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.
non convex, non linear optimization involving matrix differential equation solution
eigenvalues;constrained optimization;nonlinear programming;stochastic;nonconvex