# Linear Programming - Restating a System as a Canonical Primal

1. Sep 3, 2011

### rockofeller

1. The problem statement, all variables and given/known data
State the linear system Ax = b as a canonical minimum problem. What is the dual program?

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

3. 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 ui$\geq$0, vi$\geq$0, st. ui - vi=xi $\forall$xi$\in$x.

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

Then we define a new row vector x* (whose transpose is) [u1 v1 $\cdots$ un vn]. 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?

2. Sep 5, 2011

Any ideas?