Nonlinear constraint with two variables multiplied together and raised to some power
回答済みHi there,
My optimization problem has the following non-negative variables:
V = model.addVars([i for i in range(0,4)], lb = 0, vtype = gp.GRB.CONTINUOUS)
S = model.addVars([(i,s) for x in range(0,4) for s in range(0,1)], vtype = gp.GRB.CONTINUOUS)
E = model.addVars([e for e in range(0,4)], vtype = gp.GRB.INTEGER)
I also have one constraint that looks like this:
The value for constant 2 and constant 3 are fractions, so I could not express the constraint as something similar to S_i * S_i. After checking gurobi's built-in PWL constraint documentation, I figure that it will not help since I have two decision variables multiplied together.
Is there a way for me to add this constraint to the problem? Currently, I need the variables to be defined in that exact format. But if that is the blocking issue, let me know how to get around with it.
Thanks in advance!
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Gurobi Staff
Hi Grace,
It is possible to formulate your constraint in Gurobi with a few additional variables and equality constraints as long as your exponents \(const_2, const_3\) are \(> 0\). What you have to model in Gurobi is
\[\begin{align*}
w_1 &= \sum_e E_e \\
z_{1} &= w_1^{const_2} \text{ use addGenConstrPow} \\
z_{2,i} &= S_i^{const_3} \,\,\forall i \text{ use addGenConstrPow}\\
V_i & \leq const_1 \cdot z_{1} \cdot z_{2,i} \,\, \forall i
\end{align*}\]Link to documentation of addGenConstrPow.
Best regards,
Jaromił0 -
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Hi Jaromił,
Thanks for your reply! I implemented what you suggested, and it worked. There are some follow-up questions after seeing the output message.
1. my Q matrix is not positive semi-definite, and I would like to know how gurobi gets around with it, i.e., what algorithm it invokes and any assumptions it uses. Can I get this kind of information by looking at gurobi's documentation?
2. My model now looks nonconvex and nonlinear. Are there any tricks for me to see where the nonconvexity shows up and potential ways to make it convex? (I'm assuming the nonlinearity is unavoidable since I'm constructing the constraint in this way.)
3. I'm also interested in some sensitivity analysis. I'm considering adjusting the constraints to see whether I can still obtain the same solution or when the problem becomes infeasible. Does gurobi have built-in functions for that?
Thanks much!
Grace
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Hi Grace
1. my Q matrix is not positive semi-definite, and I would like to know how gurobi gets around with it, i.e., what algorithm it invokes and any assumptions it uses. Can I get this kind of information by looking at gurobi's documentation?
I recommend having a look at our non-convex webinar. This should be a good starting point and give you enough ideas to search for more in the literature.
2. My model now looks nonconvex and nonlinear. Are there any tricks for me to see where the nonconvexity shows up and potential ways to make it convex? (I'm assuming the nonlinearity is unavoidable since I'm constructing the constraint in this way.)
Any nonlinear equality constraint is automatically nonconvex. Thus, simply the reformulation into a quadratic form can already make the problem nonconvex. As long as the solver does not handle higher degree multilinear terms natively, there is no way to get around it.
In your particular case, the nonconvexity comes from the bilinear terms \(z_1 \cdot z_{2,i}\). The power functions are approximated via piecewise-linear formulations and thus are linear.
3. I'm also interested in some sensitivity analysis. I'm considering adjusting the constraints to see whether I can still obtain the same solution or when the problem becomes infeasible. Does gurobi have built-in functions for that?
I think that Gurobi's multi-scenario feature might be helpful here.
Best regards,
Jaromił0 -
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