Solvers

Solvers can be applied to AbstractManoptProblems with solver specific AbstractManoptSolverState.

List of Algorithms

The following algorithms are currently available

Solver	Function & State	Objective
Alternating Gradient Descent	`alternating_gradient_descent` `AlternatingGradientDescentState`	$f=(f_1,\ldots,f_n)$, $\operatorname{grad} f_i$
Chambolle-Pock	`ChambollePock`, `ChambollePockState` (using `TwoManifoldProblem`)	$f=F+G(Λ\cdot)$, $\operatorname{prox}_{σ F}$, $\operatorname{prox}_{τ G^*}$, $Λ$
Conjugate Gradient Descent	`conjugate_gradient_descent`, `ConjugateGradientDescentState`	$f$, $\operatorname{grad} f$
Cyclic Proximal Point	`cyclic_proximal_point`, `CyclicProximalPointState`	$f=\sum f_i$, $\operatorname{prox}_{\lambda f_i}$
Difference of Convex Algorithm	`difference_of_convex_algorithm`, `DifferenceOfConvexState`	$f=g-h$, $∂h$, and e.g. $g$, $\operatorname{grad} g$
Difference of Convex Proximal Point	`difference_of_convex_proximal_point`, `DifferenceOfConvexProximalState`	$f=g-h$, $∂h$, and e.g. $g$, $\operatorname{grad} g$
Douglas–Rachford	`DouglasRachford`, `DouglasRachfordState`	$f=\sum f_i$, $\operatorname{prox}_{\lambda f_i}$
Exact Penalty Method	`exact_penalty_method`, `ExactPenaltyMethodState`	$f$, $\operatorname{grad} f$, $g$, $\operatorname{grad} g_i$, $h$, $\operatorname{grad} h_j$
Frank-Wolfe algorithm	`Frank_Wolfe_method`, `FrankWolfeState`	sub-problem solver
Gradient Descent	`gradient_descent`, `GradientDescentState`	$f$, $\operatorname{grad} f$
Levenberg-Marquardt	`LevenbergMarquardt`, `LevenbergMarquardtState`	$f = \sum_i f_i$ $\operatorname{grad} f_i$ (Jacobian)
Nelder-Mead	`NelderMead`, `NelderMeadState`	$f$
Augmented Lagrangian Method	`augmented_Lagrangian_method`, `AugmentedLagrangianMethodState`	$f$, $\operatorname{grad} f$, $g$, $\operatorname{grad} g_i$, $h$, $\operatorname{grad} h_j$
Particle Swarm	`particle_swarm`, `ParticleSwarmState`	$f$
Primal-dual Riemannian semismooth Newton Algorithm	`primal_dual_semismooth_Newton`, `PrimalDualSemismoothNewtonState` (using `TwoManifoldProblem`)	$f=F+G(Λ\cdot)$, $\operatorname{prox}_{σ F}$ & diff., $\operatorname{prox}_{τ G^*}$ & diff., $Λ$
Quasi-Newton Method	`quasi_Newton`, `QuasiNewtonState`	$f$, $\operatorname{grad} f$
Steihaug-Toint Truncated Conjugate-Gradient Method	`truncated_conjugate_gradient_descent`, `TruncatedConjugateGradientState`	$f$, $\operatorname{grad} f$, $\operatorname{Hess} f$
Subgradient Method	`subgradient_method`, `SubGradientMethodState`	$f$, $∂ f$
Stochastic Gradient Descent	`stochastic_gradient_descent`, `StochasticGradientDescentState`	$f = \sum_i f_i$, $\operatorname{grad} f_i$
The Riemannian Trust-Regions Solver	`trust_regions`, `TrustRegionsState`	$f$, $\operatorname{grad} f$, $\operatorname{Hess} f$

Note that the solvers (their AbstractManoptSolverState, to be precise) can also be decorated to enhance your algorithm by general additional properties, see debug output and recording values. This is done using the debug= and record= keywords in the function calls. Similarly, since 0.4 we provide a (simple) caching of the objective function using the cache= keyword in any of the function calls..

Technical Details

The main function a solver calls is

Manopt.solve! — Method

solve!(p::AbstractManoptProblem, s::AbstractManoptSolverState)

run the solver implemented for the AbstractManoptProblemp and the AbstractManoptSolverStates employing initialize_solver!, step_solver!, as well as the stop_solver! of the solver.

