# Constraints¶

A constraint is an equality or inequality that restricts the domain of an optimization problem. CVXPY has seven types of constraints: non-positive, equality or zero, positive semidefinite, second-order cone, exponential cone, 3-dimensional power cones, and N-dimensional power cones. The vast majority of users will need only create constraints of the first three types. Additionally, most users need not know anything more about constraints other than how to create them. The constraint APIs do nonetheless provide methods that advanced users may find useful; for example, some of the APIs allow you to inspect dual variable values and residuals.

## Constraint¶

class cvxpy.constraints.constraint.Constraint(args, constr_id=None)[source]

Bases: Canonical

The base class for constraints.

A constraint is an equality, inequality, or more generally a generalized inequality that is imposed upon a mathematical expression or a list of thereof.

Parameters:
• args (list) – A list of expression trees.

• constr_id (int) – A unique id for the constraint.

abstract is_dcp(dpp: bool = False) bool[source]

Checks whether the constraint is DCP.

Returns:

True if the constraint is DCP, False otherwise.

Return type:

bool

value(tolerance: float = 1e-08)[source]

Checks whether the constraint violation is less than a tolerance.

Parameters:

tolerance (float) – The absolute tolerance to impose on the violation.

Returns:

True if the violation is less than tolerance, False otherwise.

Return type:

bool

Raises:

ValueError – If the constrained expression does not have a value associated with it.

violation()[source]

The numeric residual of the constraint.

The violation is defined as the distance between the constrained expression’s value and its projection onto the domain of the constraint:

$||\Pi(v) - v||_2^2$

where $$v$$ is the value of the constrained expression and $$\Pi$$ is the projection operator onto the constraint’s domain .

Returns:

The residual value.

Return type:

NumPy.ndarray

Raises:

ValueError – If the constrained expression does not have a value associated with it.

## NonPos¶

class cvxpy.constraints.nonpos.NonPos(expr, constr_id=None)[source]

Bases: Constraint

A constraint of the form $$x \leq 0$$.

The preferred way of creating a NonPos constraint is through operator overloading. To constrain an expression x to be non-positive, simply write x <= 0; to constrain x to be non-negative, write x >= 0. The former creates a NonPos constraint with x as its argument, while the latter creates one with -x as its argument. Strict inequalities are not supported, as they do not make sense in a numerical setting.

Parameters:
• expr (Expression) – The expression to constrain.

• constr_id (int) – A unique id for the constraint.

property dual_value

The value of the dual variable.

Type:

NumPy.ndarray

is_dcp(dpp: bool = False) bool[source]

A non-positive constraint is DCP if its argument is convex.

property shape

The shape of the constrained expression.

Type:

int

property size

The size of the constrained expression.

Type:

int

value(tolerance: float = 1e-08)

Checks whether the constraint violation is less than a tolerance.

Parameters:

tolerance (float) – The absolute tolerance to impose on the violation.

Returns:

True if the violation is less than tolerance, False otherwise.

Return type:

bool

Raises:

ValueError – If the constrained expression does not have a value associated with it.

violation()[source]

The numeric residual of the constraint.

The violation is defined as the distance between the constrained expression’s value and its projection onto the domain of the constraint:

$||\Pi(v) - v||_2^2$

where $$v$$ is the value of the constrained expression and $$\Pi$$ is the projection operator onto the constraint’s domain .

Returns:

The residual value.

Return type:

NumPy.ndarray

Raises:

ValueError – If the constrained expression does not have a value associated with it.

## Zero¶

class cvxpy.constraints.zero.Zero(expr, constr_id=None)[source]

Bases: Constraint

A constraint of the form $$x = 0$$.

The preferred way of creating a Zero constraint is through operator overloading. To constrain an expression x to be zero, simply write x == 0. The former creates a Zero constraint with x as its argument.

is_dcp(dpp: bool = False) bool[source]

A zero constraint is DCP if its argument is affine.

value(tolerance: float = 1e-08)

Checks whether the constraint violation is less than a tolerance.

Parameters:

tolerance (float) – The absolute tolerance to impose on the violation.

Returns:

True if the violation is less than tolerance, False otherwise.

Return type:

bool

Raises:

ValueError – If the constrained expression does not have a value associated with it.

violation()

The numeric residual of the constraint.

