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For $g=\nabla f(x)\ne 0$, the unit steepest descent direction is $-g/\Vert g{\Vert }_2$ by the Cauchy-Schwarz inequality. The usual unnormalized direction is $\boxed{p=-g}$; choosing its scale separately as a line-search step gives the same ray. It satisfies $\boxed{g^{T}p=-\Vert g{\Vert }_2^2<0}$, so it is a descent direction. The first-order Taylor expansion then guarantees decrease for sufficiently small positive steps. The subsequent bounds use this unnormalized negative-gradient convention.

A descent direction $s$ satisfies $g^{T}s<0$, where $g=\nabla f(x)$; the first-order Taylor expansion then gives $f(x+\alpha s)<f(x)$ for sufficiently small $\alpha >0$. To compare directions one must fix their scale. Among Euclidean unit vectors, the Cauchy-Schwarz inequality givesThis is the unit steepest descent direction. The conventional unnormalized direction used in the algorithm is the same ray, $\boxed{s=-g}$. Equivalently it is the unique minimizer of $g^{T}s+\frac{1}{2}\Vert s{\Vert }_2^2=\frac{1}{2}\Vert s+g{\Vert }_2^2-\frac{1}{2}\Vert g{\Vert }_2^2$. If $g=0$ there is no strict first-order descent direction, and the conventional update is zero.

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.