rastrigin module¶
The Rastrigin analytic problem¶
- class gemseo.problems.analytical.rastrigin.Rastrigin[source]
Bases:
OptimizationProblem
Rastrigin
OptimizationProblem
uses the Rastrigin objective function with theDesignSpace
\([-0.1,0.1]^2\)From http://en.wikipedia.org/wiki/Rastrigin_function:
the Rastrigin function is a non-convex function used as a performance test problem for optimization algorithms. It is a typical example of non-linear multimodal function. It was first proposed by [Rastrigin] as a 2-dimensional function and has been generalized by [MuhlenbeinEtAl]. Finding the minimum of this function is a fairly difficult problem due to its large search space and its large number of local minima. It has a global minimum at \(x=0\) where \(f(x)=0\). It can be extended to \(n>2\) dimensions:
\[f(x) = 10n + \sum_{i=1}^n [x_i^2 - 10\cos(2\pi x_i)]\][Rastrigin] Rastrigin, L. A. “Systems of extremal control.” Mir, Moscow (1974).
[MuhlenbeinEtAl] H. Mühlenbein, D. Schomisch and J. Born. “The Parallel Genetic Algorithm as Function Optimizer “. Parallel Computing, 17, pages 619–632, 1991.
- Parameters:
design_space – The design space on which the functions are evaluated.
pb_type – The type of the optimization problem.
input_database – A database to initialize that of the optimization problem. If
None
, the optimization problem starts from an empty database.differentiation_method – The default differentiation method to be applied to the functions of the optimization problem.
fd_step – The step to be used by the step-based differentiation methods.
parallel_differentiation – Whether to approximate the derivatives in
parallel. –
use_standardized_objective – Whether to use standardized objective for logging and post-processing.
**parallel_differentiation_options – The options to approximate the derivatives in parallel.
- static get_solution()[source]
Return theoretical optimal value of Rastrigin function.
- Returns:
The design variable and objective function at optimum.
- static rastrigin(x_dv)[source]
Evaluate the 2nd order Rastrigin function.
- Parameters:
x_dv – The design variables.
- Returns:
The Rastrigin function output.
- Return type:
- static rastrigin_jac(x_dv)[source]
Compute the analytical gradient of 2nd order Rastrigin function.
- Parameters:
x_dv – The design variable vector.
- Returns:
The analytical gradient vector of Rastrigin function.
- constraints: list[MDOFunction]
The constraints.
- current_iter: int
The current iteration.
- database: Database
The database to store the optimization problem data.
- design_space: DesignSpace
The design space on which the optimization problem is solved.
- eq_tolerance: float
The tolerance for the equality constraints.
- fd_step: float
The finite differences step.
- ineq_tolerance: float
The tolerance for the inequality constraints.
- max_iter: int
The maximum iteration.
- new_iter_observables: list[MDOFunction]
The observables to be called at each new iterate.
- nonproc_constraints: list[MDOFunction]
The non-processed constraints.
- nonproc_new_iter_observables: list[MDOFunction]
The non-processed observables to be called at each new iterate.
- nonproc_objective: MDOFunction
The non-processed objective function.
- nonproc_observables: list[MDOFunction]
The non-processed observables.
- observables: list[MDOFunction]
The observables.
- pb_type: ProblemType
The type of optimization problem.
- preprocess_options: dict
The options to pre-process the functions.
- solution: OptimizationResult
The solution of the optimization problem.
- stop_if_nan: bool
Whether the optimization stops when a function returns
NaN
.
- use_standardized_objective: bool
Whether to use standardized objective for logging and post-processing.
The standardized objective corresponds to the original one expressed as a cost function to minimize. A
DriverLibrary
works with this standardized objective and theDatabase
stores its values. However, for convenience, it may be more relevant to log the expression and the values of the original objective.