Past exam of the numerical linear algebra and continuous optimisation course of the University of Oxford / 2010 / B / 4 / i / Solution

At an unconstrained local minimizer of a twice continuously differentiable objective function, the first-order optimality condition is $\boxed{\nabla f(x_{\ast })=0}$. The second-order necessary condition is $d^T{\nabla }^{2}f(x_{\ast })d\ge 0$ for every direction $d$, so the Hessian matrix is positive semidefinite.

For the Euclidean norm, the unit steepest descent direction is $-g/\Vert g{\Vert }_2$ when $g=\nabla f(x)\ne 0$; the algorithm usually uses the unnormalized direction $p=-g$. It is a descent direction because $\boxed{g^{T}p=-\Vert g{\Vert }_2^2<0}$.

Two limitations of unpreconditioned steepest descent method in high-accuracy smooth optimization are its slow zigzagging on ill-conditioned valleys and its sensitivity to scaling of the coordinates. It uses no curvature information to correct those directions, so it can require many function and gradient evaluations compared with a suitable Newton or quasi-Newton method. These are reasons to prefer curvature-aware methods in that setting, rather than a claim that gradient-based methods are never useful.