Subgradient Method

Subgradient Method

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.


  • 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 $x\in\mathcal M$


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


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


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


SubGradientMethodOptions <: Options

stories option values for a subGradientMethod solver


  • retraction – the retration to use within
  • stepsize – a Stepsize
  • stop – a StoppingCriterion
  • x – (initial or current) value the algorithm is at
  • optimalX – optimal value

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