(adsbygoogle = window.adsbygoogle || []).push({}); 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[itex]\geq[/itex]0, c[itex]\bullet[/itex]x=min.

3. The attempt at a solution

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

define u_{i}[itex]\geq[/itex]0, v_{i}[itex]\geq[/itex]0, st. u_{i}- v_{i}=x_{i}[itex]\forall[/itex]x_{i}[itex]\in[/itex]x.

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

Then we define a new row vector x^{*}(whose transpose is) [u_{1}v_{1}[itex]\cdots[/itex] u_{n}v_{n}]. Then x^{*}is 2n[itex]\times[/itex]1 and our new constraints are A^{*}x^{*}= b, x^{*}[itex]\geq[/itex]0.

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

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# Linear Programming - Restating a System as a Canonical Primal

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