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Functions

Functions¶

The table below lists all the atomic functions available in CVXPY.

Atoms table¶

Function	Meaning	Domain	DCP Properties	Curvature
dotsort(X,W) constant \(W \in \mathbf{R}^{o \times p}\)	\(\text{dot product of}\) \(\operatorname{sort}\operatorname{vec}(X) \text{ and}\) \(\operatorname{sort}\operatorname{vec}(W)\)	\(X \in \mathbf{R}^{m \times n}\)	sign depends on \(X\), \(W\) for \(\min(W) \geq 0\)	convex
geo_mean(x) geo_mean(x, p) \(p \in \mathbf{R}^n_{+}\) \(p \neq 0\)	\(x_1^{1/n} \cdots x_n^{1/n}\)	\(x \in \mathbf{R}^n_{+}\)	positive incr.	concave
harmonic_mean(x)	\(\frac{n}{\frac{1}{x_1} + \cdots + \frac{1}{x_n}}\)	\(x \in \mathbf{R}^n_{+}\)	positive incr.	concave
inv_prod(x)	\((x_1\cdots x_n)^{-1}\)	\(x \in \mathbf{R}^n_+\)	positive decr.	convex
lambda_max(X)	\(\lambda_{\max}(X)\)	\(X \in \mathbf{S}^n\)	unknown sign	convex
lambda_min(X)	\(\lambda_{\min}(X)\)	\(X \in \mathbf{S}^n\)	unknown sign	concave
lambda_sum_largest(X,k) \(k = 1,\ldots, n\)	\(\text{sum of $k$ largest}\) \(\text{eigenvalues of $X$}\)	\(X \in\mathbf{S}^{n}\)	unknown sign	convex
lambda_sum_smallest(X,k) \(k = 1,\ldots, n\)	\(\text{sum of $k$ smallest}\) \(\text{eigenvalues of $X$}\)	\(X \in\mathbf{S}^{n}\)	unknown sign	concave
log_det(X)	\(\log \left(\det (X)\right)\)	\(X \in \mathbf{S}^n_+\)	unknown sign	concave
log_sum_exp(X)	\(\log \left(\sum_{ij}e^{X_{ij}}\right)\)	\(X \in\mathbf{R}^{m \times n}\)	unknown sign incr.	convex
matrix_frac(x, P)	\(x^T P^{-1} x\)	\(x \in \mathbf{R}^n\)	positive	convex
max(X)	\(\max_{ij}\left\{ X_{ij}\right\}\)	\(X \in\mathbf{R}^{m \times n}\)	same sign as X incr.	convex
mean(X)	\(\frac{1}{m n}\sum_{ij}\left\{ X_{ij}\right\}\)	\(X \in\mathbf{R}^{m \times n}\)	same sign as X incr.	affine
min(X)	\(\min_{ij}\left\{ X_{ij}\right\}\)	\(X \in\mathbf{R}^{m \times n}\)	same sign as X incr.	concave
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
norm(x) norm(x, 2)	\(\sqrt{\sum_{i} \lvert x_{i} \rvert^2 }\)	\(X \in\mathbf{R}^{n}\)	positive for \(x_{i} \geq 0\)	convex
norm(x, 1)	\(\sum_{i}\lvert x_{i} \rvert\)	\(x \in\mathbf{R}^{n}\)	positive for \(x_{i} \geq 0\)	convex
norm(x, “inf”)	\(\max_{i} \{\lvert x_{i} \rvert\}\)	\(x \in\mathbf{R}^{n}\)	positive for \(x_{i} \geq 0\)	convex
norm(X, “fro”)	\(\sqrt{\sum_{ij}X_{ij}^2 }\)	\(X \in\mathbf{R}^{m \times n}\)	positive for \(X_{ij} \geq 0\)	convex
