# Expressions¶

CVXPY represents mathematical objects as expression trees. An expression tree is a collection of mathematical expressions linked together by one or more atoms. Expression trees are encoded as instances of the Expression class, and each Leaf in a tree is a Variable, Parameter, or Constant.

## Expression¶

class cvxpy.expressions.expression.Expression[source]

Bases: cvxpy.utilities.canonical.Canonical

A mathematical expression in a convex optimization problem.

Overloads many operators to allow for convenient creation of compound expressions (e.g., the sum of two expressions) and constraints.

T

Expression – The transpose of the expression.

__add__(other)[source]

Expression : Sum two expressions.

__div__(other)[source]

Expression : One expression divided by another.

__eq__(other)[source]

Equality : Creates a constraint self == other.

__ge__(other)[source]

NonPos : Creates an inequality constraint.

__le__(other)[source]

Inequality : Creates an inequality constraint self <= other.

__lshift__(other)[source]

PSD : Creates a negative semidefinite inequality.

__matmul__(other)[source]

Expression : Matrix multiplication of two expressions.

__mul__(other)[source]

Expression : The product of two expressions.

__pow__(power)[source]

Raise expression to a power.

Parameters: power (float) – The power to which to raise the expression. The expression raised to power. Expression
__radd__(other)[source]

Expression : Sum two expressions.

__rdiv__(other)[source]

Expression : Called for Number / Expression.

__rlshift__(other)[source]

PSD : Creates a negative semidefinite inequality.

__rmatmul__(other)[source]

Expression : Called for matrix @ Expression.

__rmul__(other)[source]

Expression : Called for Number * Expression.

__rrshift__(other)[source]

PSD : Creates a positive semidefinite inequality.

__rshift__(other)[source]

PSD : Creates a positive semidefinite inequality.

__rsub__(other)[source]

Expression : The difference of two expressions.

__rtruediv__(other)[source]

Expression : Called for Number / Expression.

__sub__(other)[source]

Expression : The difference of two expressions.

__truediv__(other)[source]

Expression : One expression divided by another.

curvature

str – The curvature of the expression.

domain

list – The constraints describing the closure of the region where the expression is finite.

grad

Gives the (sub/super)gradient of the expression w.r.t. each variable.

Matrix expressions are vectorized, so the gradient is a matrix.

Returns: A map of variable to SciPy CSC sparse matrix; None if a variable value is missing. dict
is_affine()[source]

Is the expression affine?

is_concave()[source]

Is the expression concave?

is_constant()[source]

Is the expression constant?

is_convex()[source]

Is the expression convex?

is_dcp()[source]

Checks whether the Expression is DCP.

Returns: True if the Expression is DCP, False otherwise. bool
is_dgp()[source]

Checks whether the Expression is log-log DCP.

Returns: True if the Expression is log-log DCP, False otherwise. bool
is_log_log_affine()[source]

Is the expression affine?

is_log_log_concave()[source]

Is the expression log-log concave?

is_log_log_convex()[source]

Is the expression log-log convex?

is_nonneg()[source]

Is the expression positive?

is_nonpos()[source]

Is the expression negative?

is_zero()[source]

Is the expression all zero?

name()[source]

str : The string representation of the expression.

ndim

int – The number of dimensions in the expression’s shape.

shape

tuple – The expression dimensions.

sign

str – The sign of the expression.

size

int – The number of entries in the expression.

value

NumPy.ndarray or None – The numeric value of the expression.

## Leaf¶

class cvxpy.expressions.leaf.Leaf(shape, value=None, nonneg=False, nonpos=False, complex=False, imag=False, symmetric=False, diag=False, PSD=False, NSD=False, hermitian=False, boolean=False, integer=False, sparsity=None, pos=False, neg=False)[source]

A leaf node of an expression tree; i.e., a Variable, Constant, or Parameter.

A leaf may carry attributes that constrain the set values permissible for it. Leafs can have no more than one attribute, with the exception that a leaf may be both nonpos and nonneg or both boolean in some indices and integer in others.

