I'm looking to solve some kind of generalized nonlinear least squares problem, I think.
So for some background, lets say I have an ordinary nonlinear least squares problem. That is, a set of data points ${(x_i,y_i)}$, an objective function $f(x,\beta)$ where $\beta$ is a vector of parameters that I wish to find out in order to minimize $\sum [y_i - f(x_i,\beta)]^2$. Now I can use some appropriate algorithm, e.g. Levenberg–Marquardt and I'm done.
However, my problem now is that I have several sets of data points,say $(x_{ij},y_{ij})$ (my objective function is the same for all sets, tough) and I want to find a single vector $\beta$ that minimizes (in a least-squares sense, say) the residual over all these data sets. So how do I go about solving this? Is the best way just to combine all my sets into one and then do a normal nonlinear least squares fit?
EDIT: More info about the problem
I have a bunch of "components", lets call them A, B, and C. I also have the functional form $f(x,\beta)$ that each component should follow, but I need to figure out the vectors $\beta_A,\beta_B,\beta_C$. Now each set of data points does not contain a component in isolation, but a combination of them, e.g. one data set for A+B, another set has A+B+C, a third B+B+C, and so on. Note that the "+" in the previous sentence is not exact; I do know that the contributions are approximately additive. So from these data sets I want to figure out an approximation of the contribution of each component.
EDIT #2: "Solution"
In the end I just did the simple approach and combined all my sets into one. My Jacobian ended up looking a bit funny (sort-of blocked) but it all worked out decently in the end. Thanks for all the suggestions.