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Homework Statement
(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.
Homework Equations
Linearizing absolute expressions:
z = | x |; where z >= x, z >= -x
Linearizing free variables:
let x = x+ - x-; where x+, x- >= 0 ---- (1)
The Attempt at a Solution
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(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 )
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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
... ?