How to prove the first part is convex function.

Question

I have an objective function such that

$$\min_{x,y} \left\{\frac {A}{f(x,y)}- Bf(x,y)\right\}$$

The function $f(x,y)$ is jointly concave in $x$ and $y$ as the Hessian matrix is positive semi definite. $A$ and $B$ are constant. All the constraints are linear and affine,I didn't show it here. Can I say for the first term of the objective function that $A / f(x,y)$ is convex as $f(x,y)$ is concave.

Answer 1

Your statement is true if $A$ is nonnegative and $f(x,y) > 0$ (see Boyd&Vandenberghe, page 84). Otherwise, a counterexample is $A=1$ and $f(x,y) = -\exp(x)$, since $A/f(x,y) = -\exp(-x)$ is convex.