I am dealing with the following function $f:\mathbb{R}^n \rightarrow \mathbb{R}$, how can I prove or disprove the convexity of the following function? $$f(x)=\left\|x-\frac{Ax}{\langle x,b\rangle}\right\|_2$$ Where $A$ is a constant matrix $A\in\mathbb{R}^{n\times n}$ and $b$ a constant vector $b\in\mathbb{R}^n$.
Thanks.
It's quite obvious this is not convex. After all, it is undefined whenever $\langle b, x \rangle = 0$. So its domain is the complement of that hyperplane, and therefore a non-convex set. All convex functions have convex domains, so this is definitely non-convex.
I really can't see a way to define the domain of this function in a sensible way that preserves convexity. I suppose you could require $\langle b, x \rangle > 0$, but do you really want to do that?
Even if you do I still doubt this function is convex. Convexity is what I personally call a fragile property. It's the exception, not the rule. Even when you take known convex functions and combine them together, you must do so according to specific rules, or you are likely to produce a nonconvex result. So for instance, in this case, the ratio $Ax/\langle x, b\rangle$ is a dead giveaway.