Supported operations

Base.:*(x::Convex.AbstractExpr, y::Convex.AbstractExpr)

The binary multiplication operator $x \times y$.

Examples

ulia> x = Variable();

julia> 2 * x
* (affine; real)
├─ [2;;]
└─ real variable (id: 709…007)

julia> x = Variable(3);

julia> y = [1, 2, 3];

julia> x' * y
* (affine; real)
├─ reshape (affine; real)
│  └─ * (affine; real)
│     ├─ 3×3 SparseArrays.SparseMatrixCSC{Int64, Int64} with 3 stored entries
│     └─ reshape (affine; real)
│        └─ …
└─ [1; 2; 3;;]

Base.:+(x::Convex.AbstractExpr, y::Convex.AbstractExpr)
Base.:+(x::Convex.Value, y::Convex.AbstractExpr)
Base.:+(x::Convex.AbstractExpr, y::Convex.Value)

The addition operator $x + y$.

Examples

Applies to scalar expressions:

julia> x = Variable();

julia> x + 1
+ (affine; real)
├─ real variable (id: 110…477)
└─ [1;;]

And element-wise to a matrix of expressions:

julia> x = Variable(3);

julia> y = [1, 2, 3];

julia> atom = x + y
+ (affine; real)
├─ 3-element real variable (id: 458…482)
└─ [1; 2; 3;;]

julia> size(atom)
(3, 1)

Base.:-(x::Convex.AbstractExpr)

The univariate negation operator $-x$.

Examples

Applies to scalar expressions:

julia> x = Variable();

julia> -x
Convex.NegateAtom (affine; real)
├─ real variable (id: 161…677)

And element-wise to a matrix of expressions:

julia> x = Variable(3);

julia> atom = -x
Convex.NegateAtom (affine; real)
└─ 3-element real variable (id: 137…541)

julia> size(atom)
(3, 1)

Base.:-(x::Convex.AbstractExpr, y::Convex.AbstractExpr)
Base.:-(x::Convex.Value, y::Convex.AbstractExpr)
Base.:-(x::Convex.AbstractExpr, y::Convex.Value)

The subtraction operator $x - y$.

Examples

Applies to scalar expressions:

julia> x = Variable();

julia> x - 1
+ (affine; real)
├─ real variable (id: 161…677)
└─ [-1;;]

And element-wise to a matrix of expressions:

julia> x = Variable(3);

julia> y = [1, 2, 3];

julia> atom = y - x
+ (affine; real)
├─ [1; 2; 3;;]
└─ Convex.NegateAtom (affine; real)
   └─ 3-element real variable (id: 242…661)

Base.:/(x::Convex.AbstractExpr, y::Convex.Value)

The binary division operator $\frac{x}{y}$.

Examples

Applies to a scalar expression:

ulia> x = Variable();

julia> x / 2

and element-wise to a matrix:

julia> x = Variable(3);

julia> atom = x / 2
* (affine; real)
├─ 3-element real variable (id: 129…611)
└─ [0.5;;]

julia> size(atom)
(3, 1)

x::Convex.AbstractExpr .* y::Convex.AbstractExpr

Element-wise multiplication between matrices x and y.

Examples

julia> x = Variable(2);

julia> atom = x .* 2
* (affine; real)
├─ 2-element real variable (id: 197…044)
└─ [2;;]

julia> atom = x .* [2, 4]
.* (affine; real)
├─ 2-element real variable (id: 197…044)
└─ [2; 4;;]

julia> size(atom)
(2, 1)

x::Convex.AbstractExpr ./ y::Convex.AbstractExpr

Element-wise division between matrices x and y.

Examples

julia> x = Variable(2);

julia> atom = x ./ 2
* (affine; real)
├─ 2-element real variable (id: 875…859)
└─ [0.5;;]

julia> atom = x ./ [2, 4]
.* (affine; real)
├─ 2-element real variable (id: 875…859)
└─ [0.5; 0.25;;]

julia> size(atom)
(2, 1)

x::Convex.AbstractExpr .^ k::Int

Element-wise exponentiation of x to the power of k.

Examples

julia> x = Variable(2);

julia> atom = x .^ 2
qol_elem (convex; positive)
├─ 2-element real variable (id: 131…737)
└─ [1.0; 1.0;;]

julia> size(atom)
(2, 1)

Base.abs(x::Convex.AbstractExpr)

The epigraph of $|x|$.

Examples

Applies to a single expression:

julia> x = Variable();

julia> abs(x)
abs (convex; positive)
└─ real variable (id: 103…720)

And element-wise to a matrix of expressions:

julia> x = Variable(3);

julia> atom = abs(x)
abs (convex; positive)
└─ 3-element real variable (id: 389…882)

julia> size(atom)
(3, 1)

Base.abs2(x::Convex.AbstractExpr)

The epigraph of $|x|^2$.

Examples

Applies to a single expression:

julia> x = Variable();

julia> abs2(x)
qol_elem (convex; positive)
├─ abs (convex; positive)
│  └─ real variable (id: 319…413)
└─ [1.0;;]

And element-wise to a matrix of expressions:

julia> x = Variable(3);

julia> atom = abs2(x)
qol_elem (convex; positive)
├─ abs (convex; positive)
│  └─ 3-element real variable (id: 123…996)
└─ [1.0; 1.0; 1.0;;]

julia> size(atom)
(3, 1)

LinearAlgebra.adjoint(x::AbstractExpr)

The transpose of the conjugated matrix x.

