I have a question regarding the "classic" Lagrange optimization in a special matrix case:
If I need to optimize an objective function wrt to a $K\times N$-matrix $X$ such that row and column sums of $X$ remain unchanged, how do I formulate that as a Lagrange optimization problem?
The problem can be written as follows:
$$\min_{x_{ki}\ge0} \quad f(X)$$
s.t.
$$h_i = \sum_k x_{ki} - s_i = 0 \quad \forall i$$
$$h_k = \sum_i x_{ki} - t_k = 0 \quad \forall k$$
where $x_{ki}$ are the corresponding elements of $X$.
In order to fulfill the Karush-Kuhn-Tucker conditions, the constraints need to be lin. independent. This is clearly not the case here as the $k$th-row sum is known, given the column sums and $K-1$ other row sums.
So I am wondering: Is it enough to remove one row/column sum and then formulate the problem as usual, i.e. $L(X, \mu, \lambda) = f(X) + \sum_{i=1}^N \mu_i h_i + \sum_{k=1}^{K-1}\lambda_k h_k$ ??