Subgradient method

subgradient_method(M, f, ∂f, p=rand(M); kwargs...)
subgradient_method(M, sgo, p=rand(M); kwargs...)

perform a subgradient method $p_{k+1} = \mathrm{retr}(p_k, s_k∂f(p_k))$,

where $\mathrm{retr}$ is a retraction, $s_k$ is a step size, usually the ConstantStepsize but also be specified. Though the subgradient might be set valued, the argument ∂f should always return one element from the subgradient, but not necessarily deterministic. For more details see [FO98].

Input

• M: a manifold $\mathcal M$
• f: a cost function $f:\mathcal M→ℝ$ to minimize
• ∂f: the (sub)gradient $∂ f: \mathcal M→ T\mathcal M$ of f restricted to always only returning one value/element from the subdifferential. This function can be passed as an allocation function (M, p) -> X or a mutating function (M, X, p) -> X, see evaluation.
• p: (rand(M)) an initial value $p_0=p ∈ \mathcal M$

alternatively to f and ∂f a ManifoldSubgradientObjective sgo can be provided.

Optional

• evaluation: (AllocatingEvaluation) specify whether the subgradient works by allocation (default) form ∂f(M, y) or InplaceEvaluation in place of the form ∂f!(M, X, x).
• retraction: (default_retraction_method(M, typeof(p))) a retraction to use.
• stepsize: (ConstantStepsize(M)) specify a Stepsize
• stopping_criterion: (StopAfterIteration(5000)) a functor, seeStoppingCriterion, indicating when to stop.

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

Output

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

subgradient_method!(M, f, ∂f, p)
subgradient_method!(M, sgo, p)

perform a subgradient method $p_{k+1} = \mathrm{retr}(p_k, s_k∂f(p_k))$,

Input

• M: a manifold $\mathcal M$
• f: a cost function $f:\mathcal M→ℝ$ to minimize
• ∂f: the (sub)gradient $∂f: \mathcal M→ T\mathcal M$ of F restricted to always only returning one value/element from the subdifferential. This function can be passed as an allocation function (M, p) -> X or a mutating function (M, X, p) -> X, see evaluation.
• p: an initial value $p_0=p ∈ \mathcal M$

alternatively to f and ∂f a ManifoldSubgradientObjective sgo can be provided.

for more details and all optional parameters, see subgradient_method.

SubGradientMethodState <: AbstractManoptSolverState

stores option values for a subgradient_method solver

Fields

• retraction_method: the retraction to use within
• stepsize: (ConstantStepsize(M)) a Stepsize
• stop: (StopAfterIteration(5000))a [StoppingCriterion`](@ref)
• p: (initial or current) value the algorithm is at
• p_star: optimal value (initialized to a copy of p.)
• X: (zero_vector(M, p)) the current element from the possible subgradients at p that was last evaluated.

Constructor

SubGradientMethodState(M::AbstractManifold, p; kwargs...)

with keywords for all fields besides p_star which obtains the same type as p. You can use X= to specify the type of tangent vector to use