Let $u$ be a $n_{max} \times m$ matrix. Let $z$ be a $n_{max} \times s_{max} \times n_{max}$ cube. Let $w$ be a $n_{max} \times 1$ vector. All the three matrices can have values from the set $\{ 0, 1\}$, i.e., they are all binary vectors.

Let vector $v$ of size $m \times 1$, $n_{max}$, $m$, $s_{max} = \max (v)$ are known.

We need to find all the feasible solutions to $u$, $z$ and $w$ subject to the following constraints:

My solutions is as follows:

To satisfy constraints 1, we need to make sure that the number of non zero elements in $u$ should be equal to or less than $n_{max}$. I did that by finding all the combinations of positions in the matrix $u$, taken $n$ at a time, where $n \leq n_{max}$. Similarly, by assigning zeros to first $n$ terms in $w$ and remaining to $1$, we satisfy constraints 4, 5 and 6.

Since $m$ and $n_{max}$ can be large, it is still very hard to go through all the possible combinations of $u$ like this. So I tried to include the constraint 10 into the creation of $u$ as well. The problem however is that I am unable to find any suitable MATLAB-based solution to generate $u$ that satisfies constraint 10 as well. My question are as follows:

- Can we come up with an initial, partially feasible solution to $u$ that I can check with other constraints to make sure that its fully feasible.
- Can someone guide me how to generate all the feasible solutions subject to these constraints.
- Is there any way I can pose this problem with a constant objective function and using MATLAB's mixed integer linear program routine. Please note I want all the possible solutions and not just one of them.