# Biconvex problem whose objective function depends on only one variable

I am solving the following biconvex problem: $$\min_{x,y} f(y)$$ $$s.t. ~~ g(x) \leq 0$$ $$~~~~~h(x,y) = 0$$ $$x \in X, y \in Y$$ where $$X$$ and $$Y$$ are compact convex sets, $$g(x)$$ and $$f(y)$$ are convex and $$h(x,y)$$ is affine for fixed $$x$$ or $$y$$. I wonder if there is any efficient algorithm to solve this type of optimization problem, not restricted to algorithms for general biconvex optimization problems.