Let us consider the following optimization problem:
for fixed $y$ ($y$ can be vector, possible from compact set) find $x$ such that
\begin{equation*} \begin{aligned} & \underset{x}{\text{maximize}} & & f_0(x,y) \\ & \text{s.t.} & & f_i(x) \leq 0, \; i = 1, \ldots, m\\ & & & h_i(x) = 0, \; i = 1, \ldots, p \end{aligned} \end{equation*}
Assume there is another function $g_{0}(x, y) \neq f_0(x,y)$ such that for any fixed $y$
\begin{equation*} \begin{aligned} & \text{argmax}_{x}f_{0}(x, y) = \text{argmin}_{x}g_{0}(x, y),\\ & \text{s. t. }\,\,\,\, h_i(x) = 0, \; i = 1, \ldots, p \end{aligned} \end{equation*}
The question: when the following would work for any $y$ \begin{equation*} \begin{aligned} & \text{argmax}_{x}f_{0}(x, y) = \text{argmin}_{x}g_{0}(x,y) \\ & \text{s. t. } \, f_i(x) \leq 0, \; i = 1, \ldots, m\\ & h_i(x) = 0, \; i = 1, \ldots, p \end{aligned} \end{equation*} given some conditions imposed on $f, g, f_{i}$ and $h_{i}$ (linearity, differentiability, convexity etc.)?