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extensions.md

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Extensions

LineSearches.jl

Manopt can be used with line search algorithms implemented in LineSearches.jl. This can be illustrated by the following example of optimizing Rosenbrock function constrained to the unit sphere.

using Manopt, Manifolds, LineSearches

# define objective function and its gradient
p = [1.0, 100.0]
function rosenbrock(::AbstractManifold, x)
    val = zero(eltype(x))
    for i in 1:(length(x) - 1)
        val += (p[1] - x[i])^2 + p[2] * (x[i + 1] - x[i]^2)^2
    end
    return val
end
function rosenbrock_grad!(M::AbstractManifold, storage, x)
    storage .= 0.0
    for i in 1:(length(x) - 1)
        storage[i] += -2.0 * (p[1] - x[i]) - 4.0 * p[2] * (x[i + 1] - x[i]^2) * x[i]
        storage[i + 1] += 2.0 * p[2] * (x[i + 1] - x[i]^2)
    end
    project!(M, storage, x, storage)
    return storage
end
# define constraint
n_dims = 5
M = Manifolds.Sphere(n_dims)
# set initial point
x0 = vcat(zeros(n_dims - 1), 1.0)
# use LineSearches.jl HagerZhang method with Manopt.jl quasiNewton solver
ls_hz = Manopt.LineSearchesStepsize(M, LineSearches.HagerZhang())
x_opt = quasi_Newton(
    M,
    rosenbrock,
    rosenbrock_grad!,
    x0;
    stepsize=ls_hz,
    evaluation=InplaceEvaluation(),
    stopping_criterion=StopAfterIteration(1000) | StopWhenGradientNormLess(1e-6),
    return_state=true,
)

In general this defines the following new [stepsize](@ref Stepsize) with helper functions for setting and getting the maximum step size:

Manopt.LineSearchesStepsize
Manopt.linesearches_get_max_alpha
Manopt.linesearches_set_max_alpha

Manifolds.jl

Loading Manifolds.jl introduces the following additional functions

Manopt.max_stepsize(::FixedRankMatrices, ::Any)
Manopt.max_stepsize(::Hyperrectangle, ::Any)
Manopt.max_stepsize(::TangentBundle, ::Any)
mid_point

Internally, Manopt.jl provides the two additional functions to choose some Euclidean space when needed as

Manopt.Rn
Manopt.Rn_default

[JuMP.jl](@extref JuMP :std:doc:index)

Manopt can be used from within [JuMP.jl](@extref JuMP :std:doc:index). The manifold is provided in the @variable macro. Note that until now, only variables (points on manifolds) are supported, that are arrays, especially structs do not yet work. The algebraic expression of the objective function is specified in the @objective macro. The descent_state_type attribute specifies the solver.

using JuMP, Manopt, Manifolds
model = Model(Manopt.JuMP_Optimizer)
# Change the solver with this option, `GradientDescentState` is the default
set_attribute(model, "descent_state_type", GradientDescentState)
@variable(model, U[1:2, 1:2] in Stiefel(2, 2), start = 1.0)
@objective(model, Min, sum((A - U) .^ 2))
optimize!(model)
solution_summary(model)

Several functions from the [Mathematical Optimization Interface (MOI)](@extref JuMP :std🏷️The-MOI-interface) are extended when both Manopt.jl and [JuMP.jl](@extref JuMP :std:doc:index) are loaded:

Manopt.JuMP_Optimizer

Internal functions

JuMP.build_variable
MOI.add_constrained_variables
MOI.copy_to
MOI.empty!
MOI.dimension
MOI.supports_add_constrained_variables
MOI.get
MOI.is_valid
MOI.supports
MOI.supports_incremental_interface
MOI.set

Internal wrappers and their functions

Modules = [Base.get_extension(Manopt, :ManoptJuMPExt)]
Order   = [:type, :function]