Atomic Functions

This section of the tutorial describes the atomic functions that can be applied to CVXPY expressions. CVXPY uses the function information in this section and the DCP rules to mark expressions with a sign and curvature.

Operators

The infix operators +, -, *, /, @ are treated as functions. The operators + and - are always affine functions. The expression expr1*expr2 is affine in CVXPY when one of the expressions is constant, and expr1/expr2 is affine when expr2 is a scalar constant.

Historically, CVXPY used expr1 * expr2 to denote matrix multiplication. This is now deprecated. Starting with Python 3.5, users can write expr1 @ expr2 for matrix multiplication and dot products. As of CVXPY version 1.1, we are adopting a new standard:

  • @ should be used for matrix-matrix and matrix-vector multiplication,

  • * should be matrix-scalar and vector-scalar multiplication

Elementwise multiplication can be applied with the multiply function.

Indexing and slicing

Indexing in CVXPY follows exactly the same semantics as NumPy ndarrays. For example, if expr has shape (5,) then expr[1] gives the second entry. More generally, expr[i:j:k] selects every kth element of expr, starting at i and ending at j-1. If expr is a matrix, then expr[i:j:k] selects rows, while expr[i:j:k, r:s:t] selects both rows and columns. Indexing drops dimensions while slicing preserves dimensions. For example,

x = cvxpy.Variable(5)
print("0 dimensional", x[0].shape)
print("1 dimensional", x[0:1].shape)
O dimensional: ()
1 dimensional: (1,)

Transpose

The transpose of any expression can be obtained using the syntax expr.T. Transpose is an affine function.

Power

For any CVXPY expression expr, the power operator expr**p is equivalent to the function power(expr, p).

Scalar functions

A scalar function takes one or more scalars, vectors, or matrices as arguments and returns a scalar.

Function

Meaning

Domain

Sign

Curvature 

Monotonicity

dotsort(X,W)

constant \(W \in \mathbf{R}^{o \times p}\)

\(\langle sort\left(vec(X)\right), sort\left(vec(W)\right) \rangle\)

\(X \in \mathbf{R}^{m \times n}\)

depends on \(X\), \(W\)

convex convex

incr for \(\min(W) \geq 0\)

decr for \(\max(W) \leq 0\)

geo_mean(x)

geo_mean(x, p)

\(p \in \mathbf{R}^n_{+}\)

\(p \neq 0\)

\(x_1^{1/n} \cdots x_n^{1/n}\)

\(\left(x_1^{p_1} \cdots x_n^{p_n}\right)^{\frac{1}{\mathbf{1}^T p}}\)

\(x \in \mathbf{R}^n_{+}\)

positive positive

concave concave

incr incr.

harmonic_mean(x)

\(\frac{n}{\frac{1}{x_1} + \cdots + \frac{1}{x_n}}\)

\(x \in \mathbf{R}^n_{+}\)

positive positive

concave concave

incr incr.

inv_prod(x)

\((x_1\cdots x_n)^{-1}\)

\(x \in \mathbf{R}^n_+\)

positive positive

convex convex

decr decr.

lambda_max(X)

\(\lambda_{\max}(X)\)

\(X \in \mathbf{S}^n\)

unknown unknown

convex convex

None

lambda_min(X)

\(\lambda_{\min}(X)\)

\(X \in \mathbf{S}^n\)

unknown unknown

concave concave

None

lambda_sum_largest(X,k)

\(k = 1,\ldots, n\)

\(\text{sum of $k$ largest}\\ \text{eigenvalues of $X$}\)

\(X \in\mathbf{S}^{n}\)

unknown unknown

convex convex

None

lambda_sum_smallest(X,k)

\(k = 1,\ldots, n\)

\(\text{sum of $k$ smallest}\\ \text{eigenvalues of $X$}\)

\(X \in\mathbf{S}^{n}\)

unknown unknown

concave concave

None

log_det(X)

\(\log \left(\det (X)\right)\)

\(X \in \mathbf{S}^n_+\)

unknown unknown

concave concave

None

log_sum_exp(X)

\(\log \left(\sum_{ij}e^{X_{ij}}\right)\)

\(X \in\mathbf{R}^{m \times n}\)

unknown unknown

convex convex

incr incr.

matrix_frac(x, P)