norm(X, 1)	\(\max_{j} \\|X_{:,j}\\|_1\)	\(X \in\mathbf{R}^{m \times n}\)	positive for \(X_{ij} \geq 0\)	convex
norm(X, “inf”)	\(\max_{i} \\|X_{i,:}\\|_1\)	\(X \in\mathbf{R}^{m \times n}\)	positive for \(X_{ij} \geq 0\)	convex
norm(X, “nuc”)	\(\mathrm{tr}\left(\left(X^T X\right)^{1/2}\right)\)	\(X \in\mathbf{R}^{m \times n}\)	positive	convex
norm(X) norm(X, 2)	\(\sqrt{\lambda_{\max}\left(X^T X\right)}\)	\(X in\mathbf{R}^{m \times n}\)	positive	convex
perspective(f(x),s)	\(sf(x/s)\)	\(x \in \mathop{\bf dom} f\) \(s \geq 0\)	same sign as f	/ same as \(f\)
pnorm(X, p) \(p \geq 1\) or `p = 'inf'`	\(\left(\sum_{ij} \|X_{ij}\|^p \right)^{1/p}\)	\(X \in \mathbf{R}^{m \times n}\)	positive for \(X_{ij} \geq 0\)	convex
pnorm(X, p) \(p < 1\), \(p \neq 0\)	\(\left(\sum_{ij} X_{ij}^p \right)^{1/p}\)	\(X \in \mathbf{R}^{m \times n}_+\)	positive incr.	concave
ptp(X)	\(\max_{ij} X_{ij} - \min_{ij} X_{ij}\)	\(X \in \mathbf{R}^{m \times n}\)	positive	convex
quad_form(x, P) constant \(P \in \mathbf{S}^n_+\)	\(x^T P x\)	\(x \in \mathbf{R}^n\)	positive for \(x_i \geq 0\)	convex
quad_form(x, P) constant \(P \in \mathbf{S}^n_-\)	\(x^T P x\)	\(x \in \mathbf{R}^n\)	negative for \(x_i \geq 0\)	concave
quad_form(c, X) constant \(c \in \mathbf{R}^n\)	\(c^T X c\)	\(X \in\mathbf{R}^{n \times n}\)	sign depends on c, X monotonicity depends on c	affine
quad_over_lin(X, y)	\(\left(\sum_{ij}X_{ij}^2\right)/y\)	\(x \in \mathbf{R}^n\) \(y > 0\)	positive for \(X_{ij} \geq 0\) for \(X_{ij} \leq 0\) decr. in \(y\)	convex
std(X)	analog to numpy.std	\(X \in\mathbf{R}^{m \times n}\)	positive	convex
sum(X)	\(\sum_{ij}X_{ij}\)	\(X \in\mathbf{R}^{m \times n}\)	same sign as X incr.	affine
sum_largest(X, k) \(k = 1,2,\ldots\)	\(\text{sum of } k\) \(\text{largest }X_{ij}\)	\(X \in\mathbf{R}^{m \times n}\)	same sign as X incr.	convex
sum_smallest(X, k) \(k = 1,2,\ldots\)	\(\text{sum of } k\) \(\text{smallest }X_{ij}\)	\(X \in\mathbf{R}^{m \times n}\)	same sign as X incr.	concave
sum_squares(X)	\(\sum_{ij}X_{ij}^2\)	\(X \in\mathbf{R}^{m \times n}\)	positive for \(X_{ij} \geq 0\) for \(X_{ij} \leq 0\)	convex
trace(X)	\(\mathrm{tr}\left(X \right)\)	\(X \in\mathbf{R}^{n \times n}\)	same sign as X incr.	affine
tr_inv(X)	\(\mathrm{tr}\left(X^{-1} \right)\)	\(X \in\mathbf{S}^n_{++}\)	positive	convex
tv(x)	\(\sum_{i}\|x_{i+1} - x_i\|\)	\(x \in \mathbf{R}^n\)	positive	convex
