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We need to solve the following least square problem $$\min_x (Y-Ax)^TW(Y-Ax)$$ $$s.t. x^TA^TAx=1$$ $$c^Tx=0$$ in a closed form, where $Y \in \mathbb{R}^{n\times 1}$, $A \in \mathbb{R}^{n\times n}$, $W \in \mathbb{R}^{n\times n}$, and $c \in \mathbb{R}^{n\times 1}$ are given.

Indeed we try to solve this optimization problem by employing Lagrange multipliers and obtaining the following equations $$A^TWAx+{\alpha}A^TAx=A^TWY-{\frac{1}{2}}{\beta}c$$ $$x^TA^TAx=1$$ $$c^Tx=0$$ where ${\alpha}$ and ${\beta}$ are Lagrange multipliers. However, we failed to solve these equations analytically.

So, can the above optimization problem be solved analytically? If not, what method should be used to deal with this optimization problem? Can anyone help us? Thanks a lot!

user1034188
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    If $A$ is full rank, you can simplity the problem a little bit by setting $z=Ax$ and solve in $z$ first. – Damien Mar 09 '22 at 13:27
  • You can find some comments on numerical tools in this thread https://mathoverflow.net/questions/294606/nearest-matrix-orthogonally-similar-to-a-given-matrix. Like this link https://www.manopt.org/tutorial.html#costdescription. You can find more on SearchOnMath. – José C Ferreira Mar 09 '22 at 16:47

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