Solvers

Solvers can be applied to Problems with solver specific Options.

List of Algorithms

The following algorithms are currently available

SolverFileProblem & Option
steepest Descentsteepest_descent.jlGradientProblem, GradientDescentOptions
Cyclic Proximal Pointcyclic_proximal_point.jlProximalProblem, CyclicProximalPointOptions
Douglas–RachfordDouglasRachford.jlProximalProblem, DouglasRachfordOptions
Nelder-MeadNelderMead.jlCostProblem, NelderMeadOptions
Subgradient Methodsubgradient_method.jlSubGradientProblem, SubGradientMethodOptions
Steihaug-Toint Truncated Conjugate-Gradient Methodtruncated_conjugate_gradient_descent.jlHessianProblem,

TruncatedConjugateGradientOptions The Riemannian Trust-Regions Solver | trust_regions.jl | HessianProblem, TrustRegionsOptions

Note that the Options can also be decorated to enhance your algorithm by general additional properties.

StoppingCriteria

Stopping criteria are implemented as a functor, i.e. inherit from the base type

Manopt.StoppingCriterionType
StoppingCriterion

An abstract type for the functors representing stoping criteria, i.e. they are callable structures. The naming Scheme follows functions, see for example StopAfterIteration.

Every StoppingCriterion has to provide a constructor and its function has to have the interface (p,o,i) where a Problem as well as Options and the current number of iterations are the arguments and returns a Bool whether to stop or not.

By default each StoppingCriterion should provide a fiels reason to provide details when a criteion is met (and that is empty otherwise).

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Manopt.StoppingCriterionSetType
StoppingCriterionGroup <: StoppingCriterion

An abstract type for a Stopping Criterion that itself consists of a set of Stopping criteria. In total it acts as a stopping criterion itself. Examples are StopWhenAny and StopWhenAll that can be used to combine stopping criteria.

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Manopt.StopAfterType
StopAfter <: StoppingCriterion

store a threshold when to stop looking at the complete runtime. It uses time_ns() to measure the time and you provide a Period as a time limit, i.e. Minute(15)

Constructor

StopAfter(t)

initialize the stopping criterion to a Period t to stop after.

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Manopt.StopAfterIterationType
StopAfterIteration <: StoppingCriterion

A functor for an easy stopping criterion, i.e. to stop after a maximal number of iterations.

Fields

  • maxIter – stores the maximal iteration number where to stop at
  • reason – stores a reason of stopping if the stopping criterion has one be reached, see get_reason.

Constructor

StopAfterIteration(maxIter)

initialize the stopafterIteration functor to indicate to stop after maxIter iterations.

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Manopt.StopWhenAllType
StopWhenAll <: StoppingCriterion

store an array of StoppingCriterion elements and indicates to stop, when all indicate to stop. The reseason is given by the concatenation of all reasons.

Constructor

StopWhenAll(c::Array{StoppingCriterion,1})
StopWhenAll(c::StoppingCriterion,...)
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Manopt.StopWhenAnyType
StopWhenAny <: StoppingCriterion

store an array of StoppingCriterion elements and indicates to stop, when any single one indicates to stop. The reseason is given by the concatenation of all reasons (assuming that all non-indicating return "").

Constructor

StopWhenAny(c::Array{StoppingCriterion,1})
StopWhenAny(c::StoppingCriterion,...)
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Manopt.StopWhenChangeLessType
StopWhenChangeLess <: StoppingCriterion

stores a threshold when to stop looking at the norm of the change of the optimization variable from within a Options, i.e o.x. For the storage a StoreOptionsAction is used

Constructor

StopWhenChangeLess(ε[, a])

initialize the stopping criterion to a threshold ε using the StoreOptionsAction a, which is initialized to just store :x by default.

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Manopt.StopWhenCostLessType
StopWhenCostLess <: StoppingCriterion

store a threshold when to stop looking at the cost function of the optimization problem from within a Problem, i.e get_cost(p,o.x).

Constructor

StopWhenCostLess(ε)

initialize the stopping criterion to a threshold ε.

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as well as the functions

Manopt.get_active_stopping_criteriaFunction
get_active_stopping_criteria(c)

returns all active stopping criteria, if any, that are within a StoppingCriterion c, and indicated a stop, i.e. their reason is nonempty. To be precise for a simple stopping criterion, this returns either an empty array if no stop is incated or the stopping criterion as the only element of an array. For a StoppingCriterionSet all internal (even nested) criteria that indicate to stop are returned.

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further stopping criteria might be available for individual Solvers.

Decorated Solvers

The following decorators are available.

Debug Solver

The decorator to print debug during the iterations can be activated by decorating the Options with DebugOptions and implementing your own DebugActions. For example printing a gradient from the GradientDescentOptions is automatically available, as explained in the steepest_descent solver.

Record Solver

The decorator to record certain values during the iterations can be activated by decorating the Options with RecordOptions and implementing your own RecordActions. For example recording the gradient from the GradientDescentOptions is automatically available, as explained in the steepest_descent solver.

Technical Details

The main function a solver calls is

which is a framework, that you in general should not change or redefine. It uses the following methods, which also need to be implemented on your own algorithm, if you want to provide one.