The violation is defined as the distance between the constrained expression’s value and its projection onto the domain of the constraint:

$||\Pi(v) - v||_2^2$

where $$v$$ is the value of the constrained expression and $$\Pi$$ is the projection operator onto the constraint’s domain .

Returns:

The residual value.

Return type:

NumPy.ndarray

Raises:

ValueError – If the constrained expression does not have a value associated with it.

## PSD¶

class cvxpy.constraints.psd.PSD(expr, constr_id=None)[source]

Bases: Constraint

A constraint of the form $$\frac{1}{2}(X + X^T) \succcurlyeq_{S_n^+} 0$$

Applying a PSD constraint to a two-dimensional expression X constrains its symmetric part to be positive semidefinite: i.e., it constrains X to be such that

$z^T(X + X^T)z \geq 0,$

for all $$z$$.

The preferred way of creating a PSD constraint is through operator overloading. To constrain an expression X to be PSD, write X >> 0; to constrain it to be negative semidefinite, write X << 0. Strict definiteness constraints are not provided, as they do not make sense in a numerical setting.

Parameters:
• expr (Expression.) – The expression to constrain; must be two-dimensional.

• constr_id (int) – A unique id for the constraint.

is_dcp(dpp: bool = False) bool[source]

A PSD constraint is DCP if the constrained expression is affine.

value(tolerance: float = 1e-08)

Checks whether the constraint violation is less than a tolerance.

Parameters:

tolerance (float) – The absolute tolerance to impose on the violation.

Returns:

True if the violation is less than tolerance, False otherwise.

Return type:

bool

Raises:

ValueError – If the constrained expression does not have a value associated with it.

violation()

The numeric residual of the constraint.

The violation is defined as the distance between the constrained expression’s value and its projection onto the domain of the constraint:

$||\Pi(v) - v||_2^2$

where $$v$$ is the value of the constrained expression and $$\Pi$$ is the projection operator onto the constraint’s domain .

Returns:

The residual value.

Return type:

NumPy.ndarray

Raises:

ValueError – If the constrained expression does not have a value associated with it.

## SOC¶

class cvxpy.constraints.second_order.SOC(t, X, axis: int = 0, constr_id=None)[source]

Bases: Constraint

A second-order cone constraint for each row/column.

Assumes t is a vector the same length as X’s columns (rows) for axis == 0 (1).

t

The scalar part of the second-order constraint.

X

A matrix whose rows/columns are each a cone.

axis

Slice by column 0 or row 1.

is_dcp(dpp: bool = False) bool[source]

An SOC constraint is DCP if each of its arguments is affine.

value(tolerance: float = 1e-08)

Checks whether the constraint violation is less than a tolerance.

Parameters:

tolerance (float) – The absolute tolerance to impose on the violation.

Returns:

True if the violation is less than tolerance, False otherwise.

Return type:

bool

Raises:

ValueError – If the constrained expression does not have a value associated with it.

violation()

The numeric residual of the constraint.

The violation is defined as the distance between the constrained expression’s value and its projection onto the domain of the constraint:

$||\Pi(v) - v||_2^2$

where $$v$$ is the value of the constrained expression and $$\Pi$$ is the projection operator onto the constraint’s domain .

Returns:

The residual value.

Return type:

NumPy.ndarray

Raises:

ValueError – If the constrained expression does not have a value associated with it.

## ExpCone¶

class cvxpy.constraints.exponential.ExpCone(x: Expression, y: Expression, z: Expression, constr_id=None)[source]

Bases: Constraint

A reformulated exponential cone constraint.

Operates elementwise on $$x, y, z$$.

Original cone:

$K = \{(x,y,z) \mid y > 0, ye^{x/y} <= z\} \cup \{(x,y,z) \mid x \leq 0, y = 0, z \geq 0\}$

Reformulated cone:

$K = \{(x,y,z) \mid y, z > 0, y\log(y) + x \leq y\log(z)\} \cup \{(x,y,z) \mid x \leq 0, y = 0, z \geq 0\}$
Parameters:
is_dcp(dpp: bool = False) bool[source]

An exponential constraint is DCP if each argument is affine.

value(tolerance: float = 1e-08)

Checks whether the constraint violation is less than a tolerance.

Parameters:

tolerance (float) – The absolute tolerance to impose on the violation.

Returns:

True if the violation is less than tolerance, False otherwise.

