Other Atoms¶
The atoms listed here are neither affine nor elementwise.
cummax¶
diff_pos¶
- cvxpy.atoms.one_minus_pos.diff_pos(x, y)[source]¶
The difference \(x - y\) with domain {x, y : x > y > 0}.
This atom is log-log concave.
- Parameters:
x (
Expression
) – An Expression.y (
Expression
) – An Expression.
dotsort¶
- class cvxpy.atoms.dotsort.dotsort(X, W)[source]¶
Bases:
Atom
Computes \(\langle sort\left(vec(X)\right), sort\left(vec(W)\right) \rangle\), where \(X\) is an expression and \(W\) is constant.
Both arguments are flattened, i.e., we define \(x=vec(X)\), \(w=vec(W)\).If the length of \(w\) is less than the length of \(x\), it is conceptually padded with zeros.When the length of \(w\) is larger than the length of \(x\), an exception is raised.dotsort is a generalization of sum_largest and sum_smallest:
sum_largest(x, 3) is equivalent to dotsort(x,[1,1,1])sum_largest(x, 3.5) is equivalent to dotsort(x,[1,1,1,0.5])sum_smallest(x,3) is equivalent to -dotsort(x, [-1,-1,-1])When the constant argument is not a boolean vector, dotsort can be considered as a weighted sum of \(x\), where the largest weight is assigned to the largest entry in \(x\), etc..
eye_minus_inv¶
- class cvxpy.atoms.eye_minus_inv.eye_minus_inv(X)[source]¶
Bases:
Atom
The unity resolvent of a positive matrix, \((I - X)^{-1}\).
For an elementwise positive matrix \(X\), this atom represents
\[(I - X)^{-1},\]and it enforces the constraint that the spectral radius of \(X\) is at most \(1\).
This atom is log-log convex.
- Parameters:
X (cvxpy.Expression) – A positive square matrix.
geo_mean¶
- class cvxpy.atoms.geo_mean.geo_mean(x, p: List[int] | None = None, max_denom: int = 1024)[source]¶
Bases:
Atom
The (weighted) geometric mean of vector
x
, with optional powers given byp
:\[\left(x_1^{p_1} \cdots x_n^{p_n} \right)^{\frac{1}{\mathbf{1}^Tp}}\]The powers
p
can be alist
,tuple
, ornumpy.array
of nonnegativeint
,float
, orFraction
objects with nonzero sum.If not specified,
p
defaults to a vector of all ones, giving the unweighted geometric mean\[x_1^{1/n} \cdots x_n^{1/n}.\]The geometric mean includes an implicit constraint that \(x_i \geq 0\) whenever \(p_i > 0\). If \(p_i = 0\), \(x_i\) will be unconstrained.
The only exception to this rule occurs when
p
has exactly one nonzero element, say,p_i
, in which casegeo_mean(x, p)
is equivalent tox_i
(without the nonnegativity constraint). A specific case of this is when \(x \in \mathbf{R}^1\).Note
Generally,
p
cannot be represented exactly, so a rational, i.e., fractional, approximation must be made.Internally,
geo_mean
immediately computes an approximate normalized weight vector \(w \approx p/\mathbf{1}^Tp\) and thegeo_mean
atom is represented as\[x_1^{w_1} \cdots x_n^{w_n},\]where the elements of
w
areFraction
objects that sum to exactly 1.The maximum denominator used in forming the rational approximation is given by
max_denom
, which defaults to 1024, but can be adjusted to modify the accuracy of the approximations.The approximating
w
and the approximation error can be found through the attributesgeo_mean.w
andgeo_mean.approx_error
.Examples
The weights
w
can be seen from the string representation of thegeo_mean
object, or through thew
attribute.>>> from cvxpy import Variable, geo_mean, Problem, Maximize >>> x = Variable(3, name='x') >>> print(geo_mean(x)) geo_mean(x, (1/3, 1/3, 1/3)) >>> g = geo_mean(x, [1, 2, 1]) >>> g.w (Fraction(1, 4), Fraction(1, 2), Fraction(1, 4))
Floating point numbers with few decimal places can sometimes be represented exactly. The approximation error between
w
andp/sum(p)
is given by theapprox_error
attribute.>>> import numpy as np >>> x = Variable(4, name='x') >>> p = np.array([.12, .34, .56, .78]) >>> g = geo_mean(x, p) >>> g.w (Fraction(1, 15), Fraction(17, 90), Fraction(14, 45), Fraction(13, 30)) >>> g.approx_error 0.0
In general, the approximation is not exact.
