Cyclic proximal point

cyclic_proximal_point(M, f, proxes_f, p; kwargs...)
cyclic_proximal_point(M, mpo, p; kwargs...)
cyclic_proximal_point!(M, f, proxes_f, p; kwargs...)
cyclic_proximal_point!(M, mpo, p; kwargs...)

perform a cyclic proximal point algorithm. This can be done in-place of p.

Input

• M::AbstractManifold: a Riemannian manifold $\mathcal{M}$
• f: a cost function $f: \mathcal{M}→ℝ$ to minimize
• proxes_f: an Array of proximal maps (Functions) (M,λ,p) -> q or (M, q, λ, p) -> q for the summands of $f$ (see evaluation)

where f and the proximal maps proxes_f can also be given directly as a ManifoldProximalMapObjective mpo

Keyword arguments

• callbacks::D = Dict{Symbol,Function}(): provided callback functions given either as a single function (symbol, problem, state, k) called in every hook or as a (vector of) pairs :hook => function, which are processed by process_callbacks_arg. As key you can either pass single symbol or an array of symbols to indicate a callback should be added in multiple places
• evaluation::AbstractEvaluationType=AllocatingEvaluation(): specify whether the functions that return an array, for example a point or a tangent vector, work by allocating their result (AllocatingEvaluation) or whether they modify their input argument to return the result therein (InplaceEvaluation). Since usually the first argument is the manifold, the modified argument is the second.
• evaluation_order=:Linear: specify whether to use a fixed randomly permuted sequence (:FixedRandom), a per cycle newly permuted sequence (:Random), or the default :Linear order.
• λ=k -> 1/k: a function returning the (square summable but not summable) sequence of $λ_k$
• stopping_criterion::StoppingCriterion=StopAfterIteration(5000)|StopWhenChangeLess(1e-12): a functor indicating that the stopping criterion is fulfilled

All other keyword arguments are passed to decorate_state! for state decorators or decorate_objective! for objective decorators, respectively.

Output

The obtained approximate minimizer $p^*$. To obtain the whole final state of the solver, see get_solver_return for details, especially the return_state= keyword.

cyclic_proximal_point(M, f, proxes_f, p; kwargs...)
cyclic_proximal_point(M, mpo, p; kwargs...)
cyclic_proximal_point!(M, f, proxes_f, p; kwargs...)
cyclic_proximal_point!(M, mpo, p; kwargs...)

perform a cyclic proximal point algorithm. This can be done in-place of p.

Input

• M::AbstractManifold: a Riemannian manifold $\mathcal{M}$
• f: a cost function $f: \mathcal{M}→ℝ$ to minimize
• proxes_f: an Array of proximal maps (Functions) (M,λ,p) -> q or (M, q, λ, p) -> q for the summands of $f$ (see evaluation)

where f and the proximal maps proxes_f can also be given directly as a ManifoldProximalMapObjective mpo

Keyword arguments

• callbacks::D = Dict{Symbol,Function}(): provided callback functions given either as a single function (symbol, problem, state, k) called in every hook or as a (vector of) pairs :hook => function, which are processed by process_callbacks_arg. As key you can either pass single symbol or an array of symbols to indicate a callback should be added in multiple places
• evaluation::AbstractEvaluationType=AllocatingEvaluation(): specify whether the functions that return an array, for example a point or a tangent vector, work by allocating their result (AllocatingEvaluation) or whether they modify their input argument to return the result therein (InplaceEvaluation). Since usually the first argument is the manifold, the modified argument is the second.
• evaluation_order=:Linear: specify whether to use a fixed randomly permuted sequence (:FixedRandom), a per cycle newly permuted sequence (:Random), or the default :Linear order.
• λ=k -> 1/k: a function returning the (square summable but not summable) sequence of $λ_k$
• stopping_criterion::StoppingCriterion=StopAfterIteration(5000)|StopWhenChangeLess(1e-12): a functor indicating that the stopping criterion is fulfilled

All other keyword arguments are passed to decorate_state! for state decorators or decorate_objective! for objective decorators, respectively.

Output

The obtained approximate minimizer $p^*$. To obtain the whole final state of the solver, see get_solver_return for details, especially the return_state= keyword.

CyclicProximalPointState <: AbstractManoptSolverState

stores options for the cyclic_proximal_point algorithm. These are the

Fields

• callbacks::D: provided callback functions given as a dictionary with symbols as keys
• p::P: a point on the manifold $\mathcal{M}$ storing the current iterate
• stop::StoppingCriterion: a functor indicating that the stopping criterion is fulfilled
• λ: a function for the values of $λ_k$ per iteration (cycle) $k$
• order_type: specify whether to use a fixed randomly permuted sequence (:FixedRandom), a per cycle newly permuted sequence (:Random), or the default :Linear order.

Constructor

CyclicProximalPointState(M::AbstractManifold; kwargs...)

Generate the options

Input

• M::AbstractManifold: a Riemannian manifold $\mathcal{M}$

Keyword arguments

• callbacks::D = Dict{Symbol,Function}(): provided callback functions given as a dictionary with symbols as keys
• evaluation_order=:Linear: specify whether to use a fixed randomly permuted sequence (:FixedRandom), a per cycle newly permuted sequence (:Random), or the default :Linear order.
• λ=k -> 1.0 / k: a function to compute $λ_k$ for $k ∈ ℕ$,
• p::P =rand(M): a point on the manifold $\mathcal{M}$ to specify the initial value
• stopping_criterion::StoppingCriterion=StopAfterIteration(2000): a functor indicating that the stopping criterion is fulfilled

See also

cyclic_proximal_point