Is there a way to change a strict inequality (e.g. >) into a non-strict one? (e.g. greater than or equals to)?
If not, how would I deal with this problem? I have been attempting this and reached the following answer but I am unsure if this is correct.
I am trying to solve the following basic quadratic program for a constrained optimisation problem:
min f(x) = 2(x1)^2 + 4(x1)(x2) + 4(x1) + (x2)^2
subject to x1 > 1
The method I used to solve this was using the Lagrangian Function & Kuhn-Tucker conditions and I got an answer of:
(I changed the constraint to x1 - 1 > 0 and considered it as an equality constraint)
x1 = 1
x2 = -2
min f(x) = 2
\lambda (lagrangian multiplier) = 0 therefore, constraint is inactive.
Since my solution is x1 = 1, I am unsure if this solves the problem, as the question requires x1 > 1 which is clearly not the case.
Is there a way I can change the strict inequality constraint x1 > 1 into a normal inequality constraint? (Which is the form I am usually used to dealing with, and have had no issues solving) This is the first time I have come across this sort of issue and have tried to solve it but with no success!
Thank you very much for your help.
The minimum does not exist because $$f=(2x_1+x_2)^2+4x_1-2x_1^2\rightarrow-\infty$$ for $x_2=-2x_1$ and $x_1\rightarrow+\infty.$