Complexity of a parametrized min-cost flow problem
this is too long for a comment, but it's not an answer

One reformulation of your problem might prove to be useful in terms of finding related work. 

Note that $f$ and $g$ essentially define probability distributions over the $s_i$ and $t_i$. If we treat the $c(e)$ as a "length", then the mincost flow is basically the earthmover distance between the two distributions $f$ and $g$.

Now your maximization over $f$ is really saying: find the farthest neighbor to $g$ lying in a box defined over the simplex. 

Formulating the problem this way suggests that you want some kind of projection from $g$ to the box.