tv(X) \(Y = \left[\begin{matrix} X_{i+1,j} - X_{ij} \\ X_{i,j+1} -X_{ij} \end{matrix}\right]\)	\(\sum_{ij}\left\\| Y \right\\|_2\)	\(X \in \mathbf{R}^{m \times n}\)	positive	convex
tv([X1,…,Xk]) \(Y = \left[\begin{matrix} X_{i+1,j}^{(1)} - X_{ij}^{(1)} \\ X_{i,j+1}^{(1)} -X_{ij}^{(1)} \\ \vdots \\ X_{i+1,j}^{(k)} - X_{ij}^{(k)} \\ X_{i,j+1}^{(k)} -X_{ij}^{(k)} \end{matrix}\right]\)	\(\sum_{ij}\left\\| Y \right\\|_2\)	\(X^{(i)} \in\mathbf{R}^{m \times n}\)	positive	convex
var(X)	analog to numpy.var	\(X \in\mathbf{R}^{m \times n}\)	positive	convex
abs(x)	\(\lvert x \rvert\)	\(x \in \mathbf{C}\)	positive for \(x \geq 0\)	convex
conj(x)	complex conjugate	\(x \in \mathbf{C}\)	unknown sign	affine
entr(x)	\(-x \log (x)\)	\(x > 0\)	unknown sign	concave
exp(x)	\(e^x\)	\(x \in \mathbf{R}\)	positive incr.	convex
huber(x, M=1) \(M \geq 0\)	\(\begin{aligned} & \text{if } \|x\| \leq M\colon \\& x^2 \end{aligned}\) \(\begin{aligned} & \text{if } \|x\| > M\colon \\& 2M\|x\| - M^2 \end{aligned}\)	\(x \in \mathbf{R}\)	positive for \(x \geq 0\) for \(x \leq 0\)	convex
imag(x)	imaginary part of a complex number	\(x \in \mathbf{C}\)	unknown sign	affine
inv_pos(x)	\(1/x\)	\(x > 0\)	positive decr.	convex
kl_div(x, y)	\(x \log(x/y) - x + y\)	\(x > 0\) \(y > 0\)	positive	convex
log(x)	\(\log(x)\)	\(x > 0\)	unknown sign incr.	concave
log_normcdf(x)	approximate log of the standard normal CDF	\(x \in \mathbf{R}\)	negative incr.	concave
log1p(x)	\(\log(x+1)\)	\(x > -1\)	same sign as x incr.	concave
loggamma(x)	approximate log of the Gamma function	\(x > 0\)	unknown sign	convex
logistic(x)	\(\log(1 + e^{x})\)	\(x \in \mathbf{R}\)	positive incr.	convex
maximum(x, y)	\(\max \left\{x, y\right\}\)	\(x,y \in \mathbf{R}\)	sign depends on x,y incr.	convex
minimum(x, y)	\(\min \left\{x, y\right\}\)	\(x, y \in \mathbf{R}\)	sign depends on x,y incr.	concave
multiply(c, x) \(c \in \mathbf{R}\)	c*x	\(x \in\mathbf{R}\)	\(\mathrm{sign}(cx)\) monotonicity depends on c	affine
neg(x)	\(\max \left\{-x, 0 \right\}\)	\(x \in \mathbf{R}\)	positive decr.	convex
pos(x)	\(\max \left\{x, 0 \right\}\)	\(x \in \mathbf{R}\)	positive incr.	convex
power(x, 0)	\(1\)	\(x \in \mathbf{R}\)	positive	constant
power(x, 1)	\(x\)	\(x \in \mathbf{R}\)	same sign as x incr.	affine
power(x, p) \(p = 2, 4, 8, \ldots\)	\(x^p\)	\(x \in \mathbf{R}\)	positive for \(x \geq 0\) for \(x \leq 0\)	convex
power(x, p) \(p < 0\)	\(x^p\)	\(x > 0\)	positive decr.	convex
