[Math] Minimize linear objective subject to unit norm constraint

Question

I have a linear function with unit norm constraint that I need to minimize.

$$\begin{array}{ll} \underset{w}{\text{minimize}} & w^\top x\\ \text{subject to} & \|w\| = 1 \end{array}$$

Is there a way to do this analytically?

My current thinking is that:

$$ \frac{d}{d w} w^\top x = x $$

$$ w = \|x\| $$

But this doesn't seem to make sense at all.

Thanks

Answer 1

Taking the lagrangian we have equivalently

$$ L(w,\lambda) = w^\top x + \lambda (w^\top w -1) $$

and the stationary points are the solutions for

$$ \nabla L = 0 = \cases{x+2\lambda w = 0\\ w^\top w - 1 = 0} $$

As $w = -\frac{1}{2\lambda}x$ we have

$$ \frac{1}{4\lambda^2}x^\top x = 1 $$

or

$$ \lambda = \pm\frac 12||x|| $$

and finally

$$ w = \pm \frac{x}{||x||} $$