Quadratic Programs

Gurobi can handle both convex and non-convex quadratic constraints. The default algorithms in Gurobi only accept a few forms of quadratic constraints that are known to have convex feasible regions. Constraints of the following forms are always accepted:

• x’Qx <= e where e is a linear expression and Q is positive semi-definite (PSD),

• x’Qx <= y² where y >= 0 and Q is PSD,

• x’Qx <= yz where y >= 0 , z >= 0, and Q is PSD.

A model containing a constraint of one of the last two types is a second-order cone programming (SOCP) model. The model can contain continuous and/or integer variables.

If all quadratic constraints are convex then Gurobi can obtain the dual values (i.e., shadow prices) and reduced costs of QCP and SOCP models if the option QCP Dual Values is switched on.

Solving a model with non-convex quadratic constraints is typically much more expensive. To avoid accidentally solving a much harder problem than may have been intended, Gurobi rejects such constrains by default. To solve a model with non-convex quadratic constraints the option Nonconvex Strategy should be used.

Learn more about

• Nonconvex Strategy

• QCP Dual Values