I'm currently following this material

Optimization Theory: Chapter 2 Theory of Constrained Optimization

And I can't understand why the following statement is true, between the equations (2.9) and (2.10):

The only way that such a direction does not exist is that $∇f(x)$ and $∇h(x)$ are parallel

I can grasp the idea intuitively but not in the mathematical sense

Edit: the problem actually reduces to:

Let $a$, $b$ be two vectors belonging to $R^2$ and $c$ a scalar. There is an equivalence:

There is no $d$ such that $a^Td<0$ and $b^Td=0$ $\iff$ $a = cb$


1 Answer 1


Regarding your equivalence,

The direction $\impliedby$ is easy. Suppose $a$ and $b$ are parallel with $a=cb$ and $a'd<0$, where $c$ is a scalar and $a,b,d$ are vectors. Then $a'd=cb'd=0$, but this contradicts $a'd<0$, so there is no $d$ satisfying the conditions.

The direction $\implies$ can by using the fact that $a'd = ||a||\: ||d||\cos\theta$ where $\theta$ is the angle between the vectors. If $a,b$ are not parallel then we can find a $d$ such that $b$ and $\pm d$ are orthogonal and $a$ and $d$ or $-d$ form an obtuse angle (can't be orthogonal or else the original vectors were parallel).


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