Gurobi generating bilinear constraints for a QP model and solving it as MIP

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Pedro Pinheiro

2025年06月06日 20:08

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I am new at quadratic programming and I'm in doubt about the following.

 

Why is gurobi using MIP to solve my quadratic objective function model, when it only has continuos variables and linear constraints?

 

Does a quadratic objective function affect the convexity of the model?

Where did these bilinear constraints come from? In the original model there are only linear constraints (in continuous variables).

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  Maliheh Aramon

  • Gurobi Staff

  2025年06月06日 21:05 編集済み

  Hi Pedro, 

  Does a quadratic objective function affect the convexity of the model?

  Yes, a quadratic objective function is not necessarily convex. For instance, the quadratic objective \(x_1^2 - x_2 ^2\) is not convex. A quadratic function in the form \(x^T Q x + c^T x + d\) is convex if and only if the \(Q\) matrix is positive semidefinite.

  Where did these bilinear constraints come from? In the original model there are only linear constraints (in continuous variables).

  The Gurobi presolve has converted the quadratic objective into constraints, this explains why there are bilinear constraints in the presolved model. Gurobi presolve has also concluded that the model is non-convex and that's why it is solved as an instance of MIQCP.  
  
  Best regards,
  Maliheh

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