How to perform a constrained optimisation of a log likelihood function

Question

I have 3 binomial random variables $x_1, x_2, x_3$ with parameters $\theta_1, \theta_2, \theta_3$

If I have a log likelihood function $L = \sum_{i=1}^{3} (\log(^{N_i} C_{x_i} \times \theta_i^{x_i} \times(1-\theta_i)^{N_i-x_i} ) $

I need to maximize this function subject to constrain that $\theta_1 < \theta_2 < \theta_3$

Is this something which can be done using some known method like simulations or Bayesian approaches? I know about simulated annealing method but how do I make constraints in that algorithm?

Answer 1

You are effectively maximizing $\sum_{i=1}^3 x_i log(\theta_i) + (N_i-x_i)\log(1-\theta_i)$ which is convex in $\theta$. Hence any standard nonlinear solver should work fine, and if you want to be extra fancy you can use a dedicated exponential-cone solver.