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Linear Optimization Conversion (nested free variable)

  1. Sep 12, 2016 #1
    1. The problem statement, all variables and given/known data
    (This is a mock-up)
    Given
    maximize(x,y) Ax + | x - y |

    s.t.
    x + By - C >= 0
    y >= 0

    A, B, and C are some coefficients.

    2. Relevant equations
    Linearizing absolute expressions:
    z = | x |; where z >= x, z >= -x

    Linearizing free variables:
    let x = x+ - x-; where x+, x- >= 0 ---- (1)

    3. The attempt at a solution
    -------------------------------------------------------------------------------------------------------
    (I think?) x is not constrained like y (as in y >= 0),
    Let x = x+ - x-; where x+, x- >= 0

    To linearize the expression | x - y |
    Let z = | x - y |; such that z >= ( x - y ) and z >= -( x - y )

    So... From (1):
    Let z = | x+ - x- - y |; such that z >= ( x+ - x- - y ) and z >= -( x+ - x- - y )
    -------------------------------------------------------------------------------------------------------
    Therefore, the LP for this problem is
    maximize(x,y) A(x+ - x-) + z

    s.t.
    x+ - x- + By - C >= 0
    y >= 0
    z >= ( x+ - x- - y )
    z >= -( x+ - x- - y )
    x+, x- >= 0

    ... ?
     
  2. jcsd
  3. Sep 17, 2016 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
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