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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.

Manopt.initialize_solver! — Function

initialize_solver!(ams::AbstractManoptProblem, amp::AbstractManoptSolverState)

Initialize the solver to the optimization AbstractManoptProblem amp by initializing the necessary values in the AbstractManoptSolverState amp.

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initialize_solver!(amp::AbstractManoptProblem, dss::DebugSolverState)

Extend the initialization of the solver by a hook to run debug that were added to the :Start and :All entries of the debug lists.

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initialize_solver!(ams::AbstractManoptProblem, rss::RecordSolverState)

Extend the initialization of the solver by a hook to run records that were added to the :Start entry.

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Manopt.step_solver! — Function

step_solver!(amp::AbstractManoptProblem, ams::AbstractManoptSolverState, i)

Do one iteration step (the ith) for an AbstractManoptProblemp by modifying the values in the AbstractManoptSolverState ams.

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step_solver!(amp::AbstractManoptProblem, dss::DebugSolverState, i)

Extend the ith step of the solver by a hook to run debug prints, that were added to the :Step and :All entries of the debug lists.

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step_solver!(amp::AbstractManoptProblem, rss::RecordSolverState, i)

Extend the ith step of the solver by a hook to run records, that were added to the :Iteration entry.

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Manopt.get_solver_result — Function

get_solver_result(ams::AbstractManoptSolverState)
get_solver_result(tos::Tuple{AbstractManifoldObjective,AbstractManoptSolverState})
get_solver_result(o::AbstractManifoldObjective, s::AbstractManoptSolverState)

Return the final result after all iterations that is stored within the AbstractManoptSolverState ams, which was modified during the iterations.

For the case the objective is passed as well, but default, the objective is ignored, and the solver result for the state is called.

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Manopt.get_solver_return — Function

get_solver_return(s::AbstractManoptSolverState)
get_solver_return(o::AbstractManifoldObjective, s::AbstractManoptSolverState)

determine the result value of a call to a solver. By default this returns the same as get_solver_result, i.e. the last iterate or (approximate) minimizer.

get_solver_return(s::ReturnSolverState)
get_solver_return(o::AbstractManifoldObjective, s::ReturnSolverState)

return the internally stored state of the ReturnSolverState instead of the minimizer. This means that when the state are decorated like this, the user still has to call get_solver_result on the internal state separately.

get_solver_return(o::ReturnManifoldObjective, s::AbstractManoptSolverState)

return both the objective and the state as a tuple.

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Manopt.stop_solver! — Method

stop_solver!(amp::AbstractManoptProblem, ams::AbstractManoptSolverState, i)

depending on the current AbstractManoptProblem amp, the current state of the solver stored in AbstractManoptSolverState ams and the current iterate i this function determines whether to stop the solver, which by default means to call the internal StoppingCriterion. ams.stop

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API for solvers

this is a short overview of the different types of high-level functions are usually available for a solver. Let's assume the solver is called new_solver and requires a cost f and some first order information df as well as a starting point p on M. f and df form the objective together called obj.

Then there are basically two different variants to call

The easy to access call

new_solver(M, f, df, p=rand(M); kwargs...)
new_solver!(M, f, df, p; kwargs...)

Where the start point should be optional. Keyword arguments include the type of evaluation, decorators like debug= or record= as well as algorithm specific ones. If you provide an immutable point p or the rand(M) point is immutable, like on the Circle() this method should turn the point into a mutable one as well.

The third variant works in place of p, so it is mandatory.

This first interface would set up the objective and pass all keywords on the the objective based call.

The objective-based call

new_solver(M, obj, p=rand(M); kwargs...)
new_solver!(M, obj, p; kwargs...)

Here the objective would be created beforehand, e.g. to compare different solvers on the same objective, and for the first variant the start point is optional. Keyword arguments include decorators like debug= or record= as well as algorithm specific ones.

this variant would generate the problem and the state and check validity of all provided keyword arguments that affect the state. Then it would call the iterate process.

The manual call

If you generate the correctsponding problem and state as the previous step does, you can also use the third (lowest level) and just call

solve!(problem, state)