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Math Help - Multi-objective recursive least squares

  1. #1
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    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.
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  2. #2
    MHF Contributor
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    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.
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  3. #3
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    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)}<br />
\end{pmatrix}

    Then then the cost function can be rewritten

    C(k) = [\vec{y(k)} -A(k) \vec{w(k)}]^2
    Last edited by Superfish; May 11th 2010 at 07:57 PM.
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