Linear Programming double inequalities

[zcd]

Homework Statement

Find the dual of
$$-d \leq Ax-b \leq d$$
$$x \geq 0; c \cdot x = min$$
where A is mxn matrix and $$x,d,b \in \mathbb{R}^n$$

Homework Equations

dual of canonical is of the form
maximize $$b \cdot y$$
$$A^{T}y \leq$$
where $$y \in \mathbb{R}^m$$

The Attempt at a Solution

I tried converting it to the canonical LP and then applying transpose to A, but it turned out to be a huge mess; is there a simpler way?

 

[Ray Vickson]

Homework Statement

Find the dual of
$$-d \leq Ax-b \leq d$$
$$x \geq 0; c \cdot x = min$$
where A is mxn matrix and $$x,d,b \in \mathbb{R}^n$$

Homework Equations

dual of canonical is of the form
maximize $$b \cdot y$$
$$A^{T}y \leq$$
where $$y \in \mathbb{R}^m$$

The Attempt at a Solution

I tried converting it to the canonical LP and then applying transpose to A, but it turned out to be a huge mess; is there a simpler way?

Do you know how to write the dual of an LP of the form
$$\min \; c \cdot x$$
s.t.
$$F x \geq f$$
$$x \geq 0 ?$$
Just re-write your LP in that form and use what you already know.

RGV

 

[zcd]

How would I combine two separate matrix inequalities to one?
$$Ax \geq b-d$$
$$-Ax \geq -b-d$$

 

[Ray Vickson]

How would I combine two separate matrix inequalities to one?
$$Ax \geq b-d$$
$$-Ax \geq -b-d$$

You've just done it. Can't you see what the matrix F is?

RGV

 

[zcd]

My guess would be to make a block matrix on the left with A and -A vertically and a block vector with b-d and -b-d on the right. Could it be this simple?

 

[Ray Vickson]

My guess would be to make a block matrix on the left with A and -A vertically and a block vector with b-d and -b-d on the right. Could it be this simple?

Yes.

RGV