\(x^T P^{-1} x\)

\(x \in \mathbf{R}^n\)

\(P \in\mathbf{S}^n_{++}\)

positive positive

convex convex

None

max(X)

\(\max_{ij}\left\{ X_{ij}\right\}\)

\(X \in\mathbf{R}^{m \times n}\)

same as X

convex convex

incr incr.

mean(X)

\(\frac{1}{m n}\sum_{ij}\left\{ X_{ij}\right\}\)

\(X \in\mathbf{R}^{m \times n}\)

same as X

affine affine

incr incr.

min(X)

\(\min_{ij}\left\{ X_{ij}\right\}\)

\(X \in\mathbf{R}^{m \times n}\)

same as X

concave concave

incr incr.

mixed_norm(X, p, q)

\(\left(\sum_k\left(\sum_l\lvert x_{k,l}\rvert^p\right)^{q/p}\right)^{1/q}\)

\(X \in\mathbf{R}^{n \times n}\)

positive positive

convex convex

None

norm(x)

norm(x, 2)

\(\sqrt{\sum_{i} \lvert x_{i} \rvert^2 }\)

\(X \in\mathbf{R}^{n}\)

positive positive

convex convex

incr for \(x_{i} \geq 0\)

decr for \(x_{i} \leq 0\)

norm(x, 1)

\(\sum_{i}\lvert x_{i} \rvert\)

\(x \in\mathbf{R}^{n}\)

positive positive

convex convex

incr for \(x_{i} \geq 0\)

decr for \(x_{i} \leq 0\)

norm(x, “inf”)

\(\max_{i} \{\lvert x_{i} \rvert\}\)

\(x \in\mathbf{R}^{n}\)

positive positive

convex convex

incr for \(x_{i} \geq 0\)

decr for \(x_{i} \leq 0\)

norm(X, “fro”)

\(\sqrt{\sum_{ij}X_{ij}^2 }\)

\(X \in\mathbf{R}^{m \times n}\)

positive positive

convex convex

incr for \(X_{ij} \geq 0\)

decr for \(X_{ij} \leq 0\)

norm(X, 1)

\(\max_{j} \|X_{:,j}\|_1\)

\(X \in\mathbf{R}^{m \times n}\)

positive positive

convex convex

incr for \(X_{ij} \geq 0\)

decr for \(X_{ij} \leq 0\)

norm(X, “inf”)

\(\max_{i} \|X_{i,:}\|_1\)

\(X \in\mathbf{R}^{m \times n}\)

positive positive

convex convex

incr for \(X_{ij} \geq 0\)

decr for \(X_{ij} \leq 0\)

norm(X, “nuc”)

\(\mathrm{tr}\left(\left(X^T X\right)^{1/2}\right)\)

\(X \in\mathbf{R}^{m \times n}\)

positive positive

convex convex

None

norm(X)

norm(X, 2)

\(\sqrt{\lambda_{\max}\left(X^T X\right)}\)

\(X \in\mathbf{R}^{m \times n}\)

positive positive

convex convex

None

perspective(f(x),s)

\(sf(x/s)\)

\(x \in \mathop{\bf dom} f\)

\(s \geq 0\)

same as f

convex / concave

same as \(f\)

None

pnorm(X, p)

\(p \geq 1\)

or p = 'inf'

\(\|X\|_p = \left(\sum_{ij} |X_{ij}|^p \right)^{1/p}\)

\(X \in \mathbf{R}^{m \times n}\)

positive positive

convex convex

incr for \(X_{ij} \geq 0\)

decr for \(X_{ij} \leq 0\)

pnorm(X, p)

\(p < 1\), \(p \neq 0\)

\(\|X\|_p = \left(\sum_{ij} X_{ij}^p \right)^{1/p}\)

\(X \in \mathbf{R}^{m \times n}_+\)

positive positive

concave concave

incr incr.

ptp(X)

\(\max_{ij} X_{ij} - \min_{ij} X_{ij}\)

\(X \in \mathbf{R}^{m \times n}\)

positive positive

convex convex

None

quad_form(x, P)

constant \(P \in \mathbf{S}^n_+\)

\(x^T P x\)

\(x \in \mathbf{R}^n\)

positive positive

convex convex

incr for \(x_i \geq 0\)

decr for \(x_i \leq 0\)

quad_form(x, P)

constant \(P \in \mathbf{S}^n_-\)

