# Biconvex optimization problems

Consider minimization of a biconvex function over a biconvex set. Is the biconvex optimization problems polynomially solvable?

For the general audience: a biconvex optimization problem is a problem of the form: $$\begin{array}{ll} \text{minimize} & f(x,y) \\ \text{subject to} & (X,Y)\in\mathcal{B} \end{array}$$ which is convex in $X$ for any fixed $Y$, convex in $Y$ for any fixed $X$, but is not convex in both $(X,Y)$ jointly.
I suspect that the standard, obvious heuristic for a biconvex problem is to fix $X$ and minimize over $Y$, then fix $Y$ and minimize over $X$, and repeat.