An error is raised if a leaf is assigned a value that contradicts one or more of its attributes. See the project method for a convenient way to project a value onto a leaf’s domain.

Parameters: shape (tuple or int) – The leaf dimensions. Either an integer n for a 1D shape, or a tuple where the semantics are the same as NumPy ndarray shapes. Shapes cannot be more than 2D. value (numeric type) – A value to assign to the leaf. nonneg (bool) – Is the variable constrained to be nonnegative? nonpos (bool) – Is the variable constrained to be nonpositive? complex (bool) – Is the variable complex valued? symmetric (bool) – Is the variable symmetric? diag (bool) – Is the variable diagonal? PSD (bool) – Is the variable constrained to be positive semidefinite? NSD (bool) – Is the variable constrained to be negative semidefinite? Hermitian (bool) – Is the variable Hermitian? boolean (bool or list of tuple) – Is the variable boolean? True, which constrains the entire Variable to be boolean, False, or a list of indices which should be constrained as boolean, where each index is a tuple of length exactly equal to the length of shape. integer (bool or list of tuple) – Is the variable integer? The semantics are the same as the boolean argument. sparsity (list of tuplewith) – Fixed sparsity pattern for the variable. pos (bool) – Is the variable positive? neg (bool) – Is the variable negative?
T

Expression – The transpose of the expression.

ndim

int – The number of dimensions in the expression’s shape.

project(val)[source]

Project value onto the attribute set of the leaf.

A sensible idiom is leaf.value = leaf.project(val).

Parameters: val (numeric type) – The value assigned. The value rounded to the attribute type. numeric type
project_and_assign(val)[source]

Project and assign a value to the variable.

shape

tuple – The dimensions of the expression.

size

int – The number of entries in the expression.

value

NumPy.ndarray or None – The numeric value of the parameter.

## Variable¶

class cvxpy.expressions.variable.Variable(shape=(), name=None, var_id=None, **kwargs)[source]

The optimization variables in a problem.

T

Expression – The transpose of the expression.

name()[source]

str : The name of the variable.

ndim

int – The number of dimensions in the expression’s shape.

project(val)

Project value onto the attribute set of the leaf.

A sensible idiom is leaf.value = leaf.project(val).

Parameters: val (numeric type) – The value assigned. The value rounded to the attribute type. numeric type
project_and_assign(val)

Project and assign a value to the variable.

shape

tuple – The dimensions of the expression.

size

int – The number of entries in the expression.

value

NumPy.ndarray or None – The numeric value of the parameter.

## Parameter¶

class cvxpy.expressions.constants.parameter.Parameter(shape=(), name=None, value=None, **kwargs)[source]

Parameters in optimization problems.

Parameters are constant expressions whose value may be specified after problem creation. The only way to modify a problem after its creation is through parameters. For example, you might choose to declare the hyper-parameters of a machine learning model to be Parameter objects; more generally, Parameters are useful for computing trade-off curves.

T

Expression – The transpose of the expression.

ndim

int – The number of dimensions in the expression’s shape.

project(val)

Project value onto the attribute set of the leaf.

A sensible idiom is leaf.value = leaf.project(val).

Parameters: val (numeric type) – The value assigned. The value rounded to the attribute type. numeric type
project_and_assign(val)

Project and assign a value to the variable.

shape

tuple – The dimensions of the expression.

size

int – The number of entries in the expression.

value

NumPy.ndarray or None – The numeric value of the parameter.

## Constant¶

class cvxpy.expressions.constants.Constant(value)[source]

A constant value.

Raw numerical constants (Python primite types, NumPy ndarrays, and NumPy matrices) are implicitly cast to constants via Expression operator overloading. For example, if x is an expression and c is a raw constant, then x + c creates an expression by casting c to a Constant.

T

Expression – The transpose of the expression.

ndim

int – The number of dimensions in the expression’s shape.

shape

Returns the (row, col) dimensions of the expression.

size

int – The number of entries in the expression.

value

NumPy.ndarray or None – The numeric value of the constant.