Descent direction
= Descent direction
{title2=$\nabla f(x)^Ts<0$}
A <vector> $s$ is a <descent direction> for a <differentiable function> $f$ at $x$ if its <directional derivative> $\nabla f(x)^Ts$ is negative. The first-order expansion $f(x+\alpha s)=f(x)+\alpha\nabla f(x)^Ts+o(\alpha)$ then gives $f(x+\alpha s)<f(x)$ for sufficiently small positive $\alpha$. The negative <gradient> is a <descent direction> whenever the <gradient> is nonzero.