What is the best numerical optimizer for this minimization problem?

I am trying to mimimize the objective function in the attachment in order to calibrate the Libor Market Model, where the market sigma is given and sigma(a,b,c,d,beta) 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 >= 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.

Answer 1

[Alan Weiss]

If you have Optimization Toolbox™, as I suppose you do because you used lsqnonlin, then take a look at the Optimization Decision Table. There you find for nonlinear functions with linear constraints you should use fmincon.

Alan Weiss

MATLAB mathematical toolbox documentation