Adding Nonlinear Constraints

Gurobi 12 supports adding nonlinear constraints, using the NLExpr object to capture nonlinear expressions. gurobipy-pandas supports adding a Series of nonlinear constraints to a model via add_constrs. Note that gurobipy only supports nonlinear constraints of the form $y=f(\overline{x})$ and gurobipy-pandas applies the same restriction.

Note

To add nonlinear constraints, you must have at least gurobipy version 12.0.0 installed.

Nonlinear Equality Constraints

This example builds the constraint set $y_i=\log (\frac{1}{x_i})$, for each $i$ in the index.

>>> import pandas as pd
>>> import gurobipy as gp
>>> from gurobipy import GRB
>>> from gurobipy import nlfunc
>>> import gurobipy_pandas as gppd
>>> gppd.set_interactive()

>>> model = gp.Model()
>>> index = pd.RangeIndex(5)
>>> x = gppd.add_vars(model, index, lb=1.0, name="x")
>>> y = gppd.add_vars(model, index, lb=-GRB.INFINITY, name="y")

You can create nonlinear expressions using standard Python operators. A nonlinear expression is any expression which is not a polynomial of degree at most 2. For example, dividing a constant by a series of Var objects produces a series of nonlinear expressions.

>>> 1 / x
0    1.0 / x[0]
1    1.0 / x[1]
2    1.0 / x[2]
3    1.0 / x[3]
4    1.0 / x[4]
Name: x, dtype: object

The nlfunc module of gurobipy is used to create nonlinear expressions involving mathematical functions. You can use apply to construct a series representing the logarithm of the above result.

>>> (1 / x).apply(nlfunc.log)
0    log(1.0 / x[0])
1    log(1.0 / x[1])
2    log(1.0 / x[2])
3    log(1.0 / x[3])
4    log(1.0 / x[4])
Name: x, dtype: object

This series of expressions can be added as constraints using add_constrs. Note that you can only provide nonlinear constraints in the form $y=f(\overline{x})$ where $f$ may be a univariate or multivariate compound function. Therefore the lhs argument must be a series of Var objects, while the sense argument must be GRB.EQUAL.

>>> gppd.add_constrs(model, y, GRB.EQUAL, (1 / x).apply(nlfunc.log), name="log_x")
0    <gurobi.GenConstr 0>
1    <gurobi.GenConstr 1>
2    <gurobi.GenConstr 2>
3    <gurobi.GenConstr 3>
4    <gurobi.GenConstr 4>
Name: log_x, dtype: object

Nonlinear Inequality Constraints

As noted above, nonlinear constraints can only be provided in the form $y=f(\overline{x})$. To formulate inequality constraints, you must introduce bounded intermediate variables. The following example formulates the set of constraints $\log (x_i^2+1)\le 2$ by introducing intermediate variables $z_i$ with no lower bound and an upper bound of $2.0$.

>>> z = gppd.add_vars(model, index, lb=-GRB.INFINITY, ub=2.0, name="z")
>>> constrs = gppd.add_constrs(model, z, GRB.EQUAL, (x**2 + 1).apply(nlfunc.log))

To see the effect of this constraint, you can set a maximization objective $\sum _{i=0}^4{x}_i$ and solve the resulting model. You will find that the original variables $x_i$ are maximized to $\sqrt{e^2-1}$ in the optimal solution. The intermediate variables $z_i$ are at their upper bounds, indicating that the constraint is satisfied with equality.

>>> model.setObjective(x.sum(), sense=GRB.MAXIMIZE)
>>> model.optimize()  
Gurobi Optimizer ...
Optimal solution found ...
>>> x.gppd.X.round(3)
0    2.528
1    2.528
2    2.528
3    2.528
4    2.528
Name: x, dtype: float64
>>> z.gppd.X.round(3)
0    2.0
1    2.0
2    2.0
3    2.0
4    2.0
Name: z, dtype: float64