Gaussian process box constrained optimization I am have been trying to optimize the Gaussian process likelihood function (multivariate Gaussian likelihood) in R (optimx and nloptr) while having some box constraints for my hyper parameter estimates. 
I am facing the problem of the solution rushing to the edges of the parameter space. Why this happening is? And what is a good optimizer in R that is good for constrained R optimization?
 A: A strong and relatively straightforward routine to use for box-constraints is BOBYQA; it is available in R through the minqa package. It is the default optimiser for the lme4 package when it comes to box-constraints for the evaluation of (generalised) linear mixed models deviances ($-2$ log-likelihoods) so I suspect it will work well with the GP likelihood evaluations too. (I have used for this GP optimisation tasks successfully at some point but I actually found that re-writing my problem in an unconstrained form was more beneficial.)
As it has already been commented the fact the optimisation algorithm rushes to the boundaries it might well be due to the boundary values offering optimal values. I would suggest generating some data that you know coming from a known GP where the optimal parameters are not near the boundaries and then check your optimisation routine's behaviour. Please note that if the optimal parameters are indeed near the parameter space boundaries some asymptotic results might not hold.
