I have a minimax problem that I wish to convert to a single maximization problem. I am wondering if the approach is standard and, if so, if someone can point me to a reference.
Consider the following problem
$$V(x) = \min_{y}\max\{f(x,y),g(x,y)\}$$
Consider the reformulation of the inner maximization problem
$$\min_z \{z\} \text{ subject to } f(x,y)\le z, g(x,y)\le z$$
Taking a partial Lagrangian, the above can be written as
$$\max_{\lambda\ge0}\big\{\min_{z}\{z + \lambda(g(x,y)-z)\} \text{ subject to } f(x,y)\le z\big\}$$
So the problem is equivalently
$$V(x) = \max_{\lambda\ge0,\{(y,z):z\ge f(x,y)\}}\{\lambda(z-g(x,y))-z\}$$
Question: Is there something I missed?