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mathematical-optimizationlinear-programming

if A>0 then b=1 else b=0

i would like to formulate a problem in linear programming and i cannot seem to be able to do it. The problem is: If A>0 then b=1 else(if A=0) then b=0,

where b is a binary variable and A>=0(integer).

Any suggestions/recommendations?

Solution

  • Assuming:

    • A is a scalar (notation using A is not great)
    • A is bounded from above by constant M-1

    Constraints:

    (1) A <= M * b
(2) b <= M * A

    Check:

    A = 2
    (1) b = 1
    (2) b free

A = 0
    (1) b free
    (2) b = 0

b = 1
    (1) A free 
    (2) A > 0

b = 0
    (1) A = 0
    (2) A free

    This pretty much follows:

    • the usual indicator-constraints approach like presented here @ page 2
    • Propositional calculus:
      • x == y <-> (x -> y) & (y -> x)