Examples

julia> x = ComplexVariable(2, 2);

julia> atom = adjoint(x)
reshape (affine; complex)
└─ * (affine; complex)
   ├─ 4×4 SparseArrays.SparseMatrixCSC{Int64, Int64} with 4 stored entries
   └─ reshape (affine; complex)
      └─ conj (affine; complex)
         └─ …

julia> size(atom)
(2, 2)

Base.conj(x::Convex.AbstractExpr)

The complex conjugate of x.

If x is real, this function returns x.

Examples

Applies to a single expression:

julia> x = ComplexVariable();

julia> conj(x)
conj (affine; complex)
└─ complex variable (id: 180…137)

And element-wise to a matrix of expressions:

conj (affine; complex)
└─ complex variable (id: 180…137)

julia> x = ComplexVariable(3);

julia> atom = conj(x)
conj (affine; complex)
└─ 3-element complex variable (id: 104…031)

julia> size(atom)
(3, 1)

Convex.conv(x::Convex.AbstractExpr, y::Convex.AbstractExpr)

The convolution between two vectors x and y.

Examples

julia> x = Variable(2);

julia> y = [2, 4];

julia> atom = conv(x, y)
* (affine; real)
├─ 3×2 SparseArrays.SparseMatrixCSC{Int64, Int64} with 4 stored entries
└─ 2-element real variable (id: 663…363)

julia> size(atom)
(3, 1)

LinearAlgebra.diag(x::Convex.AbstractExpr, k::Int = 0)

Return the k-th diagonnal of the matrix X as a column vector.

Examples

Applies to a single square matrix:

julia> x = Variable(2, 2);

julia> atom = diag(x, 0)
diag (affine; real)
└─ 2×2 real variable (id: 724…318)

julia> size(atom)
(2, 1)

julia> atom = diag(x, 1)
diag (affine; real)
└─ 2×2 real variable (id: 147…856)

julia> size(atom)
(1, 1)

LinearAlgebra.diagm(x::Convex.AbstractExpr)

Create a diagonal matrix out of the vector x.

Examples

julia> x = Variable(2);

julia> atom = diagm(x)
diagm (affine; real)
└─ 2-element real variable (id: 541…968)

julia> size(atom)
(2, 2)

LinearAlgebra.dot(x::Convex.AbstractExpr, y::Convex.AbstractExpr)

The dot product $x \cdot y$. If x is complex, it is conjugated.

Examples

julia> x = ComplexVariable(2);

julia> y = [1, 2];

julia> atom = dot(x, y)
sum (affine; complex)
└─ .* (affine; complex)
   ├─ conj (affine; complex)
   │  └─ 2-element complex variable (id: 133…443)
   └─ [1; 2;;]

julia> size(atom)
(1, 1)

dotsort(x::Convex.AbstractExpr, y::Convex.Value)
dotsort(x::Convex.Value, y::Convex.AbstractExpr)

Computes dot(sort(x), sort(y)), where x or y is constant.

For example, if x = Variable(6) and y = [1 1 1 0 0 0], this atom computes the sum of the three largest elements of x.

Examples

julia> x = Variable(4);

julia> atom = dotsort(x, [1, 0, 0, 1])
dotsort (convex; real)
└─ 4-element real variable (id: 128…367)

julia> size(atom)
(1, 1)

LinearAlgebra.eigmax(X::Convex.AbstractExpr)

The epigraph of the maximum eigen value of $X$.

Examples

Applies to a single square matrix:

julia> x = Variable(2, 2);

julia> atom = eigmax(x)
eigmin (convex; real)
└─ 2×2 real variable (id: 428…695)

julia> size(atom)
(1, 1)

LinearAlgebra.eigmin(X::Convex.AbstractExpr)

The hypograph of the minimum eigen value of $X$.

Examples

Applies to a single square matrix:

julia> x = Variable(2, 2);

julia> atom = eigmin(x)
eigmin (concave; real)
└─ 2×2 real variable (id: 428…695)

julia> size(atom)
(1, 1)

entropy(x::Convex.AbstractExpr)

The hypograph of $\sum_i -x_i \log x_i$.

Examples

Applies to a matrix of expressions:

julia> x = Variable(3);

julia> atom = entropy(x)
sum (concave; real)
└─ entropy (concave; real)
   └─ 3-element real variable (id: 901…778)

julia> size(atom)
(1, 1)

entropy_elementwise(x::Convex.AbstractExpr)

The hypograph of $-x \log x$.

Examples

Applies to a single expression:

julia> x = Variable();

julia> entropy_elementwise(x)
entropy (concave; real)
└─ real variable (id: 172…395)

And element-wise to a matrix of expressions:

julia> x = Variable(3);

julia> atom = entropy_elementwise(x)
entropy (concave; real)
└─ 3-element real variable (id: 140…126)

julia> size(atom)
(3, 1)

Base.exp(x::Convex.AbstractExpr)

The epigraph of $e^x$.

Examples

Applies to a single expression:

julia> x = Variable();

julia> exp(x)
exp (convex; positive)
└─ real variable (id: 103…720)

And element-wise to a matrix of expressions:

julia> x = Variable(3);

julia> atom = exp(x)
exp (convex; positive)
└─ 3-element real variable (id: 389…882)

julia> size(atom)
(3, 1)

geomean(x::Convex.AbstractExpr...)

The hypograph of the geometric mean $\sqrt[n]{x_1 \cdot x_2 \cdot \ldots x_n}$.

