Mixed-integer linear programming: How to generate constraints?

I recommend writing down some other cases a bit. So it seems you want to constraint all row-sums and all column-sums:

For N=3, there are 9 vars (i'm assuming a square case here; you did not provide complete info):

x00 x01 x02
x10 x11 x12
x20 x21 x22

Now the constraint matrix looks like:

x00 x01 x02 | x10 x11 x12 | x20 x21 x22
---------------------------------------
1   1   1
              1   1   1
                            1   1   1
1             1             1
    1             1             1
        1             1             1

That's pretty regular. Not it's time to check out matlab's matrix-creation functions. Sadly i'm not much of a matlab-user, but:

the lower half of rows consist of:

• horizontal stacking of N identity-matrices each of size N

the upper half of rows consist of:

• block-diagonal matrix of N 1-row-vectors each of size N

the final matrix is a vertical stacking of both components

A full sparse-matrix python-example (sorry, no matlab here; but there should be nearly a 1:1 mapping), to be more clear would look like:

import numpy as np
import scipy.sparse as sp

N = 3
component_a = sp.hstack([sp.eye(N) for i in range(N)])
row_full_1 = sp.csr_matrix(np.ones(N))
component_b = sp.block_diag([row_full_1 for i in range(N)])  # matlab: blkdiag?
matrix = sp.vstack((component_b, component_a))

print(matrix.todense())

Output:

[[ 1.  1.  1.  0.  0.  0.  0.  0.  0.]
 [ 0.  0.  0.  1.  1.  1.  0.  0.  0.]
 [ 0.  0.  0.  0.  0.  0.  1.  1.  1.]
 [ 1.  0.  0.  1.  0.  0.  1.  0.  0.]
 [ 0.  1.  0.  0.  1.  0.  0.  1.  0.]
 [ 0.  0.  1.  0.  0.  1.  0.  0.  1.]]

Remark: depending on N, you need to think about using dense or sparse-matrices. Given N, the ratio of non-zeros in the matrix will be 1/N.