Optimization with Complex Variables

Convex.jl also supports optimization with complex variables. Below, we present a quick start guide on how to use Convex.jl for optimization with complex variables, and then list the operations supported on complex variables in Convex.jl. In general, any operation available in Convex.jl that is well defined and DCP compliant on complex variables should be available. We list these functions below. organized by the type of cone (linear, second-order, or semidefinite) used to represent that operation.

Internally, Convex.jl transforms the complex-domain problem to a larger real-domain problem using a bijective mapping. It then solves the real-domain problem and transforms the solution back to the complex domain.

Complex Variables

Complex Variables in Convex.jl are declared in the same way as the variables are declared but using the different keyword ComplexVariable.

  # Scalar complex variable
  z = ComplexVariable()

  # Column vector variable
  z = ComplexVariable(5)

  # Matrix variable
  z = ComplexVariable(4, 6)

  # Complex Positive Semidefinite variable
  z = HermitianSemidefinite(4)

Linear Program Representable Functions (complex variables)

All of the linear functions that are listed under Linear Program Representable Functions operate on complex variables as well. In addition, several specialized functions for complex variables are available:

operation	description	vexity	slope	notes
`real(z)`	real part of complex of variable	affine	increasing	none
`imag(z)`	imaginary part of complex variable	affine	increasing	none
`conj(x)`	element-wise complex conjugate	affine	increasing	none
`innerproduct(x,y)`	`real(trace(x'*y))`	affine	increasing	PR: one argument is constant

Second-Order Cone Representable Functions (complex variables)

Most of the second order cone function listed under Second-Order Cone Representable Functions operate on complex variables as well. Notable exceptions include:

inverse
square
sqrt
geomean
huber

One new function is available:

operation	description	vexity	slope	notes
`abs2(z)`	`square(abs(z))`	convex	increasing	none

Semidefinite Program Representable Functions (complex variables)

All SDP-representable functions listed under Semidefinite Program Representable Functions work for complex variables.

Exponential + SDP representable Functions (complex variables)

Complex variables also support logdet function.

Optimizing over quantum states

The complex and Hermitian matrix variables, along with the kron and partialtrace operations, enable the definition of a wide range of problems in quantum information theory. As a simple example, let us consider a state $\rho$ over a composite Hilbert space $\mathcal{H}_A\otimes\mathcal{H}_B$, where both component spaces are isomorphic to $\mathbb{C}^2$. Assume that $\rho$ is a product state, with its component in $\mathcal{H}_A$ given as $A$, a complex-valued matrix. We can optimize over the second component $B$ to meet some requirement. Here we simply fix the second component too, but via the partialtrace operator:

using Convex, SCS
A = [ 0.47213595 0.11469794+0.48586827im; 0.11469794-0.48586827im  0.52786405]
B = ComplexVariable(2, 2)
ρ = kron(A, B)
constraints = [
    partialtrace(ρ, 1, [2; 2]) == [1 0; 0 0],
    tr(ρ) == 1,
    isposdef(ρ),
  ]
p = satisfy(constraints)
solve!(p, SCS.Optimizer; silent = true)
p.status

OPTIMAL::TerminationStatusCode = 1

Since we fix both components as trace-1 positive semidefinite matrices, the last two constraints are actually redundant in this case.