Examples

Applies to a single expression:

julia> x = Variable();

julia> y = Variable();

julia> geomean(x, y)
geomean (concave; positive)
├─ real variable (id: 163…519)
└─ real variable (id: 107…393)

And element-wise to a matrix of expressions:

julia> x = Variable(3);

julia> y = Variable(3);

julia> atom = geomean(x, y)
geomean (concave; positive)
├─ 3-element real variable (id: 177…782)
└─ 3-element real variable (id: 307…913)

julia> size(atom)
(3, 1)

Base.hcat(args::AbstractExpr...)

Horizontally concatenate args.

Examples

Applies to a matrix:

julia> x = Variable(2, 2);

julia> atom = hcat(x, x)
hcat (affine; real)
├─ 2×2 real variable (id: 111…376)
└─ 2×2 real variable (id: 111…376)

julia> size(atom)
(2, 4)

You can also use the Julia [x x] syntax:

julia> x = Variable(2, 2);

julia> atom = [x x]
hcat (affine; real)
├─ 2×2 real variable (id: 111…376)
└─ 2×2 real variable (id: 111…376)

julia> size(atom)
(2, 4)

hinge_loss(x::Convex.AbstractExpr)

The epigraph of $\max(1 - x, 0)$.

Examples

Applies to a single expression:

julia> x = Variable();

julia> hinge_loss(x)
max (convex; positive)
├─ + (affine; real)
│  ├─ [1;;]
│  └─ Convex.NegateAtom (affine; real)
│     └─ real variable (id: 129…000)
└─ [0;;]

And element-wise to a matrix of expressions:

julia> x = Variable(3);

julia> atom = hinge_loss(x)
max (convex; positive)
├─ + (affine; real)
│  ├─ * (constant; positive)
│  │  ├─ [1;;]
│  │  └─ [1.0; 1.0; 1.0;;]
│  └─ Convex.NegateAtom (affine; real)
│     └─ 3-element real variable (id: 125…591)
└─ [0;;]

julia> size(atom)
(3, 1)

huber(x::Convex.AbstractExpr, M::Real = 1.0)

The epigraph of the Huber loss function:

\[\begin{cases} x^2 & |x| \le M \\ 2M|x| - M^2 & |x| > M \end{cases}\]

where $M \ge 1$.

Examples

Applies to a single expression:

julia> x = Variable();

julia> huber(x, 2.5)
huber (convex; positive)
└─ real variable (id: 973…369)

And element-wise to a matrix of expressions:

julia> x = Variable(3);

julia> atom = huber(x)
huber (convex; positive)
└─ 3-element real variable (id: 896…728)

julia> size(atom)
(3, 1)

Base.hvcat(
    rows::Tuple{Vararg{Int}},
    args::Union{AbstractExpr,Value}...,
)

Horizontally and vertically concatenate args in single call.

rows is the number of arguments to vertically concatenate into each column.

Examples

Applies to a matrix:

To make the matrix:

a    b[1] b[2]
c[1] c[2] c[3]

do:

julia> a = Variable();

julia> b = Variable(1, 2);

julia> c = Variable(1, 3);

julia> atom = [a b; c]  # Syntactic sugar for: hvcat((2, 1), a, b, c)
vcat (affine; real)
├─ hcat (affine; real)
│  ├─ real variable (id: 429…021)
│  └─ 1×2 real variable (id: 120…326)
└─ hcat (affine; real)
   └─ 1×3 real variable (id: 124…615)

julia> size(atom)
(2, 3)

Base.imag(x::Convex.AbstractExpr)

Return the imaginary component of x.

Examples

Applies to a single expression:

julia> x = ComplexVariable();

julia> imag(x)
imag (affine; real)
└─ complex variable (id: 407…692)

And element-wise to a matrix of expressions:

julia> x = ComplexVariable(3);

julia> atom = imag(x)
imag (affine; real)
└─ 3-element complex variable (id: 435…057)

julia> size(atom)
(3, 1)

inner_product(x::AbstractExpr, y::AbstractExpr)

The inner product $tr(x^\top y)$ where x and y are square matrices.

Examples

julia> x = Variable(2, 2);

julia> y = [1 3; 2 4];

julia> atom = inner_product(x, y)
real (affine; real)
└─ sum (affine; real)
   └─ diag (affine; real)
      └─ * (affine; real)
         ├─ …
         └─ …

julia> size(atom)
(1, 1)

invpos(x::Convex.AbstractExpr)

The epigraph of $\frac{1}{x}$.

Examples

Applies to a single expression:

julia> x = Variable();

julia> invpos(x)
qol_elem (convex; positive)
├─ [1.0;;]
└─ real variable (id: 139…839)

And element-wise to a matrix of expressions:

julia> x = Variable(3);

julia> atom = invpos(x)
qol_elem (convex; positive)
├─ [1.0; 1.0; 1.0;;]
└─ 3-element real variable (id: 133…285)

julia> size(atom)
(3, 1)

Base.kron(x::Convex.AbstractExpr, y::Convex.AbstractExpr)

The Kronecker (outer) product.

Examples

julia> x = Variable(2);

julia> y = [1 2];

julia> atom = kron(x, y)
vcat (affine; real)
├─ * (affine; real)
│  ├─ index (affine; real)
│  │  └─ 2-element real variable (id: 369…232)
│  └─ [1 2]
└─ * (affine; real)
   ├─ index (affine; real)
   │  └─ 2-element real variable (id: 369…232)
   └─ [1 2]

julia> size(atom)
(2, 2)

lieb_ando(
    A::Union{AbstractMatrix,Constant},
    B::Union{AbstractMatrix,Constant},
    K::Union{AbstractMatrix,Constant},
    t::Rational,
)

Returns LinearAlgebra.tr(K' * A^{1-t} * K * B^t) where A and B are positive semidefinite matrices and K is an arbitrary matrix (possibly rectangular).

lieb_ando(A, B, K, t) is concave in (A, B) for t in [0, 1], and convex in (A, B) for t in [-1, 0) or (1, 2]. K is a fixed matrix.

