# Solve non-linear Optimization Problem [closed]

I have to find $x$ that minimizes: $$\sum_{k}(x^H\textbf A_kx - b_k)^2$$ where $A_k$ are 4 x 4 positive definite matrices ($A_1, A_2,...A_k$), $x$ is 4 x 1 vector and $b_k$ are scalars ($b_1,b_2,...b_k)$. Can anyone give me some suggestion/hint inorder to solve this OP ?

## closed as off-topic by Wolfgang, Stefan Kohl, Chris Godsil, Jan-Christoph Schlage-Puchta, Ilya BogdanovDec 4 '15 at 16:46

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• You will probably have to use a numerical approach like Gauss-Newton or BFGS – Doc Brown Dec 4 '15 at 9:20
• Thank you for your suggestion. Since this nonlinear OP will have multiple local minima (multiple $x$ that would solve the OP), is it possible to solve in a way that $x$ is a global minimum? – zsha Dec 4 '15 at 9:45
• Since you did not mention any constraints: can one safely assume there are no constraints for $x$? – Doc Brown Dec 4 '15 at 10:57
• Yes, there are no constraints for $x$ in general (the only condition is that $Rank(x^Hx)=1$, which is automatically satisfied based on definition of the OP so I haven't consider that as a constraint). I am looking for an approach that would yield an $x$ which is a global minimum. Any suggestions on this would be appreciated. – zsha Dec 4 '15 at 11:19