Changing variable type from integer to continuous makes quadratic constraints non-convex
回答済みHello,
I am trying to solve a mixed-integer QP with quadratic constraints. The problem contains therefore both continuous and discrete variables to learn, and is solved efficiently using Gurobi.
However, when I attempt to relax my problem manually by changing some of the variable types to continuous instead of integer, Gurobi tells me that the quadratic constraints are non-convex. How is that possible since I just changed some of the variable types from integer to continuous and not the constraints? Isn't Gurobi supposed to relax those integrality constraints anyway in the branch-and-bound algorithm, so if the constraints are non-convex shouldn't I get the error also when there are additional integrality constraints?
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Gurobi Staff
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Hi Thibaut,
Gurobi uses presolving techniques to reformulate multiplications of binary/integer variables in a way to obtain a MILP, see e.g., multiplication of continuous and binary variables. You could say that the nonconvexity is hidden in the mixed-integer part.
However, when you make all variables continuous, then there are no reformulations and tricks available to formulate the problem as a MILP. In some cases, it is possible to reformulate the problem into a SOCP but this does not seem to be true for your problem. Thus, you end up with a nonconvex problem.
Best regards,
Jaromił0 -
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Hi Jaromił,
Thanks a lot for your quick and detailed response!
I understand now, it makes perfect sense.
Best regards,
Thibaut
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