Leastsquares sensitivity analysis

Is there any way in CVX that I can modify the leastsquares problem from A1.x1=b1 to A2.x2=b2
were A1 is part of A2 and b1 is part of b2
without solving the problem from the beginning
i.e add more constraints and use the first solution x1 to reach the new solution x2

No, there is not. CVX does not provide the ability to modify existing models or start from a previous point.

To the best of my knowledge none of the interior-point optimizers CVX can do warmstart.
Anders Skaaje, Yinyu Ye and myself have written a paper documenting warmstart is possible.
It should be in the next release of MOSEK and then maybe CVX can use it too. Doing so however is a nontrivial in CVX setting I believe.