Complexity of a parametrized min-cost flow problem Consider the following  min-cost flow variant:

Input: 


a positively-weighted complete bipartite graph $G = (S, T, c)$ and extra vertices $s, t$ 
edges from $s$ to $S$ and from $T$ to $t$.  
lower and upper capacity bounds $l_i, u_i$ on each edge $(s, s_i)$
flow values $g_i$ for each edge $(t_i, t)$ with the constraint that $\sum_i g_i = 1$


For any flow assignment $f$ on the edges $(s, s_i)$ (note that by flow conservation $\sum_i f_i = 1$), let $\text{cost}(f)$ be the cost of a min-cost flow satisfying the upper and lower bounds. 

Problem:


  Find an $f$ satisfying all capacity constraints such that
  $\text{cost}(f)$ is maximized.
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.