How to solve a minimize problem with maximize a subproblem

Question

I have a minimization problem $$\min_{x, y} \{f(x, y) + \max_{y} g(y)\}$$ which has a max subproblem inside it.

How to solve it? Will alternating optimizing converge to the optimum?

Answer 1

As it is stated, the choice of the variable $y$ in the inner $max$ problem is without meaning and could be replaced by $z$ for instance to give $\max_z{ g(z)}$, this is the case referred to by @Minus One-Twelfth. If you want to restrict the value of the $y$ solution to be the same in the outer and inner optimization problem, you should add this constraint explicitly as follows: $$\min_{x, y} \{f(x, y) + \max_{z} \{g(z):z=y\}\} \\ = \min_{x, y} \{f(x, y) + h(y)\}. $$