Return type:

bool

Raises:

ValueError – If the constrained expression does not have a value associated with it.

violation()

The numeric residual of the constraint.

The violation is defined as the distance between the constrained expression’s value and its projection onto the domain of the constraint:

$||\Pi(v) - v||_2^2$

where $$v$$ is the value of the constrained expression and $$\Pi$$ is the projection operator onto the constraint’s domain .

Returns:

The residual value.

Return type:

NumPy.ndarray

Raises:

ValueError – If the constrained expression does not have a value associated with it.

class cvxpy.constraints.exponential.RelEntrConeQuad(x: Expression, y: Expression, z: Expression, m: int, k: int, constr_id=None)[source]

Bases: Constraint

An approximate construction of the scalar relative entropy cone

Definition:

$K_{re}=\text{cl}\{(x,y,z)\in\mathbb{R}_{++}\times \mathbb{R}_{++}\times\mathbb{R}_{++}\:x\log(x/y)\leq z\}$

Since the above definition is very similar to the ExpCone, we provide a conversion method.

More details on the approximation can be found in Theorem-3 on page-10 in the paper: Semidefinite Approximations of the Matrix Logarithm.

Parameters:
• x (Expression) – x in the (approximate) scalar relative entropy cone

• y (Expression) – y in the (approximate) scalar relative entropy cone

• z (Expression) – z in the (approximate) scalar relative entropy cone

• m (Parameter directly related to the number of generated nodes for the quadrature) –

• algorithm (approximation used in the) –

• k (Another parameter controlling the approximation) –

is_dcp(dpp: bool = False) bool[source]

An exponential constraint is DCP if each argument is affine.

value(tolerance: float = 1e-08)

Checks whether the constraint violation is less than a tolerance.

Parameters:

tolerance (float) – The absolute tolerance to impose on the violation.

Returns:

True if the violation is less than tolerance, False otherwise.

Return type:

bool

Raises:

ValueError – If the constrained expression does not have a value associated with it.

## PowCone3D¶

class cvxpy.constraints.power.PowCone3D(x, y, z, alpha, constr_id=None)[source]

Bases: Constraint

An object representing a collection of 3D power cone constraints

x[i]**alpha[i] * y[i]**(1-alpha[i]) >= |z[i]| for all i x >= 0, y >= 0

If the parameter alpha is a scalar, it will be promoted to a vector matching the (common) sizes of x, y, z. The numeric value of alpha (or its components, in the vector case) must be a number in the open interval (0, 1).

We store flattened representations of the arguments (x, y, z, and alpha) as Expression objects. We construct dual variables with respect to these flattened representations.

is_dcp(dpp: bool = False) bool[source]

Checks whether the constraint is DCP.

Returns:

True if the constraint is DCP, False otherwise.

Return type:

bool

value(tolerance: float = 1e-08)

Checks whether the constraint violation is less than a tolerance.

Parameters:

tolerance (float) – The absolute tolerance to impose on the violation.

Returns:

True if the violation is less than tolerance, False otherwise.

Return type:

bool

Raises:

ValueError – If the constrained expression does not have a value associated with it.

violation()

The numeric residual of the constraint.

The violation is defined as the distance between the constrained expression’s value and its projection onto the domain of the constraint:

$||\Pi(v) - v||_2^2$

where $$v$$ is the value of the constrained expression and $$\Pi$$ is the projection operator onto the constraint’s domain .

Returns:

The residual value.

Return type:

NumPy.ndarray

Raises:

ValueError – If the constrained expression does not have a value associated with it.

## PowConeND¶

class cvxpy.constraints.power.PowConeND(W, z, alpha, axis: int = 0, constr_id=None)[source]

Bases: Constraint

Represents a collection of N-dimensional power cone constraints that is mathematically equivalent to the following code snippet (which makes incorrect use of numpy functions on cvxpy objects):

np.prod(np.power(W, alpha), axis=axis) >= np.abs(z), W >= 0

All arguments must be Expression-like, and z must satisfy z.ndim <= 1. The columns (rows) of alpha must sum to 1 when axis=0 (axis=1).

Note: unlike PowCone3D, we make no attempt to promote alpha to the appropriate shape. The dimensions of W and alpha must match exactly.

