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Fraction with quadratic numerator as objective function

Answered
Tobias Tillipaul
  • Gurobi-versary
  • First Question

Hi all,

I am currently trying to implement the following expression as objective function to my model (a, b being vectors of constant values):

I found out that I could construct the first part of the numerator as GRBQuadExpr, but subsequently I figured I could just express the second part of the numerator and the denominator as GRBLinExpr and add the pieces together - and failed to do so. Moreover, there is no "/" operator defined for GRBQuadExpr and GRBLinExpr, thus I am wondering if Gurobi has the means to construct such an expression and add them to models. Is there a predefined way to create fractional expressions or a common workaround that allows to use this expression or an abstraction of it?

Thanks in advance for any help, enjoy the holidays.

Best regards,

Tobias Tillipaul

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6 comments

Date Votes
  • Maliheh Aramon
    • Gurobi Staff

    Hi Tobias, 

    To model a fractional expression like \(\frac{x}{y}\) with \(x\) and \(y\) being quadratic and linear expression, respectively, you need to define an auxiliary variable \(z\) where \(z = \frac{x}{y}\).

    You can then have:

    \[\begin{align} & \min  ~~z \\ \notag & \mbox{s.t:}~~  zy = x \end{align}\]

    Best regards,

    Maliheh

    1
  • Nipun Tulsian
    • First Comment

    How can we model this constraints?

    0
  • Marika Karbstein
    • Gurobi Staff

    Can you give more details about what exactly is unclear to you?

    0
  • Nipun Tulsian
    • First Comment

    I am trying to solve f(x) = (xTAx)/B^Tx, so basically a quadratic/linear. To solve this I used the following code:                        m = gp.Model("Fractional model", env=env)
                       m.Params.NonConvex = 2

                       x = m.addMVar(len(B), lb=-0.1, ub=0.1, vtype=GRB.CONTINUOUS, name="x")
                       y = m.addVar(lb = -1000 , ub = 1000 ,vtype = GRB.CONTINUOUS, name = 'y')

                       denominator = B @ x
                       numerator = x @ A @ x

                       m.addConstr(y * denominator == numerator, name="Auxillary Constraint")
                       m.setObjective(y , gp.GRB.MINIMIZE)

                       m.optimize()

    I am running the above code on multiple datasets, for some I am able to get values for x but for many datasets it gives nan values for output. If I am keeping y unbounded then for all datasets its coming nan. Can you please help on how to overcome this problem

     

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  • Marika Karbstein
    • Gurobi Staff

    Does it really show “nan” or are the x values simply 0? 
    Please note that the default lower bound for all variables created in Gurobi is 0.

    I would recommend to write and check the LP-file. You can use model.write("Test.lp") before calling optimize.
    Does the LP file represent the model you want to solve?

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  • Nipun Tulsian
    • First Comment

    Yes it shows all nan for variables we are trying to optimize. I will check the LP file

    0

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