If strong duality holds in the dual problem and primal problem then while taking the Lagrangian can we ignore the constraints of the type $x_i\geq0$ where $x_i$'s are the optimization variables.
My Reason for The above:
Suppose if we do not ignore and just assume that $\lambda_i$'s (dual variables) associated with $x_i$ are greater than $0$ then it would imply that the Langrangian has a non-zero derivative at the optimal point while the primal problem has a zero derivative at the optimal point.
But this reasoning gives me dual variables which are equal to zero although I think if the strong duality holds than the dual variables should have strictly positive values. Any help in clearing this confusion will be much appreciated. Thanks in advance.
Consider $$\min x_1 + x_2$$ subject to $$x_1 \geq 0, x_2 \geq 0$$
We know that strong duality holds for linear programming. However, removing the sign constraint would change the question such that the objective value can be as negative as possible.
Hence, we can't remove the sign constraint simply because strong duality holds.
Note that dual variable is associated with primal constraint rather than primal variable.