Problems with complex values in optimization objective (JuMP)

Specific Domains Optimization (Mathematical)
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1

When using complex valued numbers in the objective function, get an error that I struggle to understand. This minimal example illustrates it very well. Let’s say we want to find parameters of a parametrised unitay matrix such that it agrees with a given unitary.

using LinearAlgebra
using JuMP
import HiGHS

const X = ComplexF64[0 1; 1 0]
const Y = ComplexF64[0 -im; im 0]
const Z = ComplexF64[1 0; 0 -1]

rot(op, angle) = exp(-im * angle / 2 * op)

function unitary(theta)
    return exp(-i * theta[3] * kron(Z, X)) + exp(-i * theta[2] * kron(Z, Y)) * exp(-i * theta[1] * kron(Z, X))
end

function opt(theta)
    u_target = exp(-im * pi / 4 * kron(Z, X))
    u = u_lstar(theta)
    fid = abs(tr(u_target' * u))^2 / 4
    return fid
end

model = Model(HiGHS.Optimizer)
@variable(model, -pi <= theta[1:3] <= pi)
@objective(model, Max, opt(theta))
JuMP.optimize!(model)
value(theta)

It gives the following error:

ERROR: Cannot build `GenericNonlinearExpr` because a term is complex-valued: `(0)::GenericAffExpr{ComplexF64, VariableRef}`
Stacktrace:
  [1] error(s::String)
    @ Base .\error.jl:44
  [2] _throw_if_not_real(x::GenericAffExpr{ComplexF64, VariableRef})
    @ JuMP C:\Users\xyz\.julia\packages\JuMP\7eD71\src\nlp_expr.jl:344
  [3] /(x::GenericAffExpr{ComplexF64, VariableRef}, y::NonlinearExpr)
    @ JuMP C:\Users\xyz\.julia\packages\JuMP\7eD71\src\nlp_expr.jl:401
  [4] eigtype(T::Type)
    @ LinearAlgebra C:\Users\xyz\.julia\juliaup\julia-1.12.4+0.x64.w64.mingw32\share\julia\stdlib\v1.12\LinearAlgebra\src\eigen.jl:319
  [5] exp(A::Matrix{GenericAffExpr{ComplexF64, VariableRef}})
    @ LinearAlgebra C:\Users\xyz\.julia\juliaup\julia-1.12.4+0.x64.w64.mingw32\share\julia\stdlib\v1.12\LinearAlgebra\src\dense.jl:671
...

The value returned by the optimization objective (and everything else except for the matrix product) is real, so I struggle to understand what causes the problem. Is it not allowed to use complex numbers altogether?

2

Hi @dnldlg, complex numbers may not appear in nonlinear expressions. I’ll update the documentation and the error message to make this clearer.

I’m away from my computer for the weekend, but I’ll see if there’s a way to rewrite your problem next week.