How to get explicit solution of the fractional minimization problem?

Question

I need to minimize $f(x) = \frac{x^tQx}{a^tx-b}$, $Q$ is positive definite, $a,b$ are constant vector. But when I take gradient and set it equal to $0$, it's hard to get an explicit expression for $x$. Anyone knows how to solve for x explicitly?

Answer 1

I assume you are implying $a^tx-b>0$ to ensure convexity (see Michael's comment). For simplicity I also assume that $Q$ is Hermitian but the solution can be easily generalized.

If $b \le 0$, then $x=\mathbf{0}$ is trivially a solution.

If $b > 0$, you can follow the approach below (forgive some sloppiness in the maths, for brevity).

Setting $\nabla f(x) = 0$, we obtain that the solution $\hat x$ must be at a point $x$ satisfying $$2 Q x = \frac{x^t Q x}{a^t x - b} a = c a \tag{1}$$ where the scalar $c$ is $$ c = f(\hat x) = \frac{\hat x^t Q \hat x}{a^t \hat x - b} \tag{2}.$$ This leads to $$\hat x = \frac c2 Q^{-1} a \tag{3}$$ i.e., the solution is along the direction of $Q^{-1} a$.

We now need to find the scalar $c$. We do so by substituting (3) in (2) and solving for $c$. We find again the solution $c=0$, which we already considered in the case $b \le 0$, as well as the following:

$$c = \frac{4b}{a^t Q^{-1} a} \tag{4}$$.