>>> p = [.123, .456, .789, .001] >>> g = geo_mean(x, p) >>> g.w (Fraction(23, 256), Fraction(341, 1024), Fraction(295, 512), Fraction(1, 1024)) >>> 1e-4 <= g.approx_error <= 1e-3 True
The weight vector
p
can contain combinations ofint
,float
, andFraction
objects.>>> from fractions import Fraction >>> x = Variable(4, name='x') >>> g = geo_mean(x, [.1, Fraction(1,3), 0, 2]) >>> print(g) geo_mean(x, (3/73, 10/73, 0, 60/73)) >>> g.approx_error <= 1e-10 True
Sequences of
Fraction
andint
powers can often be represented exactly.>>> p = [Fraction(1,17), Fraction(4,9), Fraction(1,3), Fraction(25,153)] >>> x = Variable(4, name='x') >>> print(geo_mean(x, p)) geo_mean(x, (1/17, 4/9, 1/3, 25/153))
Terms with a zero power will not have an implicit nonnegativity constraint.
>>> p = [1, 0, 1] >>> x = Variable(3, name='x') >>> obj = Maximize(geo_mean(x,p)) >>> constr = [sum(x) <= 1, -1 <= x, x <= 1] >>> val = Problem(obj, constr).solve() >>> x = np.array(x.value).flatten() >>> print(x) [ 1. -1. 1.]
- Parameters:
x (Variable) – A column or row vector whose elements we will take the geometric mean of.
p (Sequence (list, tuple, …) of
int
,float
, orFraction
objects) –A vector of weights for the weighted geometric mean
When
p
is a sequence ofint
and/orFraction
objects,w
can often be an exact representation of the weights. An exact representation is sometimes possible whenp
hasfloat
elements with only a few decimal places.max_denom (int) – The maximum denominator to use in approximating
p/sum(p)
withgeo_mean.w
. Ifw
is not an exact representation, increasingmax_denom
may offer a more accurate representation, at the cost of requiring more convex inequalities to represent the geometric mean.
- w¶
A rational approximation of
p/sum(p)
.- Type:
tuple of
Fractions
- approx_error¶
The error in approximating
p/sum(p)
withw
, given by \(\|p/\mathbf{1}^T p - w \|_\infty\)- Type:
float
gmatmul¶
- class cvxpy.atoms.gmatmul.gmatmul(A, X)[source]¶
Bases:
Atom
Geometric matrix multiplication; \(A \mathbin{\diamond} X\).
For \(A \in \mathbf{R}^{m \times n}\) and \(X \in \mathbf{R}^{n \times p}_{++}\), this atom represents
\[\begin{split}\left[\begin{array}{ccc} \prod_{j=1}^n X_{j1}^{A_{1j}} & \cdots & \prod_{j=1}^n X_{pj}^{A_{1j}} \\ \vdots & & \vdots \\ \prod_{j=1}^n X_{j1}^{A_{mj}} & \cdots & \prod_{j=1}^n X_{pj}^{A_{mj}} \end{array}\right]\end{split}\]This atom is log-log affine (in \(X\)).
- Parameters:
A (cvxpy.Expression) – A constant matrix.
X (cvxpy.Expression) – A positive matrix.
harmonic_mean¶
- cvxpy.atoms.harmonic_mean.harmonic_mean(x)[source]¶
The harmonic mean of
x
.- Parameters:
x (Expression or numeric) – The expression whose harmonic mean is to be computed. Must have positive entries.
- Returns:
- \[\frac{n}{\left(\sum_{i=1}^{n} x_i^{-1} \right)},\]
where \(n\) is the length of \(x\).
- Return type:
inv_prod¶
- cvxpy.atoms.inv_prod.inv_prod(value)[source]¶
The reciprocal of a product of the entries of a vector
x
.- Parameters:
x (Expression or numeric) – The expression whose reciprocal product is to be computed. Must have positive entries.
- Returns:
- \[\left(\prod_{i=1}^n x_i\right)^{-1},\]
where \(n\) is the length of \(x\).