Note: Dual variables are not currently implemented for this type of constraint.

is_dcp(dpp: bool = False) bool[source]

A power cone constraint is DCP if each argument is affine.

value(tolerance: float = 1e-08)

Checks whether the constraint violation is less than a tolerance.

Parameters:

tolerance (float) – The absolute tolerance to impose on the violation.

Returns:

True if the violation is less than tolerance, False otherwise.

Return type:

bool

Raises:

ValueError – If the constrained expression does not have a value associated with it.

violation()

The numeric residual of the constraint.

The violation is defined as the distance between the constrained expression’s value and its projection onto the domain of the constraint:

$||\Pi(v) - v||_2^2$

where $$v$$ is the value of the constrained expression and $$\Pi$$ is the projection operator onto the constraint’s domain .

Returns:

The residual value.

Return type:

NumPy.ndarray

Raises:

ValueError – If the constrained expression does not have a value associated with it.

## FiniteSet¶

class cvxpy.constraints.finite_set.FiniteSet(expre, vec, ineq_form: bool = False, constr_id=None)[source]

Bases: Constraint

Constrain each entry of an Expression to take a value in a given set of real numbers.

Parameters:
• expre (Expression) –

The given expression to be constrained. This Expression must be affine. If expre has multiple elements, then the constraint is applied separately to each element. I.e., after solving a problem with this constraint, we should have:

for e in expre.flatten():
print(e.value in vec) # => True


• vec (Union[Expression, np.ndarray, set]) – The finite collection of values to which each entry of expre is to be constrained.

• ineq_form (bool) –

Controls how this constraint is canonicalized into mixed integer linear constraints.

If True, then we use a formulation with vec.size - 1 inequality constraints, one equality constraint, and vec.size - 1 binary variables for each element of expre.

If False, then we use a formulation with vec.size binary variables and two equality constraints for each element of expre.

Defaults to False. The case ineq_form=True may speed up some mixed-integer solvers that use simple branch and bound methods.

property ineq_form: bool

Choose between two constraining methodologies, use ineq_form=False while working with Parameter types.

is_dcp(dpp: bool = False) bool[source]

A FiniteSet constraint is DCP if the constrained expression is affine.

property shape

The shape of the constrained expression.

Type:

int

property size

The size of the constrained expression.

Type:

int

violation()

The numeric residual of the constraint.

The violation is defined as the distance between the constrained expression’s value and its projection onto the domain of the constraint:

$||\Pi(v) - v||_2^2$

where $$v$$ is the value of the constrained expression and $$\Pi$$ is the projection operator onto the constraint’s domain .

Returns:

The residual value.

Return type:

NumPy.ndarray

Raises:

ValueError – If the constrained expression does not have a value associated with it.

class cvxpy.constraints.exponential.OpRelEntrConeQuad(X: Expression, Y: Expression, Z: Expression, m: int, k: int, constr_id=None)[source]

Bases: Constraint

An approximate construction of the operator relative entropy cone

Definition:

$K_{re}^n=\text{cl}\{(X,Y,T)\in\mathbb{H}^n_{++}\times \mathbb{H}^n_{++}\times\mathbb{H}^n_{++}\:D_{\text{op}}\succeq T\}$

More details on the approximation can be found in Theorem-3 on page-10 in the paper: Semidefinite Approximations of the Matrix Logarithm.

Parameters:
• X (Expression) – x in the (approximate) operator relative entropy cone

• Y (Expression) – y in the (approximate) operator relative entropy cone

• Z (Expression) – Z in the (approximate) operator relative entropy cone

• m (int) – Must be positive. Controls the number of quadrature nodes used in a local approximation of the matrix logarithm. Increasing this value results in better local approximations, but does not significantly expand the region of inputs for which the approximation is effective.

• k (int) – Must be positive. Sets the number of scaling points about which the quadrature approximation is performed. Increasing this value will expand the region of inputs over which the approximation is effective.

:param This approximation uses $$m + k$$ semidefinite constraints.:

is_dcp(dpp: bool = False) bool[source]

An operator relative conic constraint is DCP when (A, b, C) is affine

value(tolerance: float = 1e-08)

Checks whether the constraint violation is less than a tolerance.

Parameters:

tolerance (float) – The absolute tolerance to impose on the violation.

Returns:

True if the violation is less than tolerance, False otherwise.

Return type:

bool

Raises:

ValueError – If the constrained expression does not have a value associated with it.