sum of square roors in objective
AnsweredHi!
I'm trying to set my objective function with a sum of square roots as following, where x1 … xn are my decision variables and c1... cn are constant:
min f = sqrt((x1-c1)^2) + sqrt((x2-c2)^2) + … +sqrt((xn-cn)^2)
I already tried a workaround by defining new variables using addGenConstrPow(), however as type(x1-c1) seems to be a LinExpr(), python raises the AttributeError: 'gurobipy.LinExpr' object has no attribute '__cindex__'
Is there anything i am overseeing or a smart way to solve the problem?
Thank you in advance!
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Gurobi Staff
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The first two arguments to Model.addGenConstrPow() should be Var objects, not LinExpr objects. To avoid this error, you can introduce auxiliary variables \( y_i = x_i - c_i \) for \( i = 1, \ldots, n \), then pass \( y_i \) as an argument to Model.addGenConstrPow().
However, your objective function is equivalent to
$$\begin{align}\min \sum_{i = 1}^n |x_i - c_i|.\end{align}$$
After introducing auxiliary variables \( y_i \) for \( i = 1, \ldots, n\), you can reformulate the above problem as follows:
$$\begin{alignat}{3}\min\ && \sum_{i = 1}^n & \ y_i && \\ \textrm{s.t.}\ && y_i &\geq x_i - c_i &&\quad i = 1, \ldots, n \\ && y_i &\geq c_i - x_i &&\quad i = 1, \ldots, n.\end{alignat}$$
Alternatively, you can use Model.addGenConstrAbs() or the abs_() helper function to model the absolute value terms. Like Model.addGenConstrPow(), using an absolute value general constraint requires you to introduce auxiliary variables for the \( x_i - c_i \) terms.
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Thank you Eli, that helped
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