Subgradient Method

Manopt.subgradient_methodFunction
subgradient_method(M, f, ∂f, p)

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.

Input

  • M – a manifold $\mathcal M$
  • f – a cost function $f:\mathcal M→ℝ$ to minimize
  • ∂f– the (sub)gradient $\partial f: \mathcal M→ T\mathcal M$ of F restricted to always only returning one value/element from the subgradient. 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 $p_0=p ∈ \mathcal M$

Optional

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

Output

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

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Manopt.subgradient_method!Function
subgradient_method!(M, f, ∂f, x)

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 $\partial f: \mathcal M→ T\mathcal M$ of F restricted to always only returning one value/element from the subgradient. 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 $p_0=p ∈ \mathcal M$

for more details and all optional parameters, see subgradient_method.

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State

Manopt.SubGradientMethodStateType
SubGradientMethodState <: AbstractManoptSolverState

stories option values for a subgradient_method solver

Fields

  • retraction_method – the retration 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 above besides p_star which obtains the same type as p. You can use e.g. X= to specify the type of tangent vector to use

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For DebugActions and RecordActions to record (sub)gradient, its norm and the step sizes, see the steepest Descent actions.