I want to solve
$$\min_{x,y} \quad -f(x,y) \qquad \text{subject to} \quad xy C - f(x,y)K -\kappa = 0$$
where
$x, y$ are from a compact and convex subset of $\mathbb{R}^2$
$f(x,y)$ is concave, in $C^{\infty}$ and $f_{xx}=f_{yy}=-2$ $f_{xy}=f_{xxx}=f_{yyy}=0$
$C, K, \kappa$ are non-negative constants.
which implies that function $f$ is biconvex in that for any $y$ the constraint is convex in $x$ and vice versa.
I have 3 questions:
Is it correct that the solution to the related problem in which I replace the constraint with its convex envelope provides an lower bound on the optimal $-f(x,y)$. That bound is tight if at the optimizer the convex envelope coincides with the actual constraint?
Given that the constraint is the sum of a convex function and a bilinear function is there a simple way to show when the non-convexity is not an issue?
Is there a simple way to construct the convex envelope of the Lagrangian problem (i.e. to construct the "closest" convex optimization problem)?