Seems numerically unstable when t is on the endpoints of these ranges.

Reference

Ported from CVXQUAD which is based on the paper: "Lieb's concavity theorem, matrix geometric means and semidefinite optimization" by Hamza Fawzi and James Saunderson (arXiv:1512.03401)

Examples

Note that lieb_ando is implemented as a subproblem, so the returned atom is a Convex.Problem object. The Problem atom can still be used as a regular 1x1 atom in other expressions.

julia> A = Semidefinite(2, 2);

julia> B = Semidefinite(3, 3);

julia> K = [1 2 3; 4 5 6];

julia> atom = lieb_ando(A, B, K, 1 // 2)
Problem statistics
  problem is DCP         : true
  number of variables    : 3 (49 scalar elements)
  number of constraints  : 4 (157 scalar elements)
  number of coefficients : 76
  number of atoms        : 26

Solution summary
  termination status : OPTIMIZE_NOT_CALLED
  primal status      : NO_SOLUTION
  dual status        : NO_SOLUTION

Expression graph
  maximize
   └─ real (affine; real)
      └─ sum (affine; real)
         └─ diag (affine; real)
            └─ …
  subject to
   ├─ GeometricMeanHypoConeSquare constraint (convex)
   │  └─ vcat (affine; real)
   │     ├─ reshape (affine; real)
   │     │  └─ …
   │     ├─ reshape (affine; real)
   │     │  └─ …
   │     └─ reshape (affine; real)
   │        └─ …
   ├─ PSD constraint (convex)
   │  └─ 6×6 real variable (id: 173…902)
   ├─ PSD constraint (convex)
   │  └─ 6×6 real variable (id: 173…902)
   ⋮


julia> size(atom)
(1, 1)

Base.log(x::Convex.AbstractExpr)

The hypograph of $\log(x)$.

Examples

Applies to a single expression:

julia> x = Variable();

julia> log(x)
log (concave; real)
└─ real variable (id: 103…720)

And element-wise to a matrix of expressions:

julia> x = Variable(3);

julia> atom = log(x)
log (concave; real)
└─ 3-element real variable (id: 161…499)

julia> size(atom)
(3, 1)

log_perspective(x::Convex.AbstractExpr, y::Convex.AbstractExpr)

The hypograph the perspective of of the log function: $\sum y_i*\log \frac{x_i}{y_i}$.

Examples

Applies to a single expression:

julia> x = Variable();

julia> y = Variable();

julia> log_perspective(x, y)
Convex.NegateAtom (concave; real)
└─ relative_entropy (convex; real)
   ├─ real variable (id: 136…971)
   └─ real variable (id: 131…344)

And to a matrix of expressions:

julia> x = Variable(3);

julia> y = Variable(3);

julia> atom = log_perspective(x, y)
Convex.NegateAtom (concave; real)
└─ relative_entropy (convex; real)
   ├─ 3-element real variable (id: 854…248)
   └─ 3-element real variable (id: 111…174)

julia> size(atom)
(1, 1)

LinearAlgebra.logdet(X::Convex.AbstractExpr)

The hypograph of $\log(\det(X))$.

Examples

Applies to a single matrix expression:

julia> X = Variable(2, 2);

julia> atom = logdet(X)
logdet (concave; real)
└─ 2×2 real variable (id: 159…883)

julia> size(atom)
(1, 1)

logisticloss(x::Convex.AbstractExpr)

Reformulation for epigraph of the logistic loss: $\sum_i \log(e^x_i + 1)$.

This reformulation uses logsumexp.

Examples

Applies to a single expression:

julia> x = Variable();

julia> logisticloss(x)
logsumexp (convex; real)
└─ vcat (affine; real)
   ├─ real variable (id: 444…892)
   └─ [0;;]

And to a matrix of expressions:

julia> x = Variable(3);

julia> atom = logisticloss(x)
+ (convex; real)
├─ logsumexp (convex; real)
│  └─ vcat (affine; real)
│     ├─ index (affine; real)
│     │  └─ …
│     └─ [0;;]
├─ logsumexp (convex; real)
│  └─ vcat (affine; real)
│     ├─ index (affine; real)
│     │  └─ …
│     └─ [0;;]
└─ logsumexp (convex; real)
   └─ vcat (affine; real)
      ├─ index (affine; real)
      │  └─ …
      └─ [0;;]

julia> size(atom)
(1, 1)

logsumexp(x::Convex.AbstractExpr)

The epigraph of $\log\left(\sum_i e^{x_i}\right)$.

Examples

Applies to a single expression:

julia> x = Variable(2, 3);

julia> atom = logsumexp(x)
logsumexp (convex; real)
└─ 2×3 real variable (id: 121…604)

julia> size(atom)
(1, 1)

julia> atom = logsumexp(x; dims = 1)
logsumexp (convex; real)
└─ 2×3 real variable (id: 121…604)

julia> size(atom)
(1, 3)

julia> atom = logsumexp(x; dims = 2)
logsumexp (convex; real)
└─ 2×3 real variable (id: 121…604)

julia> size(atom)
(2, 1)

matrixfrac(x::AbstractExpr, P::AbstractExpr)

The epigraph of $x^\top P^{-1} x$.

