Steepest descent direction ($p=-\nabla f(x)/\Vert \nabla f(x){\Vert }_2$)

For a nonzero gradient, this unit direction minimizes the directional derivative among Euclidean unit directions. The Cauchy-Schwarz inequality gives $\nabla f(x{)}^{T}p\ge -\Vert \nabla f(x){\Vert }_2$, with equality for the displayed direction. The steepest descent method usually uses the equivalent unnormalized direction $-\nabla f(x)$ and chooses a separate step length. A different choice of norm can change the steepest direction.