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

The cost function is as follows

$\displaystyle C(k) = [\vec{d(k)} - X(k)\*\vec{w(k)}]^2 + \lambda \* [\vec{h(k)} - G_{XX'}(k) \* \vec{w(k)}]^2 $

$\displaystyle \vec{d}$ is the desired vector, $\displaystyle X$ is the information matrix, $\displaystyle \vec{w}$ is the weight vector

For derivation purposes i dont think the meaning of the second objective function symbols is necessary other than that $\displaystyle G_{XX'}(k)$ 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.