumptions it does not assert convergence of the iterates or attainment of a global minimum.