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I am trying to mimimize the following function in order to calibrate the Libor Market Model

$$\sum_{i=1}^{n} \left(\sigma_i^{market}-\sigma_i^{Reb}\left(a,b,c,d,\beta\right)/\sqrt{T_i}\right)^2,$$

where $\sigma_i^{market}$ is given and $\sigma_i^{Reb}\left(a,b,c,d,\beta\right)$ is a nonlinear function in the unknown parameters $a,b,c,d,\beta$.

The parameters should have the following constraints: $$a+d>0$$ $$d>0$$ $$c>0$$ $$\beta \geq 0$$ Which numerical optimizer method in Matlab would be a good candidate such that the constraints are satisfied?

I tried to implement the calibration with the unconstraint local optimizer fminsearch based on the Downhill-simplex algorithm. Nevertheless, with different choices of initial values of the parameters, the algorithm converges to parameters that do not respect the constraints.

So I decided to try a constrained numerical optimizer, the non linear least squares method (lsqnonlin) in Matlab with as lowerbounds 0 for the last three constraints. However, I do not know how to impose constraint number 1 with this method. Does someone have any suggestions?

Would the (unconstrained?) Levenberg Marquardt be a good candidate? Other ideas?

Thank you in advance.

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You might want to set $a= \epsilon - d$ and write $\epsilon>0$ as a constraint. I guess $\textbf{lsqnonlin}$ is the suitable fonction for what you intend to do. I personnally like to use and play around with $\textbf{fmincon}$, which allows more flexibility and performs well, if you are willing to provide Jacobian and/or Hessian in algorithms options

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