Douglas—Rachford algorithm

DouglasRachford(M, f, proxes_f, p)
DouglasRachford(M, mpo, p)

Compute the Douglas-Rachford algorithm on the manifold $\mathcal M$, initial data $p$ and the (two) proximal maps proxMaps, see [BPS16].

For $k>2$ proximal maps, the problem is reformulated using the parallel Douglas Rachford: a vectorial proximal map on the power manifold $\mathcal M^k$ is introduced as the first proximal map and the second proximal map of the is set to the mean (Riemannian Center of mass). This hence also boils down to two proximal maps, though each evaluates proximal maps in parallel, that is, component wise in a vector.

If you provide a ManifoldProximalMapObjectivempo instead, the proximal maps are kept unchanged.

Input

• M: a Riemannian Manifold $\mathcal M$
• F: a cost function consisting of a sum of cost functions
• proxes_f: functions of the form (M, λ, p)-> q performing a proximal maps, where ⁠λ denotes the proximal parameter, for each of the summands of F. These can also be given in the InplaceEvaluation variants (M, q, λ p) -> q computing in place of q.
• p: initial data $p ∈ \mathcal M$

Optional values

• evaluation: (AllocatingEvaluation) specify whether the proximal maps work by allocation (default) form prox(M, λ, x) or InplaceEvaluation in-place
• λ: ((iter) -> 1.0) function to provide the value for the proximal parameter during the calls
• α: ((iter) -> 0.9) relaxation of the step from old to new iterate, to be precise $t_{k+1} = g(α_k; t_k, s_k)$, where $s_k$ is the result of the double reflection involved in the DR algorithm
• inverse_retraction_method - (default_inverse_retraction_method(M, typeof(p))) the inverse retraction to use within
  • the reflection (ignored, if you set R directly)
  • the relaxation step
• R: method employed in the iteration to perform the reflection of x at the prox p. This uses by default reflect or reflect! depending on reflection_evaluation and the retraction and inverse retraction specified by retraction_method and inverse_retraction_method, respectively.
• reflection_evaluation: (AllocatingEvaluation whether R works in-place or allocating
• retraction_method: (default_retraction_metiod(M, typeof(p))) the retraction to use in
  • the reflection (ignored, if you set R directly)
  • the relaxation step
• stopping_criterion: (StopAfterIteration(200) |StopWhenChangeLess(1e-5)) a StoppingCriterion.
• parallel: (false) indicate whether to use a parallel Douglas-Rachford or not.

and the ones that are passed to decorate_state! for decorators.

Output

the obtained (approximate) minimizer $p^*$, see get_solver_return for details

DouglasRachford(M, f, proxes_f, p; kwargs...)

a doc string with some math $t_{k+1} = g(α_k; t_k, s_k)$

 DouglasRachford!(M, f, proxes_f, p)
 DouglasRachford!(M, mpo, p)

Compute the Douglas-Rachford algorithm on the manifold $\mathcal M$, initial data $p ∈ \mathcal M$ and the (two) proximal maps proxes_f in place of p.

For $k>2$ proximal maps, the problem is reformulated using the parallel Douglas Rachford: a vectorial proximal map on the power manifold $\mathcal M^k$ is introduced as the first proximal map and the second proximal map of the is set to the mean (Riemannian Center of mass). This hence also boils down to two proximal maps, though each evaluates proximal maps in parallel, that is component wise in a vector.

If you provide a ManifoldProximalMapObjectivempo instead, the proximal maps are kept unchanged.

Input

• M: a Riemannian Manifold $\mathcal M$
• f: a cost function consisting of a sum of cost functions
• proxes_f: functions of the form (M, λ, p)->q or (M, q, λ, p)->q performing a proximal map, where ⁠λ denotes the proximal parameter, for each of the summands of f.
• p: initial point $p ∈ \mathcal M$

For more options, see DouglasRachford.

DouglasRachfordState <: AbstractManoptSolverState

Store all options required for the DouglasRachford algorithm,

Fields

• p: the current iterate (result) For the parallel Douglas-Rachford, this is not a value from the PowerManifold manifold but the mean.
• s: the last result of the double reflection at the proximal maps relaxed by α.
• λ: function to provide the value for the proximal parameter during the calls
• α: relaxation of the step from old to new iterate, to be precise $x^{(k+1)} = g(α(k); x^{(k)}, t^{(k)})$, where $t^{(k)}$ is the result of the double reflection involved in the DR algorithm
• inverse_retraction_method: an inverse retraction method
• R: method employed in the iteration to perform the reflection of x at the prox p.
• reflection_evaluation: whether R works in-place or allocating
• retraction_method: a retraction method
• stop: a StoppingCriterion
• parallel: indicate whether to use a parallel Douglas-Rachford or not.

Constructor

DouglasRachfordState(M, p; kwargs...)

Generate the options for a Manifold M and an initial point p, where the following keyword arguments can be used

• λ: ((iter)->1.0) function to provide the value for the proximal parameter during the calls
• α: ((iter)->0.9) relaxation of the step from old to new iterate, to be precise $x^{(k+1)} = g(α(k); x^{(k)}, t^{(k)})$, where $t^{(k)}$ is the result of the double reflection involved in the DR algorithm
• R: (reflect or reflect!) method employed in the iteration to perform the reflection of x at the prox p, which function is used depends on reflection_evaluation.
• reflection_evaluation: (AllocatingEvaluation()) specify whether the reflection works in-place or allocating (default)
• stopping_criterion: (StopAfterIteration(300)) a StoppingCriterion
• parallel: (false) indicate whether to use a parallel Douglas-Rachford or not.

reflect(M, f, x; kwargs...)
reflect!(M, q, f, x; kwargs...)

reflect the point x from the manifold M at the point f(x) of the function $f: \mathcal M → \mathcal M$, given by

\[ \operatorname{refl}_f(x) = \operatorname{refl}_{f(x)}(x),\]

Compute the result in q.

see also reflect(M,p,x), to which the keywords are also passed to.

reflect(M, p, x, kwargs...)
reflect!(M, q, p, x, kwargs...)

Reflect the point x from the manifold M at point p, given by

\[ \operatorname{refl}_p(x) = \operatorname{retr}_p(-\operatorname{retr}^{-1}_p x).\]

where $\operatorname{retr}$ and $\operatorname{retr}^{-1}$ denote a retraction and an inverse retraction, respectively. This can also be done in place of q.

Keyword arguments

• retraction_method: (default_retraction_metiod(M, typeof(p))) the retraction to use in the reflection
• inverse_retraction_method: (default_inverse_retraction_method(M, typeof(p))) the inverse retraction to use within the reflection

and for the reflect! additionally

• X: (zero_vector(M,p)) a temporary memory to compute the inverse retraction in place. otherwise this is the memory that would be allocated anyways.