Proximal point method

proximal_point(M, prox_f, p=rand(M); kwargs...)
proximal_point(M, mpmo, p=rand(M); kwargs...)
proximal_point!(M, prox_f, p; kwargs...)
proximal_point!(M, mpmo, p; kwargs...)

Perform the proximal point algoritm from [FO02] which reads

\[p^{(k+1)} = \operatorname{prox}_{λ_kf}(p^{(k)})\]

Input

• M::AbstractManifold: a Riemannian manifold $\mathcal M$
• prox_f: a proximal map (M,λ,p) -> q or (M, q, λ, p) -> q for the summands of $f$ (see evaluation)

Keyword arguments

• evaluation=AllocatingEvaluation(): specify whether the functions that return an array, for example a point or a tangent vector, work by allocating its 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.
• f=nothing: a cost function $f: \mathcal M→ℝ$ to minimize. For running the algorithm, $f$ is not required, but for example when recording the cost or using a stopping criterion that requires a cost function.
• λ= k -> 1.0: a function returning the (square summable but not summable) sequence of $λ_i$
• stopping_criterion=StopAfterIteration(200)|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.

proximal_point(M, prox_f, p=rand(M); kwargs...)
proximal_point(M, mpmo, p=rand(M); kwargs...)
proximal_point!(M, prox_f, p; kwargs...)
proximal_point!(M, mpmo, p; kwargs...)

Perform the proximal point algoritm from [FO02] which reads

\[p^{(k+1)} = \operatorname{prox}_{λ_kf}(p^{(k)})\]

Input

• M::AbstractManifold: a Riemannian manifold $\mathcal M$
• prox_f: a proximal map (M,λ,p) -> q or (M, q, λ, p) -> q for the summands of $f$ (see evaluation)

Keyword arguments

• evaluation=AllocatingEvaluation(): specify whether the functions that return an array, for example a point or a tangent vector, work by allocating its 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.
• f=nothing: a cost function $f: \mathcal M→ℝ$ to minimize. For running the algorithm, $f$ is not required, but for example when recording the cost or using a stopping criterion that requires a cost function.
• λ= k -> 1.0: a function returning the (square summable but not summable) sequence of $λ_i$
• stopping_criterion=StopAfterIteration(200)|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.

ProximalPointState{P} <: AbstractGradientSolverState

Fields

• 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 $k$

Constructor

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

Initialize the proximal point method solver state, where

Input

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

Keyword arguments

• λ=k -> 1.0 a function to compute the $λ_k, k ∈ \mathcal N$,
• p=rand(M): a point on the manifold $\mathcal M$ to specify the initial value
• stopping_criterion=StopAfterIteration(100): a functor indicating that the stopping criterion is fulfilled

See also

proximal_point