# Convexity of product of two convex functions

I hope to prove convexity of a product of two (specific) convex functions when only one of them is monotonic. The problem can be written as $F(x) = f(x) g(x)$, where $f:\mathbb{R^n} \rightarrow \mathbb{R}$, $g:\mathbb{R^n} \rightarrow \mathbb{R}$, $$f(x) = (1 - k \cdot 1^T x )^{-\alpha}, \quad 0 < k < 1, \quad \alpha > 1,$$ $\textbf{dom } f = \left\{ x | x \geq 0, 1^T x < \frac{1}{k} \right\}$, and $$g(x) = a + b^T x + x^T C x,$$ where $C$ is positive-semidefinite, $g > 0$, and $\textbf{dom } g = \textbf{dom } f$. So, $f$ is strictly increasing, but $g$ is not monotonic. I'm solving an optimization problem of the type $$\underset{x \geq 0}{\min} F(x),$$ that behaves well numerically but I would really also like to prove convexity.

• This might help math.stackexchange.com/a/27575/168758 – dohmatob Oct 23 '16 at 9:34
• You're not going to find any general principles that deal with the product itself, I doubt. I suspect you're going to have to prove this sui generis (i.e., on its own). A derivative test, say along a line $x=x_0+tv$, is likely to be your best bet. – Michael Grant Oct 23 '16 at 16:38

$$\min_x \frac{a+b^Tx+x^TCx}{(1-k\cdot 1^Tx)^{\alpha}}$$
with $\alpha>1$. Two different solution approaches are described in http://pubsonline.informs.org/doi/pdf/10.1287/mnsc.13.7.492 and in http://link.springer.com/article/10.1007%2FBF02026600