I am working on an minimization algorithm of the form
\begin{equation} S = \sum \dfrac{1}{2} r_i^2 \end{equation}
with $r_i = y_i - f(x,\beta)$ is the residual. Here, $f(x,\beta)$ is the Euclidean distance formula:
\begin{equation} f(x,\beta) = \sqrt{(x_i - \beta_1)^2 + (y_i - \beta_2)^2 + (z_i - \beta_3)^2} \end{equation}
For such problem, using algorithms like Gauss_newton method, the summation saturates over time and a threshold, $\epsilon$, can be used to terminate the iterative process.
I need to estimate the position coordinates $\{\beta_1,\beta_2,\beta_3\}$ for this system. This is an unconstrained optimization for now. Let's say, I would like to increase accuracy in the estimation of $\beta_1$ and $\beta_2$, and do not really care about the error in $\beta_3$. Maybe for some reason, it is not possible to predict $\beta_3$ properly by using any measurement since, because of the process physics, it is not possible to know it accurately. How do I build in this requirement in my algorithm?
One way to do it is to constrain $\beta_1$ and $\beta_2$ such that the estimated parameters are bounded within a certain zone while leaving the unimportant variable to take up any value, thus, making it a constrained optimization problem. But I think there should be another more rigorous way to solve such problems.
Note: I cannot change the $f(x,\beta)$ to not include the $3^{rd}$ dimension since the measurements are in $3$-D space (basically the maths/physics involved in the model does not allow for the unimportant variable to be dropped). A discussion would also help!
Edit: Regarding the accuracy of the system, it has two solutions but if you add constraint e.g. $\beta_i > 0$, then it has just one global minima.