Gurobi quadratic expression object. A quadratic expression consists of a linear expression, plus a list of coefficient-variable-variable triples that capture the quadratic terms. Quadratic expressions are used to build quadratic objective functions and quadratic constraints. They are temporary objects that typically have short lifespans.

The GRBQuadExpr class is a sub-class of the abstract base class GRBExpr.

In .NET languages that support operator overloading, you generally build quadratic expressions using overloaded operators. For example, if x is a GRBVar object, then x * x is a GRBQuadExpr object. Expressions can be built from constants (e.g., expr = 0), variables (e.g., expr = 1 * x * x + 2 * x * y), or from other expressions (e.g., expr2 = 2 * expr1 + x, or expr3 = expr1 + 2 * expr2). You can also modify existing expressions (e.g., expr += x * x, or expr2 -= expr1).

The other option for building expressions is to start with an empty expression (using the GRBQuadExpr constructor), and then add terms. Terms can be added individually (using AddTerm) or in groups (using AddTerms or MultAdd). Terms can also be removed from an expression (using Remove).

Note that the cost of building expressions depends heavily on the approach you use. While you can generally ignore this issue when building small expressions, you should be aware of a few efficiency issues when building large expressions:

  • You should avoid using expr = expr + x*x or expr += x*x in a loop. It will lead to runtimes that are quadratic in the number of terms in the expression.
  • Using AddTerm in a loop is reasonably efficient, but it isn't the most efficient approach.
  • The most efficient way to build a large expression is to make a single call to AddTerms.

To add a quadratic constraint to your model, you generally build one or two quadratic expression objects (qexpr1 and qexpr2) and then use an overloaded comparison operator to build an argument for GRBModel.AddQConstr. To give a few examples:

\mathrm{model.AddQConstr}(qexpr1 <= qexpr2...
\mathrm{model.AddQConstr}(2*x*x + 3*y*y <= 4) \\

Once you add a constraint to your model, subsequent changes to the expression object you used to build the constraint will have no effect on that constraint.

Individual quadratic terms in a quadratic expression can be queried using the GetVar1 GetVar2, and GetCoeff methods. You can query the number of quadratic terms in the expression using the Size property. To query the constant and linear terms associated with a quadratic expression, first obtain the linear portion of the quadratic expression using LinExpr, and then use the Constant, GetCoeff, or GetVar on the resulting GRBLinExpr object.

Note that a quadratic expression may contain multiple terms that involve the same variable pair. These duplicate terms are merged when creating the model objective from an expression, but they may be visible when inspecting individual quadratic terms in the expression (e.g., when using GetVar1 and GetVar2).