Consider the grid $X =\{ 0,1,\ldots,m-1\}\times \{ 0,1,\ldots,n-1\}$ of size $m \times n$. An indicator function $f:X\mapsto \{ 0,1\}$ on this grid is defined to be monotonic, if $ i\leq s, j \leq t \Rightarrow f_{i,j} \leq f_{s,t} , \; \forall i,j,s,t \in X $. We are also given a function $v : X \mapsto [0,1]$ such that $\sum_{i,j}v_{i,j}=1$, and another function $b : X \mapsto [0,1]$. I want to find the monotonic indicator function $f$ that maximizes $$ \sum_{i,j} v_{i,j}\; b_{i,j}\;f_{i,j} $$ subject to the condition $$ %\frac{ \mathrm{Tr}(v^\mathrm{T}f)}{ \mathrm{Tr}(b^\mathrm{T}f)}= \frac{\sum_{i,j} v_{i,j} \; f_{i,j}}{\sum_{i,j} v_{i,j} \; b_{i,j} \; f_{i,j}} < c$$ for some specified positive constant $c$. How can I perform this optimization?
A brute force search is impractical as there are $\binom{m+n}{m}$ possible monotonic indicator functions. I'm wondering if there is a DP solution. If there are not fast solutions, what are some heuristic approaches? Another approach I thought of is to convert this into a continuous optimization problem by smoothening $b$, but not sure how to proceed.