Can we convert a non-nonvex function x'Dy to convex Consider a vector of variables $x$ and $y$ and a matrix $D$. The objective function $x^TDy$ is not convex. Can we make it convex by log transformation like in Geometric programming?
The actual optimization problem is equation 9 and the conditions are in equation 6,7 and 8 of the link https://hal.inria.fr/hal-01109774/document
 A: The logarithmic transformation is the key tool to convert geometric programming to convex. Let us write a posynomials function to optimize is given by $$f_0(x)=\sum_{n=1}^{N}{c_n{x{_n}^{a^n}}}$$ This can be converted by change of variable $y=\ln(x)$ to  $$f(y)=\ln{f_0(e^y)}=\ln(\sum_{n=1}^{N}{e^{a_n{y{_n}+b_n}}})$$ where $b_n=\ln(c_n)$ 
Now let us try to write $x^TDy$ in the above form. Let use try to do for a simple $2X2$ matrix. For variable $x$ and $y$ and $d_1$ $d_2$ $d_3$ and $d_4$ are element of $D$ we can explicitly write it as
$x^TDy={d_1}{x_1}{y_1}+{d_3}{x_2}{y_1}+{d_2}{x_1}{y_2}+{d_4}{x_2}{y_2}$
If we replace the variable $y_1$ and $y_2$ by $x_3$ and $x_4$ the equation is $x^TDy={d_1}{x_1}{x_3}+{d_3}{x_2}{x_3}+{d_2}{x_1}{x_4}+{d_4}{x_2}{x_4}$
and it appears naively that it is of the form $$f_0(x)=\sum_{n=1}^{N}{c_n{x{_n}^{a^n}}}$$ where some of the $a_n$ are $0$. Consider the the first of  $x^TDy$ which is ${d_1}{x_1}{x_3}$ for this term we must have $a^2=0$. But in this case we cannot write the second term of $x^TDy$ . Therefore, the bottom line is that we cannot write the $x^TDy$  in the form 
$$\sum_{n=1}^{N}{c_n{x{_n}^{a^n}}}$$
This means we cannot convert a function of the form $x^TDy$  to convex by logarithimic transformation. 
