(Sorry if I'm not formal enough) Let
$$ f(\textbf{x},\textbf{y}) = f_1(\textbf{x}) + f_2(\textbf{x},\textbf{y}) $$
a real function of vector variable $\textbf{z} = (\textbf{x},\textbf{y})$ , that admits max. You can assume both $f_1, f_2$ have max and min. Is it correct to say that
$$ \max_{\textbf{x},\textbf{y}} \left\{ f(\textbf{x},\textbf{y}) \right\} = \max_{\textbf{x}} \left\{ f_1(\textbf{x}) + \max_{\textbf{y}} f_2(\textbf{x},\textbf{y}) \right\} $$ ?
My attempt to show this is true...
Let $$\textbf{y}^*(\textbf{x}) = argmax_{\textbf{y}} {f_2(\textbf{x},\textbf{y})} \Rightarrow \forall \textbf{x},\textbf{y} \;\;f_2(\textbf{x},\textbf{y}) \leq f_2(\textbf{x},\textbf{y}^*(\textbf{x}))$$ So $$ \max_{\textbf{x}} \left\{ f_1(\textbf{x}) + \max_{\textbf{y}} f_2(\textbf{x},\textbf{y}) \right\} = \max_{\textbf{x}} \left\{ f_1(\textbf{x}) + f_2(\textbf{x},\textbf{y}^*(\textbf{x})) \right\} $$
Let's define
$$ \textbf{x}^* = argmax_{\textbf{x}} \left\{ f_1(\textbf{x}) + f_2(\textbf{x},\textbf{y}^*(\textbf{x})) \right\} \Rightarrow \forall \textbf{x} \;\; f_1(\textbf{x}) + f_2(\textbf{x},\textbf{y}^*(\textbf{x})) \leq f_1(\textbf{x}^*) + f_2(\textbf{x}^*,\textbf{y}^*(\textbf{x}^*)) \Rightarrow \forall \textbf{x}, \textbf{y} \;\; f_1(\textbf{x}) + f_2(\textbf{x},\textbf{y}) \leq f_1(\textbf{x}) + f_2(\textbf{x},\textbf{y}^*(\textbf{x})) \leq f_1(\textbf{x}^*) + f_2(\textbf{x}^*,\textbf{y}^*(\textbf{x}^*)) $$
So defining the point $(\textbf{x}_M,\textbf{y}_M) =(\textbf{x}^*,\textbf{y}^*(\textbf{x}^*))$ for such point the original function $f(\cdot,\cdot)$ achieves it's maximum.
Is it right?
The question arises because i have function like this to be minimized, something like:
$$ f(x_1,...,x_n) = f_1(x_1)+f_2(x_1,x_2)+...+f_n(x_1,...,x_n) $$
Where I know some properties of each $f_i$
