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
quad_form(x, P) constant $P \in \mathbf{S}^n_+$	$x^T P x$	$x \in \mathbf{R}^n$	positive	convex	for $x_i \geq 0$ for $x_i \leq 0$
quad_form(x, P) constant $P \in \mathbf{S}^n_-$	$x^T P x$	$x \in \mathbf{R}^n$	negative	concave	for $x_i \geq 0$ for $x_i \leq 0$
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	depends on c
quad_over_lin(X, y)	$\left(\sum_{ij}X_{ij}^2\right)/y$	$x \in \mathbf{R}^n$ $y > 0$	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)	approximate log of the Gamma function	$x > 0$	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., $y = (X[0,1{:}],\, X[1, 2{:}],\, \ldots, X[n-1, n])$.