By the help of lagrangian multipliers, I am solving a nonlinear problem with 6 variables using KKT analysis. At the beginning, I do not consider any upper-bound for the variables to see whether the optimal values are in the bounds or not. What is the best way to deal with this problem, when after solving it, I face that only one variable is out of its bound?
- Should I add all of the upper bounds of the variables as constraints of the problem with their corresponding lagrangian multipliers before starting to solve the problem?
- Should I add only the corresponding constraint of the variable which is out of its range with its only one lagrangian multiplier and solve the problem again?
- Should I set the value of that variable at its upper bound and then solve the new problem with 5 variables?
I will appreciate it if you provide a reference for your answer.
Bounds on the variables should be dealt with in the same way as other constraints. Even though only one variable is out-of-bounds, forcing that variable to be in-bounds may cause other variables to hit their bounds. So although there's nothing wrong with trying (2) (if you get an optimial solution of (2) where all variables happen to be in-bounds, it's an optimal solution of the actual problem), it's not guaranteed to work.
(3) might not work and is more dangerous: an optimal solution to the new problem might not be an optimal solution of the actual problem, and you wouldn't know it just from looking at the solution.