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Linear Programming - Restating a System as a Canonical Primal

  1. Sep 3, 2011 #1
    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 ui[itex]\geq[/itex]0, vi[itex]\geq[/itex]0, st. ui - vi=xi [itex]\forall[/itex]xi[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] = ai(2j) for [itex]\beta[/itex] even, ai([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) [u1 v1 [itex]\cdots[/itex] un vn]. 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?
  2. jcsd
  3. Sep 5, 2011 #2
    Any ideas?
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