In Boyd and Vandenberghe's textbook on Convex Optimization, is claimed that:
We always have $$ \underset{x,y}{\text{inf}\; f(x, y)} = \underset{x}{\text{inf}}\;\tilde{f}(x) $$ where $$ \tilde{f}(x) = \text{inf}_y\;f(x, y).$$ In other words, we can always minimize a function by first minimizing over some of the variables, and then minimizing over the remaining ones.
My question is the following (since it is unclear to me from the context): does this applies only to convex functions or to every function?