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Adaptive regularization with cubics

CurrentModule = Manopt

adaptive_regularization_with_cubics
adaptive_regularization_with_cubics!

State

AdaptiveRegularizationState

Sub solvers

There are several ways to approach the subsolver. The default is the first one.

[Lanczos iteration](@id arc-Lanczos)

Manopt.LanczosState

(Conjugate) gradient descent

There is a generic objective, that implements the sub problem

AdaptiveRegularizationWithCubicsModelObjective

Since the sub problem is given on the tangent space, you have to provide

arc_obj = AdaptiveRegularizationWithCubicsModelObjective(mho, σ)
sub_problem = DefaultProblem(TangentSpaceAt(M,p), arc_obj)

where mho is the Hessian objective of f to solve. Then use this for the sub_problem keyword and use your favourite gradient based solver for the sub_state keyword, for example a ConjugateGradientDescentState

Additional stopping criteria

StopWhenAllLanczosVectorsUsed
StopWhenFirstOrderProgress

[Technical details](@id sec-arc-technical-details)

The adaptive_regularization_with_cubics requires the following functions of a manifolds to be available

  • A [retract!](@extref ManifoldsBase :doc:retractions)(M, q, p, X); it is recommended to set the [default_retraction_method](@extref ManifoldsBase.default_retraction_method-Tuple{AbstractManifold}) to a favourite retraction. If this default is set, a retraction_method= does not have to be specified.
  • if you do not provide an initial regularization parameter σ, a [manifold_dimension](@extref ManifoldsBase.manifold_dimension-Tuple{AbstractManifold}) is required.
  • By default the tangent vector storing the gradient is initialized calling [zero_vector](@extref ManifoldsBase.zero_vector-Tuple{AbstractManifold, Any})(M,p).
  • [inner](@extref ManifoldsBase.inner-Tuple{AbstractManifold, Any, Any, Any})(M, p, X, Y) is used within the algorithm step

Furthermore, within the Lanczos subsolver, generating a random vector (at p) using [rand!](@extref Base.rand-Tuple{AbstractManifold})(M, X; vector_at=p) in place of X is required

Literature

Pages = ["adaptive-regularization-with-cubics.md"]
Canonical=false