I have an optimization (maximization) problem in which I need to optimize a variable (2-D matrix) under some constraints. Let say I have following entries in a $5$x$4$ matrix.
\begin{array}{|c|c|c|c|} \hline 0.42& 0.42 & 0.42 & 0 \\ \hline 0.38 & 0 & 0 & 0\\ \hline 0 & 0 & 0.69 & 0.69\\ \hline 0.78 & 0.78 & 0 & 0\\ \hline 0 & 0 & 0 & 0.41\\ \hline \end{array}
I want to optimize entries under the following $3$ conditions:
1) Sum of each column in updated matrix is $\leq$ $1$, preferably $1$
2) Sum of each row is less than or equal to a specific number. For example, $\sum1^{st}$ row $\leq$1.2, $\sum2^{nd}$ row $\leq$0.3, $\sum3^{rd}$ row $\leq$1.3, $\sum4^{th}$ row $\leq$1.2, $\sum5^{th}$ row (say)$\leq$0.3
3) All non-zero elements in a row must be same. For example, in the above case if $1^{st}$ element in $1^{st}$ row in the updated matrix is $0.35$ then $2^{nd}, and \ 3^{rd}$ elements must also be $0.35$.
how can I program that.
Thanks
Let $x_{ij}$ be the $(i,j)$ entries in the updated table.
we want the sum of each column in the updated matrix to be less than or equal to $1$.
$$\sum_{i=1}^{5} x_{ij} \leq 1, j=1,\ldots 4$$
We want the sum of each row to be less than equal to certain number, say the number is $b_i$, then we have
$$\sum_{j=1}^4x_{ij}\leq b_i, i=1,\ldots 5$$
All non-zero elements along a row has to be the same.
$$x_{11}=x_{12}=x_{13}$$ $$x_{33}=x_{34}$$ $$x_{41}=x_{42}$$
If the objective function is linear, then this is a linear programming problem.