I'm trying to develop an inferential procedure for a multivariate dependent Markov process. Basically, the procedure could be considered as a non linear regression, with a known dependence structure among observation belonging to the same time-point and independent from the others. The non linear trend correspond to a the solution of a matrix differential equation. A more detailed description follows:

Let define a matrix $\Theta$ of parameters $\theta_i$ \begin{array}{ccc} \theta_1 & \theta_4 & \theta_7 \\ \theta_2 & \theta_5 & \theta_8 \\ \theta_3 & \theta_6 & \theta_9 \end{array} each entry $\theta_i$ could be unconstrained or linearly constrained (both equality and inequality). This matrix, multiply by a given, fixed matrix $V$ and a vector of previous time point $\vec{X}$ observation, governs the derivative of a stochastic process. As a consequence, in my objective function $\sum(\vec{Y}-f(\Theta,\vec{X}))^2$, I need to calculate the solutions of the matrix differential equation, $f(\Theta,\vec{X})$. To calculate the solutions, I need to calculate a matrix $A$, combining the entries of $\Theta$ according to $V$, calculate eigen decompositions of $A$ and finally the solution $\hat Y=f(\Theta,\vec{X})$ (exponential).

The objective function is non-convex and has many local minima. Runnig some simulations studies, I verified that the global minimum is located at the true parameters values, and the objective is convex in the closed neighbourhood. Until now, I used a constrained Gauss-Newton method to estimate parameters, and, given good initial values, it works. Relaxing some conditions, the method to calculate initial values is not good enough to guarantee the Gauss-Newton method to converge to global minimum (it get stucks in a closed local minimum). So, I'm now looking for a global optimizer. All my code is in R and RCplex and I have some experience with C++. Is Cplex able to solve non-convex,linearly constrained problem and flexible enough to allow me for solve the matrix differential equation? Are there any alternative program? Any suggestion? Thanks.


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