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I want to optimize adjust the parameters $\alpha, \beta, \gamma, \sigma²$ for a nonlinear regression model with the Maximum Likelihood estimator through Simmulated annealing, this is the model:

$$y = \alpha X_1 + \frac{\beta}{X_2 -\gamma} + \epsilon , \epsilon \sim N(0, \sigma²)$$

I know that simmulated annealing is an optimization method, and in order to optimize the MLE for this model I need to build a target function, but I have no idea on how to build the target function for this case, how should I do it? Also, after having a target function I'll need to generate the SA algorithm, but I also don't have any idea on how should I do it. Any hints or advices are more than welcome.

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  • $\begingroup$ Why do you want to use simulated annealing for this problem, as compared to any of a number of algorithms that are likely to be much faster? $\endgroup$
    – jbowman
    Commented Jul 3, 2022 at 16:53

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The objective function in non-linear least squares for fitting $y\approx f(\vec{x})$ is $\sum_i(f(\vec{x}_i) - y_i)^2$, which is in your case $$Q(\alpha,\beta,\gamma) = \sum_{i=1}^n \left(\alpha x_{i1} +\frac{\beta}{x_{i2}-\gamma} -y_i\right)^2$$ As jbowman commented, for this particular (well behaved) objective function, there are better algorithms than simulated annealing. See, e.g., the docs of the GNU Scientific Library for an overview.

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