power(x, p) \(0 < p < 1\)	\(x^p\)	\(x \geq 0\)	positive incr.	concave
power(x, p) \(p > 1,\ p \neq 2, 4, 8, \ldots\)	\(x^p\)	\(x \geq 0\)	positive incr.	convex
real(x)	real part of a complex number	\(x \in \mathbf{C}\)	unknown sign incr.	affine
rel_entr(x, y)	\(x \log(x/y)\)	\(x > 0\) \(y > 0\)	unknown sign in \(y\)	convex
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 for \(x \geq 0\) for \(x \leq 0\)	convex
sqrt(x)	\(\sqrt x\)	\(x \geq 0\)	positive incr.	concave
square(x)	\(x^2\)	\(x \in \mathbf{R}\)	positive for \(x \geq 0\) for \(x \leq 0\)	convex
xexp(x)	\(x e^x\)	\(x \geq 0\)	positive incr.	convex
bmat()	\(\left[\begin{matrix} X^{(1,1)} & .. & X^{(1,q)} \\ \vdots & & \vdots \\ X^{(p,1)} & .. & X^{(p,q)} \end{matrix}\right]\)	\(X^{(i,j)} \in\mathbf{R}^{m_i \times n_j}\)	incr.	affine
convolve(c, x) \(c\in\mathbf{R}^m\)	\(c*x\)	\(x\in \mathbf{R}^n\)	monotonicity depends on c	affine
cumsum(X, axis=0)	cumulative sum along given axis.	\(X \in \mathbf{R}^{m \times n}\)	incr.	affine
diag(x)	\(\left[\begin{matrix}x_1 & & \\& \ddots & \\& & x_n\end{matrix}\right]\)	\(x \in\mathbf{R}^{n}\)	incr.	affine
diag(X)	\(\left[\begin{matrix}X_{11} \\\vdots \\X_{nn}\end{matrix}\right]\)	\(X \in\mathbf{R}^{n \times n}\)	incr.	affine
diff(X, k=1, axis=0) \(k \in 0,1,2,\ldots\)	kth order differences along given axis	\(X \in\mathbf{R}^{m \times n}\)	incr.	affine
hstack([X1, …, Xk])	\(\left[\begin{matrix}X^{(1)} \cdots X^{(k)}\end{matrix}\right]\)	\(X^{(i)} \in\mathbf{R}^{m \times n_i}\)	incr.	affine
kron(X, Y) constant \(X\in\mathbf{R}^{p \times q}\)	\(\left[\begin{matrix}X_{11}Y & .. & X_{1q}Y \\ \vdots & & \vdots \\ X_{p1}Y & .. & X_{pq}Y \end{matrix}\right]\)	\(Y \in \mathbf{R}^{m \times n}\)	monotonicity depends on \(X\)	affine
kron(X, Y) constant \(Y\in\mathbf{R}^{m \times n}\)	\(\left[\begin{matrix}X_{11}Y & .. & X_{1q}Y \\ \vdots & & \vdots \\ X_{p1}Y & .. & X_{pq}Y \end{matrix}\right]\)	\(X \in \mathbf{R}^{p \times q}\)	monotonicity depends on \(Y\)	affine
outer(x, y) constant \(y \in \mathbf{R}^m\)	\(x y^T\)	\(x \in \mathbf{R}^n\)	monotonicity depends on \(Y\)	affine
partial_trace(X, dims, axis=0)	partial trace	\(X \in\mathbf{R}^{n \times n}\)	incr.	affine
partial_transpose(X, dims, axis=0)	partial transpose	\(X \in\mathbf{R}^{n \times n}\)	incr.	affine
reshape(X, (m’, n’), order=’F’)	\(X' \in\mathbf{R}^{m' \times n'}\)	\(X \in\mathbf{R}^{m \times n}\) \(m'n' = mn\)	incr.	affine
