subGradientMethod(M, F, ∂F, x)

perform a subgradient method $x_{k+1} = \mathrm{retr}(x_k, s_k∂F(x_k))$,

where $\mathrm{retr}$ is a retraction, $s_k$ can be specified as a function but is usually set to a constant value. Though the subgradient might be set valued, the argument ∂F should always return one element from the subgradient.

Input

• M – a manifold $\mathcal M$
• F – a cost function $F\colon\mathcal M\to\mathbb R$ to minimize
• ∂F: the (sub)gradient $\partial F\colon\mathcal M\to T\mathcal M$ of F restricted to always only returning one value/element from the subgradient
• x – an initial value $x\in\mathcal M$

Optional

• stepsize – (ConstantStepsize(1.)) specify a Stepsize
• retraction – (exp) a retraction(M,x,ξ) to use.
• stoppingCriterion – (stopWhenAny(stopAfterIteration(200),stopWhenGradientNormLess(10.0^-8))) a functor, seeStoppingCriterion, indicating when to stop.
• returnOptions – (false) – if actiavated, the extended result, i.e. the complete Options re returned. This can be used to access recorded values. If set to false (default) just the optimal value xOpt if returned

... and the ones that are passed to decorateOptions for decorators.

Output

• xOpt – the resulting (approximately critical) point of gradientDescent

OR

• options - the options returned by the solver (see returnOptions)