# 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

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

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

concave

incr.

harmonic_mean(x)

$$\frac{n}{\frac{1}{x_1} + \cdots + \frac{1}{x_n}}$$

$$x \in \mathbf{R}^n_{+}$$

positive

concave

incr.

inv_prod(x)

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

$$x \in \mathbf{R}^n_+$$

positive

convex

decr.

lambda_max(X)

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

$$X \in \mathbf{S}^n$$

unknown

convex

None

lambda_min(X)

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

$$X \in \mathbf{S}^n$$

unknown

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

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

concave

None

log_det(X)

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

$$X \in \mathbf{S}^n_+$$

unknown

concave

None

log_sum_exp(X)

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

$$X \in\mathbf{R}^{m \times n}$$

unknown

convex

incr.

matrix_frac(x, P)

$$x^T P^{-1} x$$

$$x \in \mathbf{R}^n$$

$$P \in\mathbf{S}^n_{++}$$

positive

convex

None

max(X)

$$\max_{ij}\left\{ X_{ij}\right\}$$

$$X \in\mathbf{R}^{m \times n}$$

same as X

convex

incr.

mean(X)

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

$$X \in\mathbf{R}^{m \times n}$$

same as X

affine

incr.

min(X)

$$\min_{ij}\left\{ X_{ij}\right\}$$

$$X \in\mathbf{R}^{m \times n}$$

same as X

concave

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

convex

None

norm(x)

norm(x, 2)

$$\sqrt{\sum_{i} \lvert x_{i} \rvert^2 }$$

$$X \in\mathbf{R}^{n}$$

positive

convex

for $$x_{i} \geq 0$$

for $$x_{i} \leq 0$$

norm(x, 1)

$$\sum_{i}\lvert x_{i} \rvert$$

$$x \in\mathbf{R}^{n}$$

positive

convex

for $$x_{i} \geq 0$$

for $$x_{i} \leq 0$$

norm(x, “inf”)

$$\max_{i} \{\lvert x_{i} \rvert\}$$

$$x \in\mathbf{R}^{n}$$

positive

convex

for $$x_{i} \geq 0$$

for $$x_{i} \leq 0$$

norm(X, “fro”)

$$\sqrt{\sum_{ij}X_{ij}^2 }$$

$$X \in\mathbf{R}^{m \times n}$$

positive

convex

for $$X_{ij} \geq 0$$

for $$X_{ij} \leq 0$$

norm(X, 1)

$$\max_{j} \|X_{:,j}\|_1$$

$$X \in\mathbf{R}^{m \times n}$$

positive

convex

for $$X_{ij} \geq 0$$

for $$X_{ij} \leq 0$$

norm(X, “inf”)

$$\max_{i} \|X_{i,:}\|_1$$

$$X \in\mathbf{R}^{m \times n}$$

positive

convex

for $$X_{ij} \geq 0$$

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

convex

None

norm(X)

norm(X, 2)

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

$$X \in\mathbf{R}^{m \times n}$$

positive

convex

None

perspective(f(x),s)

$$sf(x/s)$$

$$x \in \mathop{\bf dom} f$$

$$s \geq 0$$

same as f

/

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

convex

for $$X_{ij} \geq 0$$

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

concave

incr.

ptp(X)

$$\max_{ij} X_{ij} - \min_{ij} X_{ij}$$

$$X \in \mathbf{R}^{m \times n}$$

positive

convex

None

constant $$P \in \mathbf{S}^n_+$$

$$x^T P x$$

$$x \in \mathbf{R}^n$$

positive

convex

for $$x_i \geq 0$$

for $$x_i \leq 0$$

constant $$P \in \mathbf{S}^n_-$$

$$x^T P x$$

$$x \in \mathbf{R}^n$$

negative

concave

for $$x_i \geq 0$$

for $$x_i \leq 0$$

constant $$c \in \mathbf{R}^n$$

$$c^T X c$$

$$X \in\mathbf{R}^{n \times n}$$

depends on c, X

affine

depends on c

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

$$x \in \mathbf{R}^n$$

$$y > 0$$

positive

convex

for $$X_{ij} \geq 0$$

for $$X_{ij} \leq 0$$

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

convex

None

sum(X)

$$\sum_{ij}X_{ij}$$

$$X \in\mathbf{R}^{m \times n}$$

same as X

affine

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

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

incr.

sum_squares(X)

$$\sum_{ij}X_{ij}^2$$

$$X \in\mathbf{R}^{m \times n}$$

positive

convex

for $$X_{ij} \geq 0$$

for $$X_{ij} \leq 0$$

trace(X)

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

$$X \in\mathbf{R}^{n \times n}$$

same as X

affine

incr.

tr_inv(X)

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

$$X \in\mathbf{S}^n_{++}$$

positive

convex

None

tv(x)

$$\sum_{i}|x_{i+1} - x_i|$$

$$x \in \mathbf{R}^n$$

positive

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

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

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

convex

None

von_neumann_entr(X)