upper_tri(X)	flatten the strictly upper-triangular part of \(X\)	\(X \in \mathbf{R}^{n \times n}\)	incr.	affine
vec(X)	\(x' \in\mathbf{R}^{mn}\)	\(X \in\mathbf{R}^{m \times n}\)	incr.	affine
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`	\(X \in\mathbf{R}^{n \times n}\)	incr.	affine
vstack([X1, …, Xk])	\(\left[\begin{matrix}X^{(1)} \\ \vdots \\X^{(k)}\end{matrix}\right]\)	\(X^{(i)} \in\mathbf{R}^{m_i \times n}\)	incr.	affine
geo_mean(x) geo_mean(x, p) \(p \in \mathbf{R}^n_{+}\) \(p \neq 0\)	\(x_1^{1/n} \cdots x_n^{1/n}\) \(\left(x_1^{p_1} \cdots x_n^{p_n}\right)^{\frac{1}{\mathbf{1}^T p}}\)	\(x \in \mathbf{R}^n_{+}\)	incr.	log-log affine
harmonic_mean(x)	\(\frac{n}{\frac{1}{x_1} + \cdots + \frac{1}{x_n}}\)	\(x \in \mathbf{R}^n_{+}\)	incr.	log-log concave
max(X)	\(\max_{ij}\left\{ X_{ij}\right\}\)	\(X \in\mathbf{R}^{m \times n}_{++}\)	incr.	log-log convex
min(X)	\(\min_{ij}\left\{ X_{ij}\right\}\)	\(X \in\mathbf{R}^{m \times n}_{++}\)	incr.	log-log concave
norm(x) norm(x, 2)	\(\sqrt{\sum_{i} \lvert x_{i} \rvert^2 }\)	\(X \in\mathbf{R}^{n}_{++}\)	incr.	log-log convex
norm(X, “fro”)	\(\sqrt{\sum_{ij}X_{ij}^2 }\)	\(X \in\mathbf{R}^{m \times n}_{++}\)	incr.	log-log convex
norm(X, 1)	\(\sum_{ij}\lvert X_{ij} \rvert\)	\(X \in\mathbf{R}^{m \times n}_{++}\)	incr.	log-log convex
norm(X, “inf”)	\(\max_{ij} \{\lvert X_{ij} \rvert\}\)	\(X \in\mathbf{R}^{m \times n}_{++}\)	incr.	log-log convex
pnorm(X, p) \(p \geq 1\) or `p = 'inf'`	\(\left(\sum_{ij} \|X_{ij}\|^p \right)^{1/p}\)	\(X \in \mathbf{R}^{m \times n}_{++}\)	incr.	log-log convex
pnorm(X, p) \(0 < p < 1\)	\(\left(\sum_{ij} X_{ij}^p \right)^{1/p}\)	\(X \in \mathbf{R}^{m \times n}_{++}\)	incr.	log-log convex
prod(X)	\(\prod_{ij}X_{ij}\)	\(X \in\mathbf{R}^{m \times n}_{++}\)	incr.	log-log affine
quad_form(x, P)	\(x^T P x\)	\(x \in \mathbf{R}^n\), \(P \in \mathbf{R}^{n \times n}_{++}\)	incr.	log-log convex
quad_over_lin(X, y)	\(\left(\sum_{ij}X_{ij}^2\right)/y\)	\(x \in \mathbf{R}^n_{++}\) \(y > 0\)	in \(X_{ij}\) decr. in \(y\)	log-log convex
sum(X)	\(\sum_{ij}X_{ij}\)	\(X \in\mathbf{R}^{m \times n}_{++}\)	incr.	log-log convex
sum_squares(X)	\(\sum_{ij}X_{ij}^2\)	\(X \in\mathbf{R}^{m \times n}_{++}\)	incr.	log-log convex
trace(X)	\(\mathrm{tr}\left(X \right)\)	\(X \in\mathbf{R}^{n \times n}_{++}\)	incr.	log-log convex
pf_eigenvalue(X)	spectral radius of \(X\)	\(X \in\mathbf{R}^{n \times n}_{++}\)	incr.	log-log convex
