Converting between (standard) primal to dual forms (LP) I'm working on a HW assignment as follows:

Given the primal canonical problem: $$min \langle c,x \rangle \text{ s.t. } Ax \geq b, x \geq 0$$
and the canonical dual problem: $$ max \langle b,y \rangle \text{ s.t. } A^Ty \leq c, y \geq 0$$

Show that converting PC (primal canonical) to DC (dual canonical) is similar when done directly or when done via canonical to standard conversion (i.e. primal canonical -> dual canonical == primal canonical -> primal standard -> dual standard -> dual canonical).



I converted the CP to SP (standard primal) by adding slack variables $s_1,\ldots,s_n\geq 0$ and adding them to each inequality.

My problem is showing the similarity between the SP and SD (standard dual).

What I've done so far is this: SD (standard dual) wants to maximize $b^Ty$ so:


$b = Ax$
$b^ty = (Ax)^Ty = (x^TA^T)y = x^T(A^Ty) \leq x^Tc$


Now, I don't know how to proceed. How can I show that minimizing $c^Tx$ is maximizing $b^Ty$? Am I on the correct path?

PS: I am pretty sure dual and primal problems can be shown where the primal is max and the dual is min, but I think it doesn't matter at all, right?
In class you have been shown several algorithms:


Algorithm A: Convert a primal canonical to a dual canonical.
Algorithm B: Convert a primal canonical to a primal standard.
Algorithm C: Convert a primal standard to a dual standard.
Algorithm D: Convert a dual standard to a dual canonical.


Notice that if you run Algorithms B,C,D in sequence, then you get another algorithm for converting a primal canonical to a dual canonical. The question is to compare the dual canonical program constructed this way to the one constructed by applying Algorithm A.