Optimization of a tabulation under constraint Given a tabulation contening in each cell some text wich need some space to be readable, how to adjust the sizes of the rows and the columns in order to have the tabulation as small as possible? I have asked myself this question, and I haven't managed to solve it so far. I can reformulate it as follow:

Let $A$ be a $m\times n$ matrix of positive real numbers. Find $T$ a $m\times n$ matrix of positive real numbers such as:
  
  
*
  
*$\text{rank}(T)=1$
  
*$\forall (i,j)\in\{1,...,m\}\times\{1,...,n\}, T_{i,j}\geq A_{i,j}$
  
*$\sum_{(i,j)\in\{1,...,m\}\times\{1,...,n\}}T_{i,j}$ is minimal.
  

For example, for $A=\begin{pmatrix}
   1 & 1 \\
   2 & 4 
\end{pmatrix}$, the answer is $T=\begin{pmatrix}
   1 & 2 \\
   2 & 4 
\end{pmatrix}$ : the sum if it's coefficients is 9, and it's impossible to have less than 9.
I guess it has been studied before, but I can't find any reference.
 A: That is just an idea. I am not sure if it is efficient or not.
I consider the problem in vector form
$$ \min_{r,c}\quad\sum_{i,j} r_i c_j  \quad \text{s.t.} \quad r_i c_j \ge a_{i,j},\, r_i, c_j > 0$$
for given $a_{i,j} \ge 0$.
Unlike that matrix formulation, the vector formulation has multiple optimal solutions. If $(r,c)$ is feasible, than $(\alpha r, \alpha^{-1} s)$ is also feasible for $\alpha > 0$, and they have the same objective value. So we need to add some kind of normalization like $r_1 = 1$, which I drop in the following for sake of convenience.
I will turn the problem into a smooth nonlinear convex problem:
Notice that $r_ic_j \ge a_{i,j}$ holds if and only if $\log(r_i) + \log(c_j) \ge \log(a_{i,j})$. By replacing $\log(r_i)$ with $\rho_i$ and $\log(c_j)$ with $\gamma_j$, we obtain
$$ \min_{\rho,\gamma}\quad \sum_{i,j} \exp(\rho_i + \gamma_j) \quad\text{s.t.}\quad \rho_i + \gamma_j \ge \log(a_{i,j}). $$
Now, the feasible set is a convex polyhedron and the objective is convex. So you can apply one of many well known smooth constrained optimization methods.