Examples

julia> x = Variable(2);

julia> P = Variable(2, 2);

julia> atom = matrixfrac(x, P)
matrixfrac (convex; positive)
├─ 2-element real variable (id: 139…388)
└─ 2×2 real variable (id: 126…414)

julia> size(atom)
(1, 1)

Base.max(x::Convex.AbstractExpr, y::Convex.AbstractExpr)
Base.max(x::Convex.AbstractExpr, y::Convex.Value)
Base.max(x::Convex.Value, y::Convex.AbstractExpr)

The hypograph of $max(x, y)$.

Examples

Applies to a single expression:

julia> x = Variable();

julia> max(x, 1)
max (convex; real)
├─ real variable (id: 183…974)
└─ [1;;]

And element-wise to a matrix of expressions:

julia> x = Variable(3);

julia> y = [1, 2, 3];

julia> atom = max(x, y)
max (convex; real)
├─ 3-element real variable (id: 153…965)
└─ [1; 2; 3;;]

julia> size(atom)
(3, 1)

Base.maximum(x::Convex.AbstractExpr)

The hypograph of $max(x...)$.

Examples

Applies to a matrix expression:

julia> x = Variable(3);

julia> atom = maximum(x)
maximum (convex; real)
└─ 3-element real variable (id: 159…219)

julia> size(atom)
(1, 1)

Base.min(x::Convex.AbstractExpr, y::Convex.AbstractExpr)
Base.min(x::Convex.Value, y::Convex.AbstractExpr)
Base.min(x::Convex.AbstractExpr, y::Convex.Value)

The epigraph of $min(x, y)$.

Examples

Applies to a single expression:

julia> x = Variable();

julia> min(x, 1)
min (concave; real)
├─ real variable (id: 183…974)
└─ [1;;]

And element-wise to a matrix of expressions:

julia> x = Variable(3);

julia> y = [1, 2, 3];

julia> atom = min(x, y)
min (concave; real)
├─ 3-element real variable (id: 153…965)
└─ [1; 2; 3;;]

julia> size(atom)
(3, 1)

Base.minimum(x::Convex.AbstractExpr)

The epigraph of $min(x...)$.

Examples

Applies to a matrix expression:

julia> x = Variable(3);

julia> atom = minimum(x)
minimum (convex; real)
└─ 3-element real variable (id: 159…219)

julia> size(atom)
(1, 1)

neg(x::Convex.AbstractExpr)

The epigraph of $\max(-x, 0)$.

Examples

Applies to a single expression:

julia> x = Variable();

julia> neg(x)
max (convex; positive)
├─ Convex.NegateAtom (affine; real)
│  └─ real variable (id: 467…111)
└─ [0;;]

And element-wise to a matrix of expressions:

julia> x = Variable(3);

julia> atom = neg(x)
max (convex; positive)
├─ Convex.NegateAtom (affine; real)
│  └─ 3-element real variable (id: 224…439)
└─ [0;;]

julia> size(atom)
(3, 1)

norm(x::AbstractExpr, p::Real = 2)

Computes the p-norm ‖x‖ₚ = (∑ᵢ |xᵢ|^p)^(1/p) of a vector expression x.

Matrices are vectorized (i.e., norm(x) is the same as norm(vec(x)).)

The return value depends on the value of p. Specialized cases are used for p = 1, p = 2, and p = Inf.

Examples

julia> x = Variable(2);

julia> atom = norm(x, 1)
sum (convex; positive)
└─ abs (convex; positive)
   └─ 2-element real variable (id: 779…899)

julia> size(atom)
(1, 1)

julia> norm(x, 2)
norm2 (convex; positive)
└─ 2-element real variable (id: 779…899)

julia> norm(x, Inf)
maximum (convex; positive)
└─ abs (convex; positive)
   └─ 2-element real variable (id: 779…899)

julia> norm(x, 3 // 2)
rationalnorm (convex; positive)
└─ 2-element real variable (id: 779…899)

LinearAlgebra.norm2(x::Convex.AbstractExpr)

The epigraph of the 2-norm $||x||_2$.

Examples

Applies to a matrix of expressions:

julia> x = Variable(3);

julia> atom = norm2(x)
norm2 (convex; positive)
└─ 3-element real variable (id: 162…975)

julia> size(atom)
(3, 1)

And to a complex:

julia> y = ComplexVariable(3);

julia> atom = norm2(y)
norm2 (convex; positive)
└─ vcat (affine; real)
   ├─ real (affine; real)
   │  └─ 3-element complex variable (id: 120…942)
   └─ imag (affine; real)
      └─ 3-element complex variable (id: 120…942)

julia> size(atom)
(1, 1)

nuclearnorm(x::Convex.AbstractExpr)

The epigraph of the nuclear norm $||X||_*$, which is the sum of the singular values of $X$.

Examples

Applies to a real-valued matrix:

julia> x = Variable(2, 2);

julia> atom = nuclearnorm(x)
nuclearnorm (convex; positive)
└─ 2×2 real variable (id: 106…758)

julia> size(atom)
(1, 1)

julia> y = ComplexVariable(2, 2);

julia> atom = nuclearnorm(y)
nuclearnorm (convex; positive)
└─ 2×2 complex variable (id: 577…313)

julia> size(atom)
(1, 1)

LinearAlgebra.opnorm(x::Convex.AbstractExpr, p::Real = 2)

The epigraph of the matrix norm $||X||_p$.

