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,
)

# Solver state for `Manopt.jl`s Quasi Newton Method
After 30 iterations

## Parameters
* direction update:        limited memory InverseBFGS (size 5), projections, and ParallelTransport() as vector transport.
* retraction method:       ExponentialRetraction()
* vector trnasport method: ParallelTransport()

## Stepsize
LineSearchesStepsize(HagerZhang{Float64, Base.RefValue{Bool}}
  delta: Float64 0.1
  sigma: Float64 0.9
  alphamax: Float64 Inf
  rho: Float64 5.0
  epsilon: Float64 1.0e-6
  gamma: Float64 0.66
  linesearchmax: Int64 50
  psi3: Float64 0.1
  display: Int64 0
  mayterminate: Base.RefValue{Bool}
; retraction_method=ExponentialRetraction(), vector_transport_method=ParallelTransport())

## Stopping Criterion
Stop When _one_ of the following are fulfilled:
    Max Iteration 1000:	not reached
    |grad f| < 1.0e-6: reached
Overall: reached
This indicates convergence: Yes

Manifolds.jl

LineSearchesStepsize <: Stepsize

LineSearchesStepsize(
    M::AbstractManifold,
    linesearch;
    retraction_method::AbstractRetractionMethod=default_retraction_method(M),
    vector_transport_method::AbstractVectorTransportMethod=default_vector_transport_method(M),
)
LineSearchesStepsize(
    linesearch;
    retraction_method::AbstractRetractionMethod=ExponentialRetraction(),
    vector_transport_method::AbstractVectorTransportMethod=ParallelTransport(),
)

mid_point(M, p, q, x)
mid_point!(M, y, p, q, x)

max_stepsize(M::TangentBundle, p)

max_stepsize(M::FixedRankMatrices, p)