Suppose that we wanted to minimize a function like $f(x,y)=(x-sin(y))^2+(y-3)^2+4$. Notice that $f$ is nonconvex. For each fixed $y$ define $g_y(x)=f(x,y)$. Notice that each $g_y$ is convex. Let $h(y)= min_x g_y(x)$. Notice that $h$ is convex and that the minimum of $h$ is the same as the minimum of $f$. My questions are as follows:
Is there a name for functions like $f$ that are not convex themselves but can be solved in this manner?
Is there a name for this technique of optimizing over one subset of variables first, and then over the complementary set of variables?