# Thread: Multi-objective recursive least squares

1. ## Multi-objective recursive least squares

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

The cost function is as follows

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

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

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

2. Wow. I haven't seen a least-squares problem that had me scratching my head in quite a long time. I will read up on it. I will also appreciate it if you post your results if you beat me to it.

3. It seems the secret lies in writing the cost function as

$A(k) = \begin{pmatrix} X(k) \\ \sqrt{\lambda} G_{XX'}(k) \end{pmatrix}$

$\vec{y(k)} = \begin{pmatrix} \vec{d(k)} \\ \sqrt{\lambda} \vec{h(k)}
\end{pmatrix}$

Then then the cost function can be rewritten

$C(k) = [\vec{y(k)} -A(k) \vec{w(k)}]^2$