Suppose we have binary variables $y_1, ..., y_n$. To make the representation simple, we show the concatenated vector as $\mathbf{y} = (y_1, ..., y_n)$. Consider the two following functions: $$ F_1(\mathbf{y}) = h(\mathbf{y})\prod_{i=1}^ng_i(y_i) $$ and $$ F_2(\mathbf{y}) = h(\mathbf{y})\prod_{i=1}^nf_i(\mathbf{y}) $$ In the first function the factors inside the $\prod$ are functions of only one variable. But in the second function, they are function of (potentially) all variables, i.e. $\mathbf{y}$.
Suppose we want to maximize these two functions with respect to $\mathbf{y}$. From computational-hardness point of view, how different are these two functions?
Update: Suppose we don't know much about properties and forms of the $g_i()$, $f_i()$, $l_i()$, $h_i()$