Constraint¶

The Constraint class is used to define a constraint in the optimization model. It is a subclass of Expression and has the same methods and properties as Expression.

PyOptInterface supports the following types of constraints:

Linear Constraint
Quadratic Constraint
Second-Order Cone Constraint
Special Ordered Set (SOS) Constraint

Note

Not all optimizers support all types of constraints. Please refer to the documentation of the optimizer you are using to see which types of constraints are supported.

import pyoptinterface as poi
from pyoptinterface import copt

model = copt.Model()

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Linear Constraint¶

It is defined as:

\[\begin{split} \begin{align} \text{expr} &= a^T x + b &\leq \text{rhs} \\ \text{expr} &= a^T x + b &= \text{rhs} \\ \text{expr} &= a^T x + b &\geq \text{rhs} \end{align} \end{split}\]

It can be added to the model using the add_linear_constraint method of the Model class.

x = model.add_variable(name="x")
y = model.add_variable(name="y")

con = model.add_linear_constraint(2.0*x + 3.0*y, poi.ConstraintSense.LessEqual, 1.0)

model.add_linear_constraint(expr, sense, rhs[, name=""])¶

add a linear constraint to the model

Parameters:

expr – the expression of the constraint
sense (pyoptinterface.ConstraintSense) – the sense of the constraint, which can be GreaterEqual, Equal, or LessEqual
rhs (float) – the right-hand side of the constraint
name (str) – the name of the constraint, optional

Returns:

the handle of the constraint

Note

PyOptInterface provides pyoptinterface.Eq, pyoptinterface.Leq, and pyoptinterface.Geq as alias of pyoptinterface.ConstraintSense to represent the sense of the constraint with a shorter name.

Quadratic Constraint¶

Like the linear constraint, it is defined as:

\[\begin{split} \begin{align} \text{expr} &= x^TQx + a^Tx + b &\leq \text{rhs} \\ \text{expr} &= x^TQx + a^Tx + b &= \text{rhs} \\ \text{expr} &= x^TQx + a^Tx + b &\geq \text{rhs} \end{align} \end{split}\]

It can be added to the model using the add_quadratic_constraint method of the Model class.

x = model.add_variable(name="x")
y = model.add_variable(name="y")

expr = x*x + 2.0*x*y + 4.0*y*y
con = model.add_quadratic_constraint(expr, poi.ConstraintSense.LessEqual, 1.0)

model.add_quadratic_constraint(expr, sense, rhs[, name=""])¶

add a quadratic constraint to the model

Parameters:

expr – the expression of the constraint
sense (pyoptinterface.ConstraintSense) – the sense of the constraint, which can be GreaterEqual, Equal, or LessEqual
rhs (float) – the right-hand side of the constraint
name (str) – the name of the constraint, optional

Returns:

the handle of the constraint

Second-Order Cone Constraint¶

It is defined as:

\[ variables=(t,x) \in \mathbb{R}^{N} : t \ge \lVert x \rVert_2 \]

It can be added to the model using the add_second_order_cone_constraint method of the Model class.

N = 6
vars = [model.add_variable() for i in range(N)]

con = model.add_second_order_cone_constraint(vars)

There is another form of second-order cone constraint called as rotated second-order cone constraint, which is defined as:

\[ variables=(t_{1},t_{2},x) \in \mathbb{R}^{N} : 2 t_1 t_2 \ge \lVert x \rVert_2^2 \]

model.add_second_order_cone_constraint(variables[, name="", rotated=False])¶

add a second order cone constraint to the model

Parameters:

variables – the variables of the constraint, can be a list of variables
name (str) – the name of the constraint, optional
rotated (bool) – whether the constraint is a rotated second-order cone constraint, optional

Returns:

the handle of the constraint

Exponential Cone Constraint¶

It is defined as:

\[ variables=(t,s,r) \in \mathbb{R}^{3} : t \ge s \exp(\frac{r}{s}), s \ge 0 \]

The dual form is:

\[ variables=(t,s,r) \in \mathbb{R}^{3} : t \ge -r \exp(\frac{s}{r} - 1), r \le 0 \]

Currently, only COPT(after 7.1.4), Mosek support exponential cone constraint. It can be added to the model using the add_exp_cone_constraint method of the Model class.

model.add_exp_cone_constraint(variables[, name="", dual=False])¶

add a second order cone constraint to the model

Parameters:

variables – the variables of the constraint, can be a list of variables
name (str) – the name of the constraint, optional
dual (bool) – whether the constraint is dual form of exponential cone, optional

Returns:

the handle of the constraint

Special Ordered Set (SOS) Constraint¶

SOS constraints are used to model special structures in the optimization problem. It contains two types: SOS1 and SOS2, the details can be found in Wikipedia.

It can be added to the model using the add_sos_constraint method of the Model class.

N = 6
vars = [model.add_variable(domain=poi.VariableDomain.Binary) for i in range(N)]

con = model.add_sos_constraint(vars, poi.SOSType.SOS1)

model.add_sos_constraint(variables, sos_type[, weights])¶

add a special ordered set constraint to the model

Parameters:

variables – the variables of the constraint, can be a list of variables
sos_type (pyoptinterface.SOSType) – the type of the SOS constraint, which can be SOS1 or SOS2
weights (list[float]) – the weights of the variables, optional, will be set to 1 if not provided

Returns:

the handle of the constraint

Constraint Attributes¶

After a constraint is created, we can query or modify its attributes. The following table lists the standard constraint attributes:

**Standard constraint attributes**¶
Attribute name	Type
Name	str
Primal	float
Dual	float

The most common attribute we will use is the Dual attribute, which represents the dual multiplier of the constraint after optimization.

# get the dual multiplier of the constraint after optimization
dual = model.get_constraint_attribute(con, poi.ConstraintAttribute.Dual)

Delete constraint¶

We can delete a constraint by calling the delete_constraint method of the model:

model.delete_constraint(con)

After a constraint is deleted, it cannot be used in the model anymore, otherwise an exception will be raised.

We can query whether a constraint is active by calling the is_constraint_active method of the model:

is_active = model.is_constraint_active(con)

Modify constraint¶

For linear and quadratic constraints, we can modify the right-hand side of a constraint by calling the set_normalized_rhs method of the model.

For linear constraints, we can modify the coefficients of the linear part of the constraint by calling the set_normalized_coefficient method of the model.

con = model.add_linear_constraint(x + y, poi.Leq, 1.0)

# modify the right-hand side of the constraint
model.set_normalized_rhs(con, 2.0)

# modify the coefficient of the linear part of the constraint
model.set_normalized_coefficient(con, x, 2.0)