# Solvers

Solvers can be applied to `AbstractManoptProblem`

s with solver specific `AbstractManoptSolverState`

.

# List of Algorithms

The following algorithms are currently available

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 `AbstractManoptProblem`

`p`

and the `AbstractManoptSolverState`

`s`

employing `initialize_solver!`

, `step_solver!`

, as well as the `stop_solver!`

of the solver.

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`

.

`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.

`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.

`Manopt.step_solver!`

— Function`step_solver!(amp::AbstractManoptProblem, ams::AbstractManoptSolverState, i)`

Do one iteration step (the `i`

th) for an `AbstractManoptProblem`

`p`

by modifying the values in the `AbstractManoptSolverState`

`ams`

.

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

Extend the `i`

th step of the solver by a hook to run debug prints, that were added to the `:Step`

and `:All`

entries of the debug lists.

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

Extend the `i`

th step of the solver by a hook to run records, that were added to the `:Iteration`

entry.

`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.

`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.

`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`

## 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)`