\(x^T P x\)

\(x \in \mathbf{R}^n\)

negative negative

concave concave

decr for \(x_i \geq 0\)

incr for \(x_i \leq 0\)

quad_form(c, X)

constant \(c \in \mathbf{R}^n\)

\(c^T X c\)

\(X \in\mathbf{R}^{n \times n}\)

depends on c, X

affine affine

depends on c

quad_over_lin(X, y)

\(\left(\sum_{ij}X_{ij}^2\right)/y\)

\(x \in \mathbf{R}^n\)

\(y > 0\)

positive positive

convex convex

incr for \(X_{ij} \geq 0\)

decr for \(X_{ij} \leq 0\)

decr decr. in \(y\)

std(X)

\(\sqrt{\frac{1}{mn} \sum_{ij}\left(X_{ij} - \frac{1}{mn}\sum_{k\ell} X_{k\ell}\right)^2}\)

\(X \in\mathbf{R}^{m \times n}\)

positive positive

convex convex

None

sum(X)

\(\sum_{ij}X_{ij}\)

\(X \in\mathbf{R}^{m \times n}\)

same as X

affine affine

incr incr.

sum_largest(X, k)

\(k = 1,2,\ldots\)

\(\text{sum of } k\text{ largest }X_{ij}\)

\(X \in\mathbf{R}^{m \times n}\)

same as X

convex convex

incr incr.

sum_smallest(X, k)

\(k = 1,2,\ldots\)

\(\text{sum of } k\text{ smallest }X_{ij}\)

\(X \in\mathbf{R}^{m \times n}\)

same as X

concave concave

incr incr.

sum_squares(X)

\(\sum_{ij}X_{ij}^2\)

\(X \in\mathbf{R}^{m \times n}\)

positive positive

convex convex

incr for \(X_{ij} \geq 0\)

decr for \(X_{ij} \leq 0\)

trace(X)

\(\mathrm{tr}\left(X \right)\)

\(X \in\mathbf{R}^{n \times n}\)

same as X

affine affine

incr incr.

tr_inv(X)

\(\mathrm{tr}\left(X^{-1} \right)\)

\(X \in\mathbf{S}^n_{++}\)

positive positive

convex convex

None

tv(x)

\(\sum_{i}|x_{i+1} - x_i|\)

\(x \in \mathbf{R}^n\)

positive positive

convex convex

None

tv(X)

\(\sum_{ij}\left\| \left[\begin{matrix} X_{i+1,j} - X_{ij} \\ X_{i,j+1} -X_{ij} \end{matrix}\right] \right\|_2\)

\(X \in \mathbf{R}^{m \times n}\)

positive positive

convex convex

None

tv([X1,…,Xk])

\(\sum_{ij}\left\| \left[\begin{matrix} X_{i+1,j}^{(1)} - X_{ij}^{(1)} \\ X_{i,j+1}^{(1)} -X_{ij}^{(1)} \\ \vdots \\ X_{i+1,j}^{(k)} - X_{ij}^{(k)} \\ X_{i,j+1}^{(k)} -X_{ij}^{(k)} \end{matrix}\right] \right\|_2\)

\(X^{(i)} \in\mathbf{R}^{m \times n}\)

positive positive

convex convex

None

var(X)

\({\frac{1}{mn} \sum_{ij}\left(X_{ij} - \frac{1}{mn}\sum_{k\ell} X_{k\ell}\right)^2}\)

\(X \in\mathbf{R}^{m \times n}\)

positive positive

convex convex

None

von_neumann_entr(X)

\(-\operatorname{tr}(X\operatorname{logm}(X))\)

\(X \in \mathbf{S}^{n}_+\)

unknown unknown

concave concave

None

Clarifications for scalar functions

The domain \(\mathbf{S}^n\) refers to the set of symmetric matrices. The domains \(\mathbf{S}^n_+\) and \(\mathbf{S}^n_-\) refer to the set of positive semi-definite and negative semi-definite matrices, respectively. Similarly, \(\mathbf{S}^n_{++}\) and \(\mathbf{S}^n_{--}\) refer to the set of positive definite and negative definite matrices, respectively.

