How to derive the dual for LP like this?

Question

I know how to derive dual for normal LPs, but what if we are unlikely to have something like this:

maximize z
s.t.   z < 3y-2
       1 < y < 2

, where the constraints are not directly related to the objective but related indeed. In this case, how can we compute dual for it?

Currently, the way I solve it is to ignore the second constraint and convert the original one into:

minimize -z
s.t.   z < 3y-2
       1 < y < 2

Firstly, I have this Lagrangian:

$L(z,\lambda) = -z + \lambda (z-3y+2)$.

Then take derivative w.r.t. $z$, and set it to be $0$:

$\frac{\partial L}{\partial z} = -1 + \lambda = 0$.

Finally, plugging in $\lambda=1$ into Lagrangian, we have:

$g(\lambda) = L(z*, \lambda) = -3y+2$.

Even if $g$ is not a function in terms of $\lambda$, but let's just write it in this way. Then I don't what to do next.

Thank you very much!

Answer 1

Rewrite as:

maximize 0y + 1z
s.t.    -3y + 1z <= -2
        -1y + 0z <= -1
         1y + 0z <=  2

Then the dual is:

minimize -2t - 1u + 2v
s.t.     -3t - 1u + 1v = 0
          1t + 0u + 0v = 1
           t,   u,   v >= 0

Equivalently:

minimize -u + 2v - 2
s.t.     -u +  v = 3
          u,   v >= 0

Equivalently:

minimize  u + 4
s.t.      u >= 0

Hence $(t,u,v)=(1,0,3)$ is the unique optimal dual solution, and complementary slackness yields optimal primal solution $(y,z)=(2,4)$.