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I'm reading the book by Nocedal and Wright on numerical optimization - http://www.bioinfo.org.cn/~wangchao/maa/Numerical_Optimization.pdf

On page 322 (the general topic here is 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$ ($c_1(x)$ 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?

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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.

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