- Return type:
lambda_max¶
lambda_min¶
lambda_sum_largest¶
lambda_sum_smallest¶
log_det¶
log_sum_exp¶
matrix_frac¶
max¶
min¶
mixed_norm¶
- cvxpy.atoms.mixed_norm.mixed_norm(X, p: int | str = 2, q: int | str = 1)[source]¶
Lp,q norm; \((\sum_k (\sum_l \lvert x_{k,l} \rvert^p)^{q/p})^{1/q}\).
- Parameters:
X (Expression or numeric constant) – The matrix to take the l_{p,q} norm of.
p (int or str, optional) – The type of inner norm.
q (int or str, optional) – The type of outer norm.
- Returns:
An Expression representing the mixed norm.
- Return type:
norm¶
- cvxpy.atoms.norm.norm(x, p: int | str = 2, axis=None, keepdims: bool = False)[source]¶
Wrapper on the different norm atoms.
- Parameters:
x (Expression or numeric constant) – The value to take the norm of. If x is 2D and axis is None, this function constructs a matrix norm.
p (int or str, optional) – The type of norm. Valid options include any positive integer, ‘fro’ (for frobenius), ‘nuc’ (sum of singular values), np.inf or ‘inf’ (infinity norm).
axis (The axis along which to apply the norm, if any.) –
keepdims (If this is set to True, the axes which are reduced are left) – in the result as dimensions with size one.
- Returns:
An Expression representing the norm.
- Return type:
norm1¶
norm2¶
- cvxpy.atoms.norm.norm2(x, axis=None)[source]¶
The 2-norm of x.
- Parameters:
x (Expression or numeric constant) – The value to take the norm of. If x is 2D and axis is None, this function constructs a matrix norm.
- Returns:
An Expression representing the norm.
- Return type:
norm_inf¶
normNuc¶
one_minus_pos¶
- class cvxpy.atoms.one_minus_pos(x)[source]¶
Bases:
Atom
The difference \(1 - x\) with domain {x : 0 < x < 1}.
This atom is log-log concave.
- Parameters:
x (
Expression
) – An Expression.
perspective¶
- class cvxpy.atoms.perspective.perspective(f: Expression, s: Variable, f_recession: Expression | None = None)[source]¶
Bases:
Atom
Implements the perspective transform of a convex or concave scalar expression. Uses the fact that given a cone form for the epigraph of \(f\) via
\(\{ (t, x) \in \mathbb{R}^{n+1} : t \geq f(x) \}\) \(= \{ (t,x) : Fx + gt + e \in K \},\)
the epigraph of the perspective transform of f can be given by
\(\{ (t, x, s) \in \mathbb{R}^{n+2} : t \geq sf(x/s) \}\) \(= \{ (t,x,s) : Fx + gt + se \in K \},\)
(see https://web.stanford.edu/~boyd/papers/pdf/sw_aff_ctrl.pdf).
Note that this is the perspective transform of a scalar expression viewed as a function of its underlying variables. The perspective atom does not return a Callable, so you cannot create compositions such as \(p(g(x),s)\), where \(p(z,s) = sf(z/s)\) is the perpective transform of \(f\).
pf_eigenvalue¶
- class cvxpy.atoms.pf_eigenvalue.pf_eigenvalue(X)[source]¶
Bases:
Atom
The Perron-Frobenius eigenvalue of a positive matrix.
For an elementwise positive matrix \(X\), this atom represents its spectral radius, i.e., the magnitude of its largest eigenvalue. Because \(X\) is positive, the spectral radius equals its largest eigenvalue, which is guaranteed to be positive.
This atom is log-log convex.
- Parameters:
X (cvxpy.Expression) – A positive square matrix.
pnorm¶
- cvxpy.atoms.pnorm.pnorm(x, p: int | str = 2, axis=None, keepdims: bool = False, max_denom: int = 1024)[source]¶
Factory function for a mathematical p-norm.
- Parameters:
p (numeric type or string) – The type of norm to construct; set this to np.inf or ‘inf’ to construct an infinity norm.
- Returns:
A norm1, norm_inf, or Pnorm object.
- Return type:
Pnorm¶
- class cvxpy.atoms.pnorm.Pnorm(x, p: int = 2, axis=None, keepdims: bool = False, max_denom: int = 1024)[source]¶
The vector p-norm, for p not equal to 1 or infinity.
If given a matrix variable,
pnorm
will treat it as a vector, and compute the p-norm of the concatenated columns. Only accepts p values that are not equal to 1 or infinity; the norm1 and norm_inf classes handle those norms.For \(p > 1\), the p-norm is given by
\[\|x\|_p = \left(\sum_i |x_i|^p \right)^{1/p},\]with domain \(x \in \mathbf{R}^n\).
