I'm reading Nocedal & Wright's Numerical Optimization. On page 322, on optimization with one inequality constraint, right after equation 12.18, the text says that when we are in a region where $ c_1(x) > 0$ (where $c_1(x)$ is the constraint), the direction ($f(x)$ is the objective function to be minimized)
$$d = -c_1(x) \frac{\nabla f(x)}{||\nabla f(x)||}$$
satisfies equation 12.18, which is
$$c_1(x) + \nabla c_1(x)^T d \geq 0$$
This should hold if
$$ \frac{\nabla c_1(x)^T \nabla f(x)}{||\nabla f(x)||} < 1$$
But I don't see why this should be true. If there were another $$||\nabla c_1(x)||$$ in the denominator, it might have held. Can someone shed light on what I might be missing?
You are right.
As mentioned on point $43$ of the Errata. The denominator should contain an extra factor $\left\| \nabla c_1(x)\right\|$.
I am attaching the Errata of the book.