I have a function $f(x,y)=f_1(x,y)+f_2(x,y)+f_3(x,y)$ where:
$$ f_1(x,y) = \frac{x(1-a)}{e^{\frac{-b}{x}}-a} $$
$$ f_2(x,y) = \frac{xy(1-ya)}{d-ya} $$
$$ f_3(x,y) = \frac{x(1-y)(1-(1-y)a)}{e^{\frac{-c}{x}}-(1-y)a} $$ $0\leq x$ and $0\leq y\leq 1$, and a, b, c and d are some positive parameters. I need to minimize $f(x,y)$ over a convex set of $x$ and $y$; I have found out that the Hessian of $f_2(x,y)$ is indefinite and therefore, $f(x,y)$ is not convex. I am not sure whether it is quasi-convex or not, but using Matlab plots, it looks like a convex function. Anyway, I can prove that for a fixed $x$, $f(x,y)$ is convex with respect to $y$ and for a fixed $y$, $f(x,y)$ is convex with respect to $x$ in the range of $x$ that I am interested.
Using an iterative algorithm through nested loops for $x$ and $y$ (where each loop is based on golden section method), I find the suboptimal values for $x$ and $y$ which have almost the same values as those I find by checking the figure and by brute-force search in a finely discrete domain. It means that my algorithm finds the values very close or equal to the optimal values, but I am unable to prove that my algorithm will always converge.
I am wondering whether there is a disciplined algorithm for such problems (i.e., when the objective function is convex with respect to each of the the variables but not when considering them together). Could you please give me some directions?
Thanks.