Linearization of a Fractional Expression

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Lennard Hansmann

March 10, 2025 11:58

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How can I linearize the expression (x * l - y) / y when x is a binary variable, l is a parameter, and y is an integer variable with a known range ymin ≤ y ≤ ymax ?

I hope that linearizing this expression will help me solve the problem more efficiently.

 

Related to

• Modeling
• Nonlinear

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  Ronald van der Velden

  • Gurobi Staff

  March 10, 2025 12:08

  Hi Lennard,

  My first reaction would be that you could separate (x * l)/y from y/y=1. Then the question is where this expression would be used - in a constraint or the objective function? For a constraint, what's on the other side of the (in)equality sign?

  Kind regards,
  Ronald

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  Lennard Hansmann

  March 10, 2025 12:33 Edited

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  It is a constraint that calculates a penalty for placing an item of length l in a box of variable length, where the variable part of the penalty depends on the degree of overfilling of the box. The penalty is represented by c, which is a real-valued variable.
  Kind regards,
  Lennard

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  Ronald van der Velden

  • Gurobi Staff

  March 11, 2025 10:38 Edited

  Ah, the fact that there's a variable on the left side does not allow a quick fix. Still, you could try multiplying both sides by \[max_{j,l,k}^{box}\] to see if this improves performance. I'm assuming you've explicitly set the bounds on that variable, which could also help.

  Kind regards,
  Ronald

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