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 \(y = f(\bar{x})\) 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
  1.0 / x[0]
  1.0 / x[1]
  1.0 / x[2]
  1.0 / x[3]
  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)
  log(1.0 / x[0])
  log(1.0 / x[1])
  log(1.0 / x[2])
  log(1.0 / x[3])
  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 \(y = f(\bar{x})\) 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")
  <gurobi.GenConstr 0>
  <gurobi.GenConstr 1>
  <gurobi.GenConstr 2>
  <gurobi.GenConstr 3>
  <gurobi.GenConstr 4>
Name: log_x, dtype: object

Nonlinear Inequality Constraints¶

As noted above, nonlinear constraints can only be provided in the form \(y= f(\bar{x})\). 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