How to define this expression?

Specific Domains Optimization (Mathematical)
,
2

One approach is this:

using JuMP, Ipopt
n = 5
n! = factorial(n)
model = Model(Ipopt.Optimizer)
@variable(model, X[1:n, 1:n])
@variable(model, y[1:n])
@expression(model, z[i=1:n], sum(X[i, :]))
@NLexpression(
    model,
    expr[i=1:n],
    z[i]^4 / n! - sum(X[i, j] * z[j]^4 for j in 1:n) / (n! * (n + 1)),
)
@NLconstraint(model, sum(y[i] * expr[i] for i in 1:n) == 1)

The trick is to realize that you can’t construct nonlinear expressions outside the macros, you can’t use linear algebra like X * z, and you can’t use function calls like factorial(n) inside the macro.

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