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

$$X \in \mathbf{S}^{n}_+$$

unknown

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

convex

for $$x \geq 0$$

for $$x \leq 0$$

conj(x)

complex conjugate

$$x \in \mathbf{C}$$

unknown

affine

None

entr(x)

$$-x \log (x)$$

$$x > 0$$

unknown

concave

None

exp(x)

$$e^x$$

$$x \in \mathbf{R}$$

positive

convex

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

convex

for $$x \geq 0$$

for $$x \leq 0$$

imag(x)

imaginary part of a complex number

$$x \in \mathbf{C}$$

unknown

affine

none

inv_pos(x)

$$1/x$$

$$x > 0$$

positive

convex

decr.

kl_div(x, y)

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

$$x > 0$$

$$y > 0$$

positive

convex

None

log(x)

$$\log(x)$$

$$x > 0$$

unknown

concave

incr.

log_normcdf(x)

approximate log of the standard normal CDF

$$x \in \mathbf{R}$$

negative

concave

incr.

log1p(x)

$$\log(x+1)$$

$$x > -1$$

same as x

concave

incr.

loggamma(x)

$$x > 0$$

unknown

convex

None

logistic(x)

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

$$x \in \mathbf{R}$$

positive

convex

incr.

maximum(x, y)

$$\max \left\{x, y\right\}$$

$$x,y \in \mathbf{R}$$

depends on x,y

convex

incr.

minimum(x, y)

$$\min \left\{x, y\right\}$$

$$x, y \in \mathbf{R}$$

depends on x,y

concave

incr.

multiply(c, x)

$$c \in \mathbf{R}$$

c*x

$$x \in\mathbf{R}$$

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

affine

depends on c

neg(x)

$$\max \left\{-x, 0 \right\}$$

$$x \in \mathbf{R}$$

positive

convex

decr.

pos(x)

$$\max \left\{x, 0 \right\}$$

$$x \in \mathbf{R}$$

positive

convex

incr.

power(x, 0)

$$1$$

$$x \in \mathbf{R}$$

positive

constant

power(x, 1)

$$x$$

$$x \in \mathbf{R}$$

same as x

affine

incr.

power(x, p)

$$p = 2, 4, 8, \ldots$$

$$x^p$$

$$x \in \mathbf{R}$$

positive

convex

for $$x \geq 0$$

for $$x \leq 0$$

power(x, p)

$$p < 0$$

$$x^p$$

$$x > 0$$

positive

convex

decr.

power(x, p)

$$0 < p < 1$$

$$x^p$$

$$x \geq 0$$

positive

concave

incr.

power(x, p)

$$p > 1,\ p \neq 2, 4, 8, \ldots$$

$$x^p$$

$$x \geq 0$$

positive

convex

incr.

real(x)

real part of a complex number

$$x \in \mathbf{C}$$

unknown

affine

incr.

rel_entr(x, y)

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

$$x > 0$$

$$y > 0$$

unknown

convex

None in $$x$$

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

convex

for $$x \geq 0$$

for $$x \leq 0$$

sqrt(x)

$$\sqrt x$$

$$x \geq 0$$

positive

concave

incr.

square(x)

$$x^2$$

$$x \in \mathbf{R}$$

positive

convex

for $$x \geq 0$$

for $$x \leq 0$$

xexp(x)

$$x e^x$$

$$x \geq 0$$

positive

convex

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

incr.

convolve(c, x)

$$c\in\mathbf{R}^m$$

$$c*x$$

$$x\in \mathbf{R}^n$$

affine

depends on c

cumsum(X, axis=0)

cumulative sum along given axis.

$$X \in \mathbf{R}^{m \times n}$$

affine

incr.

diag(x)

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

$$x \in\mathbf{R}^{n}$$

affine

incr.

diag(X)

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

$$X \in\mathbf{R}^{n \times n}$$

affine

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

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

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

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

depends on $$Y$$

outer(x, y)

constant $$y \in \mathbf{R}^m$$

$$x y^T$$

$$x \in \mathbf{R}^n$$

affine

depends on $$y$$

partial_trace(X, dims, axis=0)

partial trace

$$X \in\mathbf{R}^{n \times n}$$

affine

incr.

partial_transpose(X, dims, axis=0)

partial transpose

$$X \in\mathbf{R}^{n \times n}$$

affine

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

incr.

upper_tri(X)

flatten the strictly upper-triangular part of $$X$$

$$X \in \mathbf{R}^{n \times n}$$

affine

incr.

vec(X)

$$x' \in\mathbf{R}^{mn}$$

$$X \in\mathbf{R}^{m \times n}$$

affine

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

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

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., $$y = (X[0,1{:}],\, X[1, 2{:}],\, \ldots, X[n-1, n])$$.