Linear Programming - Restating a System as a Canonical Primal

[rockofeller]

Homework Statement

State the linear system Ax = b as a canonical minimum problem. What is the dual program?

Homework Equations

The canonical minimum problem is Ax = b, x$\geq$0, c$\bullet$x=min.

The Attempt at a Solution

I'm confused here, in part because there is no objective function c$\bullet$x=min. So far, I have:

define u_i$\geq$0, v_i$\geq$0, st. u_i - v_i=x_i $\forall$x_i$\in$x.

Then, if A is m$\times$n, define a new matrix A^* with elements a^*_{$\alpha\beta$} = a_i(2j) for $\beta$ even, a_{i($\frac{J+1}{2}$)} for $\beta$ odd. Then A^* is an m$\times$2n matrix.

Then we define a new row vector x^* (whose transpose is) [u₁ v₁ $\cdots$ u_n v_n]. Then x^* is 2n$\times$1 and our new constraints are A^*x^* = b, x^*$\geq$0.

Have I gotten this "right" so far? How do I come up with the new objective function?

 

[rockofeller]

Any ideas?