Examples

Applies to a real- or complex-valued matrix:

julia> x = Variable(2, 2);

julia> atom = LinearAlgebra.opnorm(x, 1)
maximum (convex; positive)
└─ * (convex; positive)
   ├─ [1.0 1.0]
   └─ abs (convex; positive)
      └─ 2×2 real variable (id: 106…758)

julia> atom = LinearAlgebra.opnorm(x, 2)
opnorm (convex; positive)
└─ 2×2 real variable (id: 106…758)

julia> atom = LinearAlgebra.opnorm(x, Inf)
maximum (convex; positive)
└─ * (convex; positive)
    ├─ abs (convex; positive)
    │  └─ 2×2 real variable (id: 106…758)
    └─ [1.0; 1.0;;]


julia> y = ComplexVariable(2, 2);

julia> atom = maximum (convex; positive)
└─ * (convex; positive)
   ├─ abs (convex; positive)
   │  └─ 2×2 complex variable (id: 116…943)
   └─ [1.0; 1.0;;]

julia> size(atom)
(1, 1)

partialtrace(x, sys::Int, dims::Vector)

Returns the partial trace of x over the systh system, where dims is a vector of integers encoding the dimensions of each subsystem.

partialtranspose(x, sys::Int, dims::Vector)

Returns the partial transpose of x over the systh system, where dims is a vector of integers encoding the dimensions of each subsystem.

pos(x::Convex.AbstractExpr)

The epigraph of $\max(x, 0)$.

Examples

Applies to a single expression:

julia> x = Variable();

julia> pos(x)
max (convex; positive)
├─ real variable (id: 467…111)
└─ [0;;]

And element-wise to a matrix of expressions:

julia> x = Variable(3);

julia> atom = pos(x)
max (convex; positive)
├─ 3-element real variable (id: 154…809)
└─ [0;;]

julia> size(atom)
(3, 1)

qol_elementwise(x::AbstractExpr, y::AbstractExpr)

The elementwise epigraph of $\frac{x^2}{y}$.

Examples

julia> x = Variable(3);

julia> y = Variable(3, Positive());

julia> atom = qol_elementwise(x, y)
qol_elem (convex; positive)
├─ 3-element real variable (id: 155…648)
└─ 3-element positive variable (id: 227…080)

julia> size(atom)
(3, 1)

quadform(x::AbstractExpr, A::AbstractExpr; assume_psd=false)

Represents $x^\top A x$ where either:

• x is a vector-valued variable and A is a positive semidefinite or negative semidefinite matrix (and in particular Hermitian or real symmetric). If assume_psd=true, then A will be assumed to be positive semidefinite. Otherwise, Convex._is_psd will be used to check if A is positive semidefinite or negative semidefinite.
• or A is a matrix-valued variable and x is a vector.

Examples

julia> x = Variable(2);

julia> A = [1 0; 0 1]
2×2 Matrix{Int64}:
 1  0
 0  1

julia> atom = quadform(x, A)
* (convex; positive)
├─ [1;;]
└─ qol_elem (convex; positive)
   ├─ norm2 (convex; positive)
   │  └─ * (affine; real)
   │     ├─ …
   │     └─ …
   └─ [1.0;;]

julia> size(atom)
(1, 1)

julia> x = [1, 2]

julia> A = Variable(2, 2);

julia> atom = quadform(x, A)
* (affine; real)
├─ * (affine; real)
│  ├─ [1 2]
│  └─ 2×2 real variable (id: 111…794)
└─ [1; 2;;]

julia> size(atom)
(1, 1)

quadoverlin(x::AbstractExpr, y::AbstractExpr)

The epigraph of $\frac{||x||_2^2}{y}$.

Examples

julia> x = Variable(3);

julia> y = Variable(Positive());

julia> atom = quadoverlin(x, y)
qol (convex; positive)
├─ 3-element real variable (id: 868…883)
└─ positive variable (id: 991…712)

julia> size(atom)
(1, 1)

quantum_entropy(X::AbstractExpr, m::Integer, k::Integer)

quantum_entropy returns -LinearAlgebra.tr(X*log(X)) where X is a positive semidefinite.

Note this function uses logarithm base e, not base 2, so return value is in units of nats, not bits.

Quantum entropy is concave. This function implements the semidefinite programming approximation given in the reference below. Parameters m and k control the accuracy of this approximation: m is the number of quadrature nodes to use and k the number of square-roots to take. See reference for more details.

The implementation uses the expression

\[H(X) = -tr(D_{op}(X||I))\]

where $D_{op}$ is the operator relative entropy:

\[D_{op}(X||Y) = X^{1/2}*logm(X^{1/2} Y^{-1} X^{1/2})*X^{1/2}\]

Reference

Ported from CVXQUAD which is based on the paper: "Lieb's concavity theorem, matrix geometric means and semidefinite optimization" by Hamza Fawzi and James Saunderson (arXiv:1512.03401)

Examples

Applies to a matrix:

julia> X = Variable(2, 2);

julia> atom = quantum_entropy(X)
quantum_entropy (concave; positive)
└─ 2×2 real variable (id: 700…694)

julia> size(atom)
(1, 1)

quantum_relative_entropy(
    A::AbstractExpr,
    B::AbstractExpr,
    m::Integer,
    k::Integer,
)

quantum_relative_entropy returns LinearAlgebra.tr(A*(log(A)-log(B))) where A and B are positive semidefinite matrices.

Note this function uses logarithm base e, not base 2, so return value is in units of nats, not bits.

