Find new orbits

To find new orbits, first initialize a problem using the initialize function (see Problem definition). Then, use the find_orbits function to find new orbits.

The find_orbits function receives the following arguments:

  • P: A Problem object.
  • method: An optimization method. It must be a subtype of AbstractMethod. See below for more information.
  • number_of_orbits: The number of orbits to be found. Default is 1. If number_of_orbits = Inf, the function will continue to search for orbits until it is stopped by the user.
  • starting_path_type: a Symbol that indicates the type of the starting path. Default is :random. See below for more information.
  • starting_path: a Path object that indicates the starting path. Default is nothing. If starting_path is provided, starting_path_type is ignored.
  • print_path: a Bool that indicates if the path should be printed. Default is true. The filename is the value of the action.
  • print_path_folder: a String that indicates the folder where the path should be printed. Default is "./" (the current folder).

Any additional keyword arguments are passed to the optimization method. For example, when using a CompoundMethod, you can pass the keyword max_iterations in order to specify the maximum number of times the first and second methods are executed.

Methods

To specify an optimization method, you can combine multiple AtomicMethods. The available atomic methods are:

  • First Order Methods:
    • BFGS(): BFGS method.
    • ConjugateGradient(): Conjugate gradient method.
    • GradientDescent(): Gradient descent method.
  • Second Order Methods:
    • Newton(): Newton method.

Every atomic method accepts the following options:

  • max_iter: Maximum number of iterations. Default is 200.
  • tolerance: Tolerance on the gradient norm for the stopping criterion. Default is 1e-8.

You can combine them using the following constructors:

  • OneMethod(method): Use a single method.
  • InitMinimizeMethod(init, method): Use an initialization method and then a minimization method.
  • CompoundMethod(init, first, second): Use a compound method. The init method is executed first. Then the first and second methods are executed repetedly until the stopping criterion is met.

The following are valid examples of method combinations:

  • OneMethod(BFGS()): Use the BFGS method with 200 iterations and a tolerance of 1e-8 (default options)
  • OneMethod(Newton(400)): Use the Newton method with 400 iterations (custom) and a tolerance of 1e-8 (default)
  • InitMinimizeMethod(ConjugateGradient(10), BFGS()): Use the Conjugate Gradient method with 10 iterations (custom) and then the BFGS method with 200 iterations and a tolerance of 1e-8 (default)
  • CompoundMethod(BFGS(5), Newton(300), BFGS(10)): Use the BFGS method with 5 iterations (custom) to initialize, then the Newton method with 300 iterations (custom) and finally the BFGS method with 10 iterations and a tolerance of 1e-8 (default). The last two methods are executed repeatedly until the stopping criterion is met.

Starting Path Type

The starting_path_type argument can be one of the following:

  • :random: A random path.
  • :circular: A circular path.
  • :perturbed_circular: A circular path with a small perturbation.

Example


julia> result = find_orbits(P, OneMethod(BFGS()), number_of_orbits=1)number_of_orbits = 1
#0: Searching new orbit...
  #1 
     First method: BFGS(max_iter=200, tolerance=1.0e-8)
     Action  = 7.480447856567118
     ∇action = 2.280734604662109e-9

Minimization result:
	Method: 	CompoundMethod{Nothing, BFGS, Nothing}
	Converged: 	true
	Gradient norm: 	2.2807346066240926e-9
	Action value: 	7.480447856567119
==> Optimization converged

1-element Vector{Any}:
 Minimization result:
	Method: 	CompoundMethod{Nothing, BFGS, Nothing}
	Converged: 	true
	Gradient norm: 	2.2807346066240926e-9
	Action value: 	7.480447856567119