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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.

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  • $\begingroup$ This might help math.stackexchange.com/a/27575/168758 $\endgroup$ – dohmatob Oct 23 '16 at 9:34
  • $\begingroup$ 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. $\endgroup$ – Michael Grant Oct 23 '16 at 16:38
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I do not have a solution to your problem, but still like to note the following but it's too long for a comment. The problem you want to solve is a fractional optimization problem:

$$\min_x \frac{a+b^Tx+x^TCx}{(1-k\cdot 1^Tx)^{\alpha}}$$

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

For both solution approaches, the numerator needs to be convex (which it is),while the denominator needs to be concave and nonnegative. Unfortunately, the denominator is convex, so you cannot solve it as a fractional optimization problem.

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