Quantum relative entropy is convex (jointly) in (A, B). This function implements the semidefinite programming approximation given in the reference below. Parameters m and k control the accuracy of this approximation: m is the number of quadrature nodes to use and k the number of square-roots to take. See reference for more details.

Implementation uses the expression

\[D(A||B) = e'*D_{op} (A \otimes I || I \otimes B) )*e\]

where $D_{op}$ is the operator relative entropy and e = vec(Matrix(I, n, n)).

Reference

Ported from CVXQUAD which is based on the paper: "Lieb's concavity theorem, matrix geometric means and semidefinite optimization" by Hamza Fawzi and James Saunderson (arXiv:1512.03401)

Examples

julia> A = Variable(2, 2);

julia> B = Variable(2, 2);

julia> atom = quantum_relative_entropy(A, B)
quantum_relative_entropy (convex; positive)
├─ 2×2 real variable (id: 144…849)
└─ 2×2 real variable (id: 969…693)

julia> size(atom)
(1, 1)

rationalnorm(x::AbstractExpr, k::Rational{Int})

The epigraph of ||x||_k.

Examples

Applies to a single matrix:

julia> x = Variable(2);

julia> atom = rationalnorm(x, 3 // 2)
rationalnorm (convex; positive)
└─ 2-element real variable (id: 182…293)

julia> size(atom)
(1, 1)

Base.real(x::Convex.AbstractExpr)

Return the real component of x.

Examples

Applies to a single expression:

julia> x = ComplexVariable();

julia> real(x)
real (affine; real)
└─ complex variable (id: 407…692)

And element-wise to a matrix of expressions:

julia> x = ComplexVariable(3);

julia> atom = real(x)
real (affine; real)
└─ 3-element complex variable (id: 435…057)

julia> size(atom)
(3, 1)

relative_entropy(x::Convex.AbstractExpr, y::Convex.AbstractExpr)

The epigraph of $\sum x_i*\log \frac{x_i}{y_i}$.

Examples

Applies to a single expression:

julia> x = Variable();

julia> y = Variable();

julia> relative_entropy(x, y)
relative_entropy (convex; real)
├─ real variable (id: 124…372)
└─ real variable (id: 409…346)

And element-wise to a matrix of expressions:

julia> x = Variable(3);

julia> y = Variable(3);

julia> atom = relative_entropy(x, y)
relative_entropy (convex; real)
├─ 3-element real variable (id: 906…671)
└─ 3-element real variable (id: 118…912)

julia> size(atom)
(1, 1)

Base.reshape(x::AbstractExpr, m::Int, n::Int)

Reshapes the expression x into a matrix with m rows and n columns.

Examples

Applies to a matrix:

julia> x = Variable(6, 1);

julia> size(x)
(6, 1)

julia> atom = reshape(x, 2, 3)
reshape (affine; real)
└─ 6-element real variable (id: 103…813)

julia> size(atom)
(2, 3)

Convex.rootdet(X::Convex.AbstractExpr)

The hypograph of $\det(X)^{\frac{1}{n}}$, where $n$ is the side-dimension of the square matrix $X$.

Examples

Applies to a single matrix expression:

julia> X = Variable(2, 2);

julia> atom = rootdet(X)
rootdet (concave; real)
└─ 2×2 real variable (id: 159…883)

julia> size(atom)
(1, 1)

sigmamax(x::Convex.AbstractExpr)

The epigraph of the spectral norm $||X||_2$, which is the maximum of the singular values of $X$.

Examples

Applies to a real- or complex-valued matrix:

julia> x = Variable(2, 2);

julia> atom = sigmamax(x)
opnorm (convex; positive)
└─ 2×2 real variable (id: 106…758)

julia> size(atom)
(1, 1)

julia> y = ComplexVariable(2, 2);

julia> atom = sigmamax(y)
opnorm (convex; positive)
└─ 2×2 complex variable (id: 577…313)

julia> size(atom)
(1, 1)

Base.sqrt(x::Convex.AbstractExpr)

The hypograph of $\sqrt x$.

Examples

Applies to a single expression:

julia> x = Variable();

julia> sqrt(x)
geomean (concave; positive)
├─ real variable (id: 576…546)
└─ [1.0;;]

And element-wise to a matrix of expressions:

julia> x = Variable(3);

julia> atom = sqrt(x)
geomean (concave; positive)
├─ 3-element real variable (id: 181…583)
└─ [1.0; 1.0; 1.0;;]

julia> size(atom)
(3, 1)

square(x::AbstractExpr)

The epigraph of $x^2$.

Examples

Applies elementwise to a matrix

julia> x = Variable(3);

julia> atom = square(x)
qol_elem (convex; positive)
├─ 3-element real variable (id: 438…681)
└─ [1.0; 1.0; 1.0;;]

julia> size(atom)
(3, 1)

Base.sum(x::Convex.AbstractExpr; dims = :)

Sum x, optionally along a dimension dims.

Examples

Sum all elements in an expression:

julia> x = Variable(2, 2);

julia> atom = sum(x)
sum (affine; real)
└─ 2×2 real variable (id: 263…449)

julia> size(atom)
(1, 1)

Sum along the first dimension, creating a row vector:

julia> x = Variable(2, 2);

julia> atom = sum(x; dims = 1)
* (affine; real)
├─ [1.0 1.0]
└─ 2×2 real variable (id: 143…826)

julia> size(atom)
(1, 2)

Sum along the second dimension, creating a columnn vector:

julia> atom = sum(x; dims = 2)
* (affine; real)
├─ 2×2 real variable (id: 143…826)
└─ [1.0; 1.0;;]

julia> size(atom)
(2, 1)

sumlargest(x::Convex.AbstractExpr, k::Int)

Sum the k largest values of x.

