I only found a example in which the constraind on $\beta_i, \ i=1,...k$ would make them sum up to one as well introduces a constraind such that all $\beta_i$ are all greater or equal zero. Since I dont need a condition for the summation of the $\beta_i$ I'm not quite sure how to deal with this problem.
Since in this context a matrix $C$ of constraints is introduced such that $C\beta = c$, I'm not sure how to deal with $C\beta \geq 0$.
Any suggestions are very welcome!