Choosing between Symmetric and Hermitian programmatically in JuMP

Specific Domains Optimization (Mathematical)
1

I want to switch between Symmetric and Hermitian variables depending on the input data. For data I’d like to define a variable a, that can be either and then create the variable, e.g.

a = Symmetric
model = Model()
@variable(model, x[1:2,1:2], a)

but that errors. How do I pass them correctly?

Looking at the source code of the @variable macro, it seems like it wants to have Symmetric or Hermitian as a symbol, but

a = :Symmetric
model = Model()
@variable(model, x[1:2,1:2], a)

also fails. It does work if I do

a = :Symmetric
model = Model()
@eval @variable(model, x[1:2,1:2], $a)

but that’s a bit insane. What is going on here? What kind of argument does @variable want? Is it possible to do it without @eval?

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2

Try without the @eval: @variable(model, x[1:2,1:2], $a).

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3

That gives me an error:

ERROR: syntax: “$” expression outside quote

4

Why not just use Hermitian in all cases? For real data, Hermitian is functionally (and mathematically) equivalent to Symmetric. For complex data, Symmetric is of limited utility in linear algebra.

5

What I’m doing is switching between a variable that belongs to the d(d+1)/2 dimensional vector space of real Hermitian matrices, or one that belongs to the d^2 dimensional vector space of complex Hermitian matrices. “Symmetric” and “Hermitian” are just the names JuMP uses for these vector spaces.

6

@variable is a macro, so it operates before a=:Symmetric even takes place. I guess a simple if/else

if (...)
    @variable(model, x[...], Symmetric)
else
    @variable(model, x[...], Hermitian)
end

is your best option. Is there a tangible gain from passing a variable to the @variable macro?

Another possibility is to define

function add_variable(model, dim, ::Type{Symmetric})
    @variable(model, [1:dim, 1:dim], Symmetric)
end

at the cost of dealing with anonymous variables later on:

m[:x] = add_variable(m, 2, Symmetric)
7

I am using an if-else now, that’s what I want to avoid. It leads to a lot of copy-pasting, since I need to switch between real and complex variables in several places.

8

You can do:

julia> model = Model();

julia> a = SymmetricMatrixSpace()
SymmetricMatrixSpace()

julia> @variable(model, x[1:2,1:2], set=a)
2×2 LinearAlgebra.Symmetric{VariableRef, Matrix{VariableRef}}:
 x[1,1]  x[1,2]
 x[1,2]  x[2,2]

where it’s HermitianMatrixSpace() for Hermitian matrices. The docs mention set= in this section and SymmetricMatrixSpace here, but I don’t see anywhere they mention using it for programmatic construction, maybe that would be useful to add to the docs.

4 Likes
9

Thanks a lot, that works. Testing a little bit I found a slightly nicer syntax:

model = Model()
a = HermitianMatrixSpace()
@variable(model, x[1:2,1:2] in a)
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10

I’ll update the docs: [docs] clarify how to pass PSD and Symmetric via set kwarg by odow · Pull Request #3940 · jump-dev/JuMP.jl · GitHub

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