For a vector expression x, norm(x) and norm(x, 2) give the Euclidean norm. For a matrix expression X, however, norm(X) and norm(X, 2) give the spectral norm.

The function norm(X, "fro") is called the Frobenius norm and norm(X, "nuc") the nuclear norm. The nuclear norm can also be defined as the sum of X’s singular values.

The functions max and min give the largest and smallest entry, respectively, in a single expression. These functions should not be confused with maximum and minimum (see Elementwise functions). Use maximum and minimum to find the max or min of a list of scalar expressions.

The CVXPY function sum sums all the entries in a single expression. The built-in Python sum should be used to add together a list of expressions. For example, the following code sums a list of three expressions:

expr_list = [expr1, expr2, expr3]
expr_sum = sum(expr_list)

Functions along an axis

The functions sum, norm, max, min, mean, std, var, and ptp can be applied along an axis. Given an m by n expression expr, the syntax func(expr, axis=0, keepdims=True) applies func to each column, returning a 1 by n expression. The syntax func(expr, axis=1, keepdims=True) applies func to each row, returning an m by 1 expression. By default keepdims=False, which means dimensions of length 1 are dropped. For example, the following code sums along the columns and rows of a matrix variable:

X = cvxpy.Variable((5, 4))
col_sums = cvxpy.sum(X, axis=0, keepdims=True) # Has size (1, 4)
col_sums = cvxpy.sum(X, axis=0) # Has size (4,)
row_sums = cvxpy.sum(X, axis=1) # Has size (5,)

Elementwise functions

These functions operate on each element of their arguments. For example, if X is a 5 by 4 matrix variable, then abs(X) is a 5 by 4 matrix expression. abs(X)[1, 2] is equivalent to abs(X[1, 2]).

Elementwise functions that take multiple arguments, such as maximum and multiply, operate on the corresponding elements of each argument. For example, if X and Y are both 3 by 3 matrix variables, then maximum(X, Y) is a 3 by 3 matrix expression. maximum(X, Y)[2, 0] is equivalent to maximum(X[2, 0], Y[2, 0]). This means all arguments must have the same dimensions or be scalars, which are promoted.

Function

Meaning

Domain

Sign

Curvature 

Monotonicity

abs(x)

\(\lvert x \rvert\)

\(x \in \mathbf{C}\)

positive positive

convex convex

incr for \(x \geq 0\)

decr for \(x \leq 0\)

conj(x)

complex conjugate

\(x \in \mathbf{C}\)

unknown unknown

affine affine

None

entr(x)

\(-x \log (x)\)

\(x > 0\)

unknown unknown

concave concave

None

exp(x)

\(e^x\)

\(x \in \mathbf{R}\)

positive positive

convex convex

incr incr.

huber(x, M=1)

\(M \geq 0\)

\(\begin{cases}x^2 &|x| \leq M \\2M|x| - M^2&|x| >M\end{cases}\)

\(x \in \mathbf{R}\)

positive positive

convex convex

incr for \(x \geq 0\)

decr for \(x \leq 0\)

imag(x)

imaginary part of a complex number

\(x \in \mathbf{C}\)

unknown unknown

affine affine

none

inv_pos(x)

\(1/x\)

\(x > 0\)

positive positive

convex convex

decr decr.

kl_div(x, y)

\(x \log(x/y) - x + y\)

\(x > 0\)

\(y > 0\)

positive positive

convex convex

None

log(x)

\(\log(x)\)

\(x > 0\)

unknown unknown

concave concave

incr incr.

log_normcdf(x)

approximate log of the standard normal CDF

\(x \in \mathbf{R}\)

negative negative

concave concave

incr incr.

log1p(x)

\(\log(x+1)\)

\(x > -1\)

same as x

concave concave

incr incr.

loggamma(x)

approximate log of the Gamma function

\(x > 0\)

unknown unknown

convex convex

None

logistic(x)

\(\log(1 + e^{x})\)

\(x \in \mathbf{R}\)

positive positive

convex convex

incr incr.

maximum(x, y)

\(\max \left\{x, y\right\}\)

\(x,y \in \mathbf{R}\)

depends on x,y

convex convex

incr incr.

minimum(x, y)

\(\min \left\{x, y\right\}\)

\(x, y \in \mathbf{R}\)

depends on x,y

concave concave

incr incr.

