# Creating orthogonality constraints in JuMP

I’m trying to impose some orthogonality constraints for an optimal design problem in JuMP; since these are nonlinear vector constraints I believe that I have to use the @NLconstraint macro with splatting. I can’t quite figure out how to use these for a matrix, however, because indexing is banned. An example is below, any tips would be appreciated! PS I tried to use a collection of vectors rather than a matrix for my variable x, using macros to generate the appropriate number, but that ultimately seemed to complicated…

``````using JuMP
using Ipopt
using LinearAlgebra

X = randn(10,1000)
Xn = sqrt.(sum(X.^2, dims=2))
X ./= Xn
ϵ = 1e-1
λ = 1.0
α = 1.0
n_eigs = 5
v, V = eigen(X'*X + ϵ*I)
Γ = V[:, (end-n_eigs+1):end]

μ = rand(10)
U = copy(Γ)

function Ml(μ, X, ϵ)
M = ϵ*I + X'*Diagonal(μ)*X
Mc = cholesky(Hermitian(M))
return (M, Mc)
end

Ml(μ) = Ml(μ, X, ϵ)

_, Mc = Ml(μ)

f(U) = α*sum((U.-Γ).^2) + λ*tr(U'*(Mc\U))

function g!(gvec, U, m, n)
for i = 1:n
gvec[((i-1)*m+1):i*m] .= 2*α*(U[1:m, i].-Γ[1:m, i]) .+ 2*λ*(Mc\U[1:m, i])
end
end

m, n = size(U)

L = LinearIndices((m, n))
jump_wrapper_f(x...) = f([x[L[i,j]] for i in 1:m, j in 1:n])
jump_wrapper_g!(gvec, x...) = g!(gvec, [x[L[i,j]] for i in 1:m, j in 1:n], m, n)

sumprod(x...) = sum(x[i]*x[i+m] for i in 1:m)
for i in 1:m
gvec[i] = x[i+m]
gvec[i+m] = x[i]
end
end

model = Model(optimizer_with_attributes(Ipopt.Optimizer))
JuMP.register(model, :f, m*n, jump_wrapper_f, jump_wrapper_g!)

@variable(model, x[1:m, 1:n])

set_start_value.(x, U)

# This is the problem section because of the indexing expressions being splatted

for i = 1:n
for j = i:n
@NLconstraint(model, sumprod(x[1:m, i]..., x[1:m, j]...) == ((i==j) ? 1 : 0))
end
end

@NLobjective(model, Min, f(x...))

optimize!(model)
``````

You want something like

``````for i = 1:n
for j = i:n
x_i = x[1:m, i]
x_j = x[1:m, j]
@NLconstraint(model, sumprod(xi..., xj...) == ((i==j) ? 1 : 0))
end
end
``````
2 Likes

Thanks - I feel somewhat stupid now that I didn’t realize you could create temporary variables from JuMP variables like that!