Fraction with quadratic numerator as objective function

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Tobias Tillipaul

2022年12月23日 19:44

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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Maliheh Aramon

Gurobi Staff

2022年12月24日 03:09
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

2025年12月30日 09:54
How can we model this constraints?
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Marika Karbstein

Gurobi Staff

2025年12月30日 11:18
Can you give more details about what exactly is unclear to you?
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Nipun Tulsian

2025年12月30日 11:55
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

2025年12月30日 12:54
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

2025年12月31日 00:47
Yes it shows all nan for variables we are trying to optimize. I will check the LP file
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