I have a (maybe trivial?) question on constrained optimization. Assume that I have the following maximization problem:
I setup the Lagrangian and I get the following first order conditions with the lagrange multiplier .
By the Kuhn Tucker conditions, we know that if , then the constraint is binding. However, from the first order conditions, we can see that the multiplier is negative! How can this be? Am I doing things wrong here?
Because to me, it feels like we can also say that the unconstrained maxima of the objective is outside of the feasible region, right?
Also, if I changed the constraint to
the first order conditions will still look the same, but clearly, the optimum has violated the constraint.
Please give me some pointers