I used for a non-linear optimization problem the objective function O.F.=Σ(xs-xo)^2, where xs is the simulated value and xo is the observed. The algorithm runs adequately but I would like to know if this objective function is limited only for linear problems or whether it can also be used for non-linear problems.
Note that excel uses (or at least used to apply) simplex optimization algorithm, which may not be a good idea for non-linear and/or non-convex problems. So the main problem could be the simplex optimization, i.e. not objective function.
The solver tool in excel uses the generalized reduced gradient method and in the new versions there is an option where you can choose linear or non-linear solving and other properties (see the references). The new tool has been verified even for non-linear cases and in my case works very well, the R2 between observed and predicted is 0.98. My concern from theoretical point of view is if the objective function that I use is only for linear problems.
Lasdon, L.S. and Smith, S. 1992. Solving large sparse nonlinear programs using GRG. ORSA Journal on Computing 4(1), 2-15.
Fylstra, D., Lasdon, L., Watson, J., Waren, A., 1998. Design and use of the Microsoft Excel solver. Interfaces 28(5), 29-55.
Lasdon, L.S.; Waren. A. D.; Jain, A.; and Ratner, M. 1978, "Design and testing of a generalized reduced gradient code for nonlinear programming," ACM Transactions on Mathematical Software, Vol. 4, No. 1, pp. 34-49.
Generally, an objective function is only related to your goal or that your concern, and it has little relation with the properties of the problem, which means that both linear and non-linear problems can employ the function you mentioned.
similar to MengChang, it seems to me that the least squares approach you apply is rather general and does not depend on linear or non-linear behaviour behind.
Anyway, you may want to add or refine you model (e.g. by adequate weighting or selection of sampled values) to avoid biasing your results towards a specific region or response model.
I mean, since you have your simulated values, you already have a model for the response and you know how linear or non-linear (i.e. which other pattern) it is.
Being this the case you can select also non-linear sampling, or weight differently (lower weights) for the sampled regions leading to similar responses.
Ultimately, you may also linearize your problem, and check for consistency on your solver results (which I also use quite often).
Quadratic objective function is better in Excel. If the degree is more than 2 better use either Mathematica or LINGO. But Lingo non linear solvers are not good enough o give optimal solutions
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