A maxdet problem with three variables - maximum likelihood estimation in Mathlab/Mata/R Problem
Given $n$ known vectors $y_j = (y_1, y_2, ..., y_{K_j})', \forall j=1,n$ and a constant $K$; determine three vectors ($\alpha, \beta, \theta^2)$ that maximise:
$$
\text{T}  (\alpha, \beta, \theta^2) = \sum\limits_{j=1}^{n} [\log(\det \Omega) + (y_j - \alpha)'\Omega^{-1}(y_j - \alpha)] 
$$
where $\alpha = (\alpha_1, \alpha_2, ..., \alpha_{K_j})'$, $\beta = (\beta_1, \beta_2, ..., \beta_{K_j})'$ and $\theta^2 = (\theta^2_1, \theta^2_2, ..., \theta^2_{K_j})'$; $\text{B}$, $\Theta$ and $\Omega$ are three $K_j \times K_j$ matrices as: $$\text{B} = \begin{bmatrix} \beta_1 & 0 & ... & 0 & 0 \\ 0 & \beta_2 & ... & 0 & 0 \\ ... & ... & ... & ... & ... \\ 0 & 0 & ... & 0 & \beta_{K_j} \end{bmatrix}$$ $$ \Theta = \begin{bmatrix} \theta^2_1 & 0 & ... & 0 & 0 \\ 0 & \theta^2_2 & ... & 0 & 0 \\ ... & ... & ... & ... & ... \\ 0 & 0 & ... & 0 & \theta^2_{K_j} \end{bmatrix} $$ $$ \Omega = \beta \beta' + B \Theta B' $$
Application
This maximisation is used to solve the Unobserved Component Models. I happen to know that if $\beta_i \beta_j = 0$ then Stata can run GLS to maximise $\text{T}$ given the $n$ vectors $y_j$. 
In a general case, I am not aware of any mathematical/statistical programme that can yield/perform the algorithm to maximise $\text{T}$. If anyone happens to come across this type of problem, please let me know too. 
I am from economics major and just want to understand the algorithms behind the maximum-likelihood estimators. I find this problem from a working paper of the World Bank and attempt to tackle. 
Thank you!
 A: I think you need nonnegativity constraints on all the parameters (variables).  MAXDET by Boyd et al, and its later more flexible evolution CVX http://cvxr.com/cvx/ , can NOT be used because even though the log determinant term is concave, the other term is convex, hence, you have a "nasty" non-convex problem. There could be multiple local maxima, and there may even be local minima and saddle points. Different starting points (initial values (guesses) for the parameters could result in different solutions.
You can not maximize one term at a time. It will not work.
Forget about maximum likelihood estimation software, and use a general purpose local nonlinear optimizer.  You need to be able to evaluate the objective function.  That is easy to do in MATLAB.  If you have the Optimization toolbox, you could use FMINCON. Because FMINCON only minimizes, you'll have to minimize the negative of the function you want to maximize.  Be sure to specify the lower bound constraints.  If you don't know how to evaluate the gradient, you can tell the optimizer to use finite differences for the gradient, but that will make it take longer if there are a lot of parameters to be estimated. If you can get access to KNITRO http://www.ziena.com/knitro.htm , use that instead of FMINCON, because even though the algorithms are nominally basically the same as FMINCON, it's a much higher quality implementation, which could make the difference between solving it in a minute in KNITRO vs. having FMINCON labor for hours or days and still not solve it (on the other hand, FMINCON might work as well on this problem, but likely not). You should try several different starting values.
You could also call an optimizer from C/C++, Python, Fortran, or maybe R.
Also, the question (thread) title is misleading. It is NOT a maxdet problem.  If if were a maxdet of a positive definite matrix problem, with no other term in the objective function, then it would be a concave maximization, which is considered to be a convex optimization problem, which is easy and fast to formulate and solve (unless there are several thousand or more parameters being estimated) using a tool such as CVX. But because you are adding a convex function to maxdet, it is decidedly non-convex, and not so easy, and neither maxdet nor CVX can be used at all.
If you want to understand how MLE and nonlinear optimization algorithms work, and how they SHOULD work, take a course in nonlinear optimization. 
