gemseo / problems / analytical

# rastrigin module¶

## The Rastrigin analytic problem¶

Classes:

 Rastrigin OptimizationProblem uses the Rastrigin objective function with the DesignSpace $$[-0.1,0.1]^2$$
class gemseo.problems.analytical.rastrigin.Rastrigin[source]

Rastrigin OptimizationProblem uses the Rastrigin objective function with the DesignSpace $$[-0.1,0.1]^2$$

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.

The constructor initializes the Rastrigin OptimizationProblem by defining the DesignSpace and the objective function.

Attributes:

Methods:

 add_callback(callback_func[, each_new_iter, …]) Add a callback function after each store operation or new iteration. add_constraint(cstr_func[, value, …]) Add a constraint (equality and inequality) to the optimization problem. add_eq_constraint(cstr_func[, value]) Add an equality constraint to the optimization problem. add_ineq_constraint(cstr_func[, value, positive]) Add an inequality constraint to the optimization problem. add_new_iter_listener(listener_func) Add a listener to be called when a new iteration is stored to the database. add_observable(obs_func[, new_iter]) Add a function to be observed. add_store_listener(listener_func) Add a listener to be called when an item is stored to the database. aggregate_constraint(constr_id[, method, groups]) Aggregates a constraint to generate a reduced dimension constraint. Change the objective function sign in order to minimize its opposite. Check if the optimization problem is ready for run. check_format(input_function) Check that a function is an instance of MDOFunction. Clear all the listeners. evaluate_functions([x_vect, eval_jac, …]) Compute the objective and the constraints. export_hdf(file_path[, append]) Export the optimization problem to an HDF file. export_to_dataset(name[, by_group, …]) Export the database of the optimization problem to a Dataset. get_active_ineq_constraints(x_vect[, tol]) For each constraint, indicate if its different components are active. Retrieve all the functions of the optimization problem. Retrieve the names of all the function of the optimization problem. Retrieve the best infeasible point within a given tolerance. Retrieve the names of the constraints. Retrieve the number of constraints. Retrieve the names of the design variables. Retrieve the total number of design variables. Retrieve all the equality constraints. Retrieve the number of equality constraints. Retrieve the total dimension of the equality constraints. Retrieve the feasible points within a given tolerance. Retrieve all the inequality constraints. Retrieve the number of inequality constraints. Retrieve the total dimension of the inequality constraints. Retrieve the non-processed constraints. Retrieve the