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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.

mark leeds
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  • $\theta$ are not equality constraints so how do you transform it to always meet the constraint? Suppose there's $\theta'$ that always satisfies the constraints, then there could be regions of $\theta$ that does not exist on $\theta'$ but could have contained more optimal solutions. In other words, limiting it to $\theta'$ will give a suboptimal solution to the original problem. – syockit Apr 28 '23 at 07:35
  • @syockit. thanks for your comment. would you mind providing a simple example because I don't see what you are getting at but it sounds useful and enlightening. thanks. – mark leeds Apr 29 '23 at 12:03

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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.

Apurv
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amaoagou
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    Hi Apurv: Thanks for your reply. Unfortunately, I don't follow it because for the problem I described, lagrange multipliers ( i.e: $\lambda$ ) are not involved. – mark leeds May 01 '15 at 05:20
  • This answer seems to be a non-sequitur. I don't understand what it has to do with the question. Can you edit it to explain what connection you had in mind? – D.W. Jul 12 '16 at 05:45