constraint.md

file_format	kernelspec
mystnb	name python3

Constraint

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()

Constraint Sense

The sense of a constraint can be one of the following:

• poi.Eq: equal
• poi.Leq: less than or equal
• poi.Geq: greater than or equal

They are the abbreviations of poi.ConstraintSense.Equal, poi.ConstraintSense.LessEqual or poi.ConstraintSense.GreaterEqual and can be used in the sense argument of the constraint creation functions.

Linear Constraint

It is defined as:

$$\begin{array}{rlr}\text{expr}& =a^{T}x+b& \le \text{rhs}\\ \text{expr}& =a^{T}x+b& =\text{rhs}\\ \text{expr}& =a^{T}x+b& \ge \text{rhs}\end{array}$$

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.Leq, 1.0)

add a linear constraint to the model

:param expr: the expression of the constraint
:param pyoptinterface.ConstraintSense sense: the sense of the constraint
:param float rhs: the right-hand side of the constraint
:param str name: the name of the constraint, optional
:return: the handle of the constraint

:::{note}

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

Quadratic Constraint

Like the linear constraint, it is defined as:

$$\begin{array}{rlr}\text{expr}& =x^{T}Qx+a^{T}x+b& \le \text{rhs}\\ \text{expr}& =x^{T}Qx+a^{T}x+b& =\text{rhs}\\ \text{expr}& =x^{T}Qx+a^{T}x+b& \ge \text{rhs}\end{array}$$

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)

add a quadratic constraint to the model

:param expr: the expression of the constraint
:param pyoptinterface.ConstraintSense sense: the sense of the constraint, which can be `GreaterEqual`, `Equal`, or `LessEqual`
:param float rhs: the right-hand side of the constraint
:param str name: the name of the constraint, optional
:return: the handle of the constraint

Second-Order Cone Constraint

It is defined as:

$$variables=(t,x)\in {\mathbb{R}}^N:t\ge \Vert x{\Vert }_2$$

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:

$$variables=(t_1,t_2,x)\in {\mathbb{R}}^N:2t_1{t}_2\ge \Vert x{\Vert }_2^2$$

add a second order cone constraint to the model

:param variables: the variables of the constraint, can be a list of variables
:param str name: the name of the constraint, optional
:param bool rotated: whether the constraint is a rotated second-order cone constraint, optional
:return: 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.

add a second order cone constraint to the model

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

add a special ordered set constraint to the model

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

:::{list-table} Standard constraint attributes :header-rows: 1 :widths: 20 20

• • Attribute name
  • Type
• • Name
  • str
• • Primal
  • float
• • Dual
  • float
• • IIS
  • bool :::

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)

Create constraint with comparison operator

In other modeling languages, we can create a constraint with a comparison operator, like:

model.addConsr(x + y <= 1)

This is quite convenient, so PyOptInterface now supports to create constraint with comparison operators <=, ==, >= as a shortcut to create a linear or quadratic constraint.

model.add_linear_constraint(x + y <= 1)
model.add_linear_constraint(x <= y)
model.add_quadratic_constraint(x*x + y*y <= 1)

:::{note}

Creating constraint with comparison operator may cause performance issue especially the left-hand side and right-hand side of the constraint are complex expressions. PyOptInterface needs to create a new expression by subtracting the right-hand side from the left-hand side, which may be time-consuming.

If that becomes the bottleneck of performance, it is recommended to construct the left-hand side expression with ExprBuilder and call add_linear_constraint or add_quadratic_constraint method to create constraints explicitly. :::