Examples

Applies to a matrix:

julia> x = Variable(3, 3);

julia> atom = sumlargest(x, 2)
sumlargest (convex; real)
├─ 3×3 real variable (id: 833…482)
└─ [2;;]

julia> size(atom)
(1, 1)

sumlargesteigs(x::Convex.AbstractExpr, k::Int)

Sum the k largest eigen values of x.

Examples

Applies to a matrix:

julia> x = Variable(3, 3);

julia> atom = sumlargesteigs(x, 2)
sumlargesteigs (convex; real)
├─ 3×3 real variable (id: 833…482)
└─ [2;;]

julia> size(atom)
(1, 1)

sumsmallest(x::Convex.AbstractExpr, k::Int)

Sum the k smallest values of x.

Examples

Applies to a matrix:

julia> x = Variable(3, 3);

julia> atom = sumsmallest(x, 2)
Convex.NegateAtom (concave; real)
└─ sumlargest (convex; real)
   └─ Convex.NegateAtom (affine; real)
      └─ 3×3 real variable (id: 723…082)

julia> size(atom)
(1, 1)

sumsquares(x::AbstractExpr)

The epigraph of $||x||_2^2$.

Examples

Applies to a single matrix

julia> x = Variable(3);

julia> atom = sumsquares(x)
qol (convex; positive)
├─ 3-element real variable (id: 125…181)
└─ [1;;]

julia> size(atom)
(1, 1)

LinearAlgebra.tr(x::AbstractExpr)

The trace of the matrix x.

Examples

julia> x = Variable(2, 2);

julia> atom = tr(x)
sum (affine; real)
└─ diag (affine; real)
   └─ 2×2 real variable (id: 844…180)

julia> size(atom)
(1, 1)

trace_logm(
    X::Convex.AbstractExpr,
    C::AbstractMatrix,
    m::Integer = 3,
    k::Integer = 3,
)

trace_logm(X, C) returns LinearAlgebra.tr(C*logm(X)) where X and C are positive definite matrices and C is constant.

trace_logm is concave in X.

This function implements the semidefinite programming approximation given in the reference below. Parameters m and k control the accuracy of the approximation: m is the number of quadrature nodes to use and k is the number of square-roots to take. See reference for more details.

Implementation uses the expression

\[tr(C \times logm(X)) = -tr(C \times D_{op}(I||X))\]

where D_{op} is the operator relative entropy:

\[D_{op}(X||Y) = X^{1/2}*logm(X^{1/2} Y^{-1} X^{1/2})*X^{1/2}\]

Reference

Ported from CVXQUAD which is based on the paper: "Lieb's concavity theorem, matrix geometric means and semidefinite optimization" by Hamza Fawzi and James Saunderson (arXiv:1512.03401)

Examples

Applies to a matrix:

julia> X = Variable(2, 2);

julia> C = [1 0; 0 1];

julia> atom = trace_logm(X, C)
trace_logm (concave; real)
└─ 2×2 real variable (id: 608…362)

julia> size(atom)
(1, 1)

trace_mpower(A::Convex.AbstractExpr, t::Rational, C::AbstractMatrix)

trace_mpower(A, t, C) returns LinearAlgebra.tr(C*A^t) where A and C are positive definite matrices, C is constant and t is a rational in [-1, 2].

When t is in [0, 1], trace_mpower(A, t, C) is concave in A (for fixed positive semidefinite matrix C) and convex for t in [-1, 0) or (1, 2].

Reference

Ported from CVXQUAD which is based on the paper: "Lieb's concavity theorem, matrix geometric means and semidefinite optimization" by Hamza Fawzi and James Saunderson (arXiv:1512.03401)

Examples

Applies to a matrix:

julia> A = Variable(2, 2);

julia> C = [1 0; 0 1];

julia> atom = trace_mpower(A, 1 // 2, C)
trace_mpower (concave; real)
└─ 2×2 real variable (id: 150…626)

julia> size(atom)
(1, 1)

LinearAlgebra.transpose(x::AbstractExpr)

The transpose of the matrix x.

Examples

julia> x = Variable(2, 2);

julia> atom = transpose(x)
reshape (affine; real)
└─ * (affine; real)
   ├─ 4×4 SparseArrays.SparseMatrixCSC{Int64, Int64} with 4 stored entries
   └─ reshape (affine; real)
      └─ 2×2 real variable (id: 151…193)

julia> size(atom)
(2, 2)

Base.vcat(args::AbstractExpr...)

Vertically concatenate args.

Examples

Applies to a matrix:

julia> x = Variable(2, 2);

julia> atom = vcat(x, x)
vcat (affine; real)
├─ 2×2 real variable (id: 111…376)
└─ 2×2 real variable (id: 111…376)

julia> size(atom)
(4, 2)

You can also use the Julia [x; x] syntax:

julia> x = Variable(2, 2);

julia> atom = [x; x]
vcat (affine; real)
├─ 2×2 real variable (id: 111…376)
└─ 2×2 real variable (id: 111…376)

julia> size(atom)
(4, 2)

Base.vec(x::AbstractExpr)

Reshapes the expression x into a column vector.

Examples

Applies to a matrix:

julia> x = Variable(2, 2);

julia> atom = vec(x)
reshape (affine; real)
└─ 2×2 real variable (id: 115…295)

julia> size(atom)
(4, 1)