Consider $m$ points $(x_i ,y_i )$ in the plane that can be approximated by a curve of the form $y=\frac {c_0+c_1x+c_2x^2}{d_0+d_1x+d_2x^2}$. Suppose that $||(c_0,c_1,c_2,d_0,d_1,d_2)^T||_2=1$.
I need to find an optimization problem of a quadratic form that gives the coeffecients vector $c$.
I know that once I have a symmetric matrix $A$ then $min{\{c^TAc\}}$ over all possible $c$ such that $||c||=1$ is the smallest eigenvalue of $A$. How can I use this fact to find the optimization problem?
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First of all, unless $m < 6$, your problem is overdetermined, so it is not possible to solve for coefficients that cause the curve to fit perfectly.
So, to find an approximate fit, you will need to specify which error you want to minimize in the fit. One common choice is the least-squares error $$E = \sum_i \left(y_i - y(x_i)\right)^2$$ however I see no way to turn minimization of $E$ into a quadratic optimization problem. You can instead define the error to be $$\tilde E = \sum_i \left([d_0+d_1x_i + d_2x_i^2]y_i - [c_0+c_1x_i+c_2x_i^2]\right)^2$$ which is a quadratic form.
Continuing from user7530's answer, first of all, in order to avoid degeneracy, you will need to impose one of the coefficients (say, $d_0=1$).
The model being nonlinear with respect to the parameters, you will need to use nonlinear regression which will require "reasonable" estimates. To get these, you can linearize the model doing what user7530 suggested. This corresponds to a multilinear regression $$y=c_0+c_1x+c_2x^2-d_1 xy-d_2x^2y$$ When this is done, you are ready for the nonlinear regression which is required since what is measured is $y$ and not any of its possible transforms.