I'm trying to derive a recursive solution to a two objective least squares formulation using QR decomposition.

The cost function is as follows

is the desired vector, is the information matrix, is the weight vector

For derivation purposes i dont think the meaning of the second objective function symbols is necessary other than that is a diagonal matrix.

I can get the least squares solution of this, but am finding it impossible to put into a recursive form, QR or not. Is it even possible to create a recursive multiobjective solution? I can't find any derivations of this anywhere.