I have an overdetermined linear problem of the form $A x = b$, which is solved in least squares sense using the Moore–Penrose pseudo invers. The issue now is, that over time additional constraints and thus equations arise.
Is there an algorithm for iterative update of the LS-solution w/o complete recomputation of $A^+$?
There are methods for updating the pseudo-inverse. Specifically, I'm familiar with rank-one updates, that is the case where $A' = A+uv^T$ for vectors $u$ and $v$ of appropriate dimensions.
The Matrix Cookbook has the procedure descrbied at section 3.2.5.
Two papers that deal with this problem more generally (courtesy of Wikipedia):
Meyer, Carl D., Jr. Generalized inverses and ranks of block matrices. SIAM J. Appl. Math. 25 (1973), 597—602
Meyer, Carl D., Jr. Generalized inversion of modified matrices. SIAM J. Appl. Math. 24 (1973), 315—323