Relation between arg min of two functions

Question

When is $u_F(x) = \underset{u}{\text{argmin}}(F_1(x),\cdots,F_u(x),\cdots,F_U(x))$ $\le$ $\underset{u}{\text{argmin}}(G_1(x),\cdots,G_u(x),\cdots,G_U(x)) = u_G(x)$ where $u \in \{1,2,\cdots,U\}, x \in P \subseteq R^N$. $F_u, G_u:R^N \rightarrow R$ are some continuous functions. I believe it would involve some sort of sub-modularity. But I would appreciate a specific answer or pointer to a reference.

Answer 1

$u_F=\underset{u}{\text{argmin}}\big(F_u(x)\big)$ is the $u$ giving the least value of $F_u(x)$.

$u_G=\underset{u}{\text{argmin}}\big(G_u(x)\big)$ is the $u$ giving the least value of $G_u(x)$.

$u_F\le u_G$ means the least value of $F$ comes at an smaller (or equal) point to the least value of $G$.