multiply(c, x)

\(c \in \mathbf{R}\)

c*x

\(x \in\mathbf{R}\)

\(\mathrm{sign}(cx)\)

affine affine

depends on c

neg(x)

\(\max \left\{-x, 0 \right\}\)

\(x \in \mathbf{R}\)

positive positive

convex convex

decr decr.

pos(x)

\(\max \left\{x, 0 \right\}\)

\(x \in \mathbf{R}\)

positive positive

convex convex

incr incr.

power(x, 0)

\(1\)

\(x \in \mathbf{R}\)

positive positive

constant

 

power(x, 1)

\(x\)

\(x \in \mathbf{R}\)

same as x

affine affine

incr incr.

power(x, p)

\(p = 2, 4, 8, \ldots\)

\(x^p\)

\(x \in \mathbf{R}\)

positive positive

convex convex

incr for \(x \geq 0\)

decr for \(x \leq 0\)

power(x, p)

\(p < 0\)

\(x^p\)

\(x > 0\)

positive positive

convex convex

decr decr.

power(x, p)

\(0 < p < 1\)

\(x^p\)

\(x \geq 0\)

positive positive

concave concave

incr incr.

power(x, p)

\(p > 1,\ p \neq 2, 4, 8, \ldots\)

\(x^p\)

\(x \geq 0\)

positive positive

convex convex

incr incr.

real(x)

real part of a complex number

\(x \in \mathbf{C}\)

unknown unknown

affine affine

incr incr.

rel_entr(x, y)

\(x \log(x/y)\)

\(x > 0\)

\(y > 0\)

unknown unknown

convex convex

None in \(x\)

decr in \(y\)

scalene(x, alpha, beta)

\(\text{alpha} \geq 0\)

\(\text{beta} \geq 0\)

\(\alpha\mathrm{pos}(x)+ \beta\mathrm{neg}(x)\)

\(x \in \mathbf{R}\)

positive positive

convex convex

incr for \(x \geq 0\)

decr for \(x \leq 0\)

sqrt(x)

\(\sqrt x\)

\(x \geq 0\)

positive positive

concave concave

incr incr.

square(x)

\(x^2\)

\(x \in \mathbf{R}\)

positive positive

convex convex

incr for \(x \geq 0\)

decr for \(x \leq 0\)

xexp(x)

\(x e^x\)

\(x \geq 0\)

positive positive

convex convex

incr incr.

Clarifications on elementwise functions

The functions log_normcdf and loggamma are defined via approximations. log_normcdf has highest accuracy over the range -4 to 4, while loggamma has similar accuracy over all positive reals. See CVXPY GitHub PR #1224 and CVXPY GitHub Issue #228 for details on the approximations.

Vector/matrix functions

A vector/matrix function takes one or more scalars, vectors, or matrices as arguments and returns a vector or matrix.

CVXPY is conservative when it determines the sign of an Expression returned by one of these functions. If any argument to one of these functions has unknown sign, then the returned Expression will also have unknown sign. If all arguments have known sign but CVXPY can determine that the returned Expression would have different signs in different entries (for example, when stacking a positive Expression and a negative Expression) then the returned Expression will have unknown sign.

Function

Meaning

Domain

Curvature 

Monotonicity

bmat([[X11,…,X1q], …, [Xp1,…,Xpq]])

\(\left[\begin{matrix} X^{(1,1)} & \cdots & X^{(1,q)} \\ \vdots & & \vdots \\ X^{(p,1)} & \cdots & X^{(p,q)} \end{matrix}\right]\)

\(X^{(i,j)} \in\mathbf{R}^{m_i \times n_j}\)

affine affine

incr incr.

convolve(c, x)

\(c\in\mathbf{R}^m\)

\(c*x\)

\(x\in \mathbf{R}^n\)

affine affine

depends on c

cumsum(X, axis=0)

cumulative sum along given axis.

