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# # User-defined Hessians
# In this tutorial, we explain how to write a user-defined operator (see
# [User-defined operators](@ref jump_user_defined_operators)) with a Hessian
# matrix explicitly provided by the user.
#
# For a more advanced example, see [Nested optimization problems](@ref).
# This tutorial uses the following packages:
using JuMP
import Ipopt
# ## Rosenbrock example
# As a simple example, we consider the Rosenbrock function:
rosenbrock(x...) = (1 - x[1])^2 + 100 * (x[2] - x[1]^2)^2
# which has the gradient vector:
function ∇rosenbrock(g::AbstractVector, x...)
g[1] = 400 * x[1]^3 - 400 * x[1] * x[2] + 2 * x[1] - 2
g[2] = 200 * (x[2] - x[1]^2)
return
end
# and the Hessian matrix:
function ∇²rosenbrock(H::AbstractMatrix, x...)
H[1, 1] = 1200 * x[1]^2 - 400 * x[2] + 2
## H[1, 2] = -400 * x[1] <-- not needed because Hessian is symmetric
H[2, 1] = -400 * x[1]
H[2, 2] = 200.0
return
end
# You may assume the Hessian matrix `H` is initialized with zeros, and
# because it is symmetric you need only to fill in the non-zero of the
# lower-triangular terms.
# The matrix type passed in as `H` depends on the automatic differentiation
# system, so make sure the first argument to the Hessian function supports an
# `AbstractMatrix` (it may be something other than `Matrix{Float64}`). However,
# you may assume only that `H` supports `size(H)` and `setindex!`.
# Now that we have the function, its gradient, and its Hessian, we can construct
# a JuMP model, add the operator, and use it in a macro:
model = Model(Ipopt.Optimizer)
@variable(model, x[1:2])
@operator(model, op_rosenbrock, 2, rosenbrock, ∇rosenbrock, ∇²rosenbrock)
@objective(model, Min, op_rosenbrock(x[1], x[2]))
optimize!(model)
solution_summary(model; verbose = true)