Comparing dual of a canonical primal program - Directly and by dual of the standard program

Question

I have it as a homework question to compare dual programs in the following way:

1. Take a canonical program and write its dual
2. Take the same canonical program, write it as a standard program, take the dual of the latter, then write it as a canonical program.

In the following image, option 1 is starting with the upper left square and going right, and option two is (starting with the same place and) going down and then right and then up.

I have tried to do this, both in general and both by example, but I'm getting something strange: First I add slack variables to move from inequalities to equality, and I add the constraint that all slack variables are positive.

Then I write the dual of that program, the inequality look like $[A|I]^Ty\leq [c|0]$

(here $[A|I]$ denotes a block matrix) which gives two sets of inequalities: one is $A^Ty\leq c$ and the other is $Iy\leq 0$. Which is almost like the CD form in the picture - but I got $y\leq 0$ instead of $y\geq 0$.

Why do I get a different result ? I guess I have some error and that I should actually get $y\geq 0$, but I checked multiple times and I don't see any errors.

Answer 1

Since $Ax\ge b$, you should make $Ax$ smaller to get equality. Slack variables are also non-negative, so the coefficients should be $-1$. It's $[A|-I]$ instead of $[A|I]$.