I have a question that I've always wondered about concerning the "L-BFGS-B" algorithm. I am not familar with the details of the algorithm except for the fact that it optimizes a non-linear function subject to box-constraints. My question is:
Given that box constraints of the form $a \le θ \le b$ can always be eliminated by introducing a transformed variable say thetaprime where thetaprime meets the constraints ( atleast for intervals. I realize that more complex constraints can't be handled ) why would the "L-BFGS-B" algorithm or any algorithm that handles box-constraints ever be necessary. Thanks for any wisdom.
For non-equal constraints, the dual multiplier or Lagrangian multiplier (Lambda) introduced here also need to be $\lambda \geq 0$ or $\lambda \leq 0$; they are box constraints too.