For \(p < 1,\ p \neq 0\), the p-norm is given by
\[\|x\|_p = \left(\sum_i x_i^p \right)^{1/p},\]with domain \(x \in \mathbf{R}^n_+\).
Note that the “p-norm” is actually a norm only when \(p > 1\). For these cases, it is convex.
The expression is not defined when \(p = 0\).
Otherwise, when \(p < 1\), the expression is concave, but it is not a true norm.
Note
Generally,
p
cannot be represented exactly, so a rational, i.e., fractional, approximation must be made.Internally,
pnorm
computes a rational approximation to the reciprocal \(1/p\) with a denominator up tomax_denom
. The resulting approximation can be found through the attributepnorm.p
. The approximation error is given by the attributepnorm.approx_error
. Increasingmax_denom
can give better approximations.When
p
is anint
orFraction
object, the approximation is usually exact.- Parameters:
x (cvxpy.Variable) – The value to take the norm of.
p (int, float, or Fraction) – We require that \(p > 1\), but \(p \neq \infty\). See the norm1 and norm_inf classes for these norms, or use the pnorm function wrapper to instantiate them.
- max_denomint
The maximum denominator considered in forming a rational approximation for
p
.- axis0 or 1
The axis to apply the norm to.
- Returns:
An Expression representing the norm.
- Return type:
ptp¶
prod¶
- cvxpy.atoms.prod.prod(expr, axis=None, keepdims: bool = False) Prod [source]¶
Multiply the entries of an expression.
The semantics of this atom are the same as np.prod.
This atom is log-log affine, but it is neither convex nor concave.
- Parameters:
expr (Expression or list[Expression, Numeric]) – The expression to multiply the entries of, or a list of Expressions and numeric types.
axis (int) – The axis along which to take the product; ignored if expr is a list.
keepdims (bool) – Whether to drop dimensions after taking the product; ignored if expr is a list.
quad_form¶
quad_over_lin¶
resolvent¶
- cvxpy.atoms.eye_minus_inv.resolvent(X, s: float)[source]¶
The resolvent of a positive matrix, \((sI - X)^{-1}\).
For an elementwise positive matrix \(X\) and a positive scalar \(s\), this atom computes
\[(sI - X)^{-1},\]and it enforces the constraint that the spectral radius of \(X/s\) is at most \(1\).
This atom is log-log convex.
- Parameters:
X (cvxpy.Expression) – A positive square matrix.
s (cvxpy.Expression or numeric) – A positive scalar.
sigma_max¶
std¶
sum_largest¶
sum_smallest¶
sum_squares¶
- cvxpy.atoms.sum_squares.sum_squares(expr)[source]¶
The sum of the squares of the entries.
- Parameters:
expr (Expression) – The expression to take the sum of squares of.
- Returns:
An expression representing the sum of squares.
- Return type:
SuppFuncAtom¶
tr_inv¶
tv¶
- cvxpy.atoms.total_variation.tv(value, *args)[source]¶
Total variation of a vector, matrix, or list of matrices.
Uses L1 norm of discrete gradients for vectors and L2 norm of discrete gradients for matrices.
- Parameters:
value (Expression or numeric constant) – The value to take the total variation of.
args (Matrix constants/expressions) – Additional matrices extending the third dimension of value.
- Returns:
An Expression representing the total variation.
- Return type:
var¶
von_neumann_entr¶
- class cvxpy.von_neumann_entr(X, quad_approx: Tuple[int, int] = ())[source]¶
Bases:
Atom
Computes the Von Neumann Entropy of the positive-definite matrix \(X\in\mathbb{S}^n_{+}\)
\[-\operatorname{tr}(X \operatorname{logm}(X))\]where \(\operatorname{tr}\) is the trace and \(\operatorname{logm}\) is the matrix logarithm
May alternatively be expressed as:\[\texttt{von_neumann_entr}(X) = -\textstyle\sum_{i=1}^n \lambda_i \log \lambda_i\]where \(\lambda_{i}\) are the eigenvalues of \(X\)This atom does not enforce \(\operatorname{tr}(X) = 1\) as is expected in applications from quantum mechanics.
- Parameters:
X (Expression or numeric) – A PSD matrix