\(X \in \mathbf{R}^{m \times n}\)

affine affine

incr incr.

diag(x)

\(\left[\begin{matrix}x_1 & & \\& \ddots & \\& & x_n\end{matrix}\right]\)

\(x \in\mathbf{R}^{n}\)

affine affine

incr incr.

diag(X)

\(\left[\begin{matrix}X_{11} \\\vdots \\X_{nn}\end{matrix}\right]\)

\(X \in\mathbf{R}^{n \times n}\)

affine affine

incr incr.

diff(X, k=1, axis=0)

\(k \in 0,1,2,\ldots\)

kth order differences along given axis

\(X \in\mathbf{R}^{m \times n}\)

affine affine

incr incr.

hstack([X1, …, Xk])

\(\left[\begin{matrix}X^{(1)} \cdots X^{(k)}\end{matrix}\right]\)

\(X^{(i)} \in\mathbf{R}^{m \times n_i}\)

affine affine

incr incr.

kron(X, Y)

constant \(X\in\mathbf{R}^{p \times q}\)

\(\left[\begin{matrix}X_{11}Y & \cdots & X_{1q}Y \\ \vdots & & \vdots \\ X_{p1}Y & \cdots & X_{pq}Y \end{matrix}\right]\)

\(Y \in \mathbf{R}^{m \times n}\)

affine affine

depends on \(X\)

kron(X, Y)

constant \(Y\in\mathbf{R}^{m \times n}\)

\(\left[\begin{matrix}X_{11}Y & \cdots & X_{1q}Y \\ \vdots & & \vdots \\ X_{p1}Y & \cdots & X_{pq}Y \end{matrix}\right]\)

\(X \in \mathbf{R}^{p \times q}\)

affine affine

depends on \(Y\)

outer(x, y)

constant \(y \in \mathbf{R}^m\)

\(x y^T\)

\(x \in \mathbf{R}^n\)

affine affine

depends on \(y\)

partial_trace(X, dims, axis=0)

partial trace

\(X \in\mathbf{R}^{n \times n}\)

affine affine

incr incr.

partial_transpose(X, dims, axis=0)

partial transpose

\(X \in\mathbf{R}^{n \times n}\)

affine affine

incr incr.

reshape(X, (m’, n’), order=’F’)

\(X' \in\mathbf{R}^{m' \times n'}\)

\(X \in\mathbf{R}^{m \times n}\)

\(m'n' = mn\)

affine affine

incr incr.

upper_tri(X)

flatten the strictly upper-triangular part of \(X\)

\(X \in \mathbf{R}^{n \times n}\)

affine affine

incr incr.

vec(X)

\(x' \in\mathbf{R}^{mn}\)

\(X \in\mathbf{R}^{m \times n}\)

affine affine

incr incr.

vec_to_upper_tri(X, strict=False)

\(x' \in\mathbf{R}^{n(n-1)/2}\) for strict=True

\(x' \in\mathbf{R}^{n(n+1)/2}\) for strict=False

\(X \in\mathbf{R}^{n \times n}\)

affine affine

incr incr.

vstack([X1, …, Xk])

\(\left[\begin{matrix}X^{(1)} \\ \vdots \\X^{(k)}\end{matrix}\right]\)

\(X^{(i)} \in\mathbf{R}^{m_i \times n}\)

affine affine

incr incr.

Clarifications on vector and matrix functions

The input to \(\texttt{bmat}\) is a list of lists of CVXPY expressions. It constructs a block matrix. The elements of each inner list are stacked horizontally and then the resulting block matrices are stacked vertically.

The output \(y = \mathbf{convolve}(c, x)\) has size \(n+m-1\) and is defined as \(y_k =\sum_{j=0}^{k} c[j]x[k-j]\).

The output \(y = \mathbf{vec}(X)\) is the matrix \(X\) flattened in column-major order into a vector. Formally, \(y_i = X_{i \bmod{m}, \left \lfloor{i/m}\right \rfloor }\).

The output \(Y = \mathbf{reshape}(X, (m', n'), \text{order='F'})\) is the matrix \(X\) cast into an \(m' \times n'\) matrix. The entries are taken from \(X\) in column-major order and stored in \(Y\) in column-major order. Formally, \(Y_{ij} = \mathbf{vec}(X)_{m'j + i}\). If order=’C’ then \(X\) will be read in row-major order and \(Y\) will be written to in row-major order.

The output \(y = \mathbf{upper\_tri}(X)\) is formed by concatenating partial rows of \(X\). I.e., \(y = (X[0,1{:}],\, X[1, 2{:}],\, \ldots, X[n-1, n])\).