diff_pos(x, y)	\(x - y\)	\(0 < y < x\)	incr. in \(x\) decr. in \(y\)	log-log concave
entr(x)	\(-x \log (x)\)	\(0 < x < 1\)	None	log-log concave
exp(x)	\(e^x\)	\(x > 0\)	incr.	log-log convex
log(x)	\(\log(x)\)	\(x > 1\)	incr.	log-log concave
maximum(x, y)	\(\max \left\{x, y\right\}\)	\(x,y > 0\)	incr.	log-log convex
minimum(x, y)	\(\min \left\{x, y\right\}\)	\(x, y > 0\)	incr.	log-log concave
multiply(x, y)	\(x*y\)	\(x, y > 0\)	incr.	log-log affine
one_minus_pos(x)	\(1 - x\)	\(0 < x < 1\)	decr.	log-log concave
power(x, 0)	\(1\)	\(x > 0\)	constant	constant
power(x, p)	\(x\)	\(x > 0\)	for \(p > 0\) for \(p < 0\)	log-log affine
sqrt(x)	\(\sqrt x\)	\(x > 0\)	incr.	log-log affine
square(x)	\(x^2\)	\(x > 0\)	incr.	log-log affine
xexp(x)	\(x e^x\)	\(x > 0\)	incr.	log-log convex
bmat()	\(\left[\begin{matrix} X^{(1,1)} & .. & X^{(1,q)} \\ \vdots & & \vdots \\ X^{(p,1)} & .. & X^{(p,q)} \end{matrix}\right]\)	\(X^{(i,j)} \in\mathbf{R}^{m_i \times n_j}_{++}\)	incr.	log-log affine
diag(x)	\(\left[\begin{matrix}x_1 & & \\& \ddots & \\& & x_n\end{matrix}\right]\)	\(x \in\mathbf{R}^{n}_{++}\)	incr.	log-log affine
diag(X)	\(\left[\begin{matrix}X_{11} \\\vdots \\X_{nn}\end{matrix}\right]\)	\(X \in\mathbf{R}^{n \times n}_{++}\)	incr.	log-log affine
eye_minus_inv(X)	\((I - X)^{-1}\)	\(X \in\mathbf{R}^{n \times n}_{++}\), \(\lambda_{\text{pf}}(X) < 1\)	incr.	log-log convex
gmatmul(A, x) \(A \in \mathbf{R}^{m \times n}\)	\(\left[\begin{matrix}\prod_{j=1}^n x_j^{A_{1j}} \\\vdots \\\prod_{j=1}^n x_j^{A_{mj}}\end{matrix}\right]\)	\(x \in \mathbf{R}^n_{++}\)	for \(A_{ij} \geq 0\) for \(A_{ij} \leq 0\)	log-log affine
hstack([X1, …, Xk])	\(\left[\begin{matrix}X^{(1)} \cdots X^{(k)}\end{matrix}\right]\)	\(X^{(i)} \in\mathbf{R}^{m \times n_i}_{++}\)	incr.	log-log affine
matmul(X, Y)	\(XY\)	\(X \in\mathbf{R}^{m \times n}_{++}\) \(Y \in\mathbf{R}^{n \times p}_{++}\)	incr.	log-log convex
resolvent(X)	\((sI - X)^{-1}\)	\(X \in\mathbf{R}^{n \times n}_{++}\) \(\lambda_{\text{pf}}(X) < s\)	incr.	log-log convex
reshape(X, (m’, n’))	\(X' \in\mathbf{R}^{m' \times n'}\)	\(X \in\mathbf{R}^{m \times n}_{++}\) \(m'n' = mn\)	incr.	log-log affine
vec(X)	\(x' \in\mathbf{R}^{mn}\)	\(X \in\mathbf{R}^{m \times n}_{++}\)	incr.	log-log affine
vstack([X1, …, Xk])	\(\left[\begin{matrix}X^{(1)} \\ \vdots \\X^{(k)}\end{matrix}\right]\)	\(X^{(i)} \in\mathbf{R}^{m_i \times n}_{++}\)	incr.	log-log affine