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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