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.
[Math] Convexity of product of two convex functions
convex optimizationconvex-analysis
Best Answer
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.
Update a little late, but here is how you can solve the problem: consider the biobjective problem with objectives $\min_x \{ a+b^Tx+x^TCx \}$ and $\max_x \{ (1-k\cdot 1^Tx)^{\alpha} \}$, create the Pareto frontier, and derive the optimal solution from there. To find a point on the Pareto curve, you can optimize one objective while constraining the other, nothing that for maximizing or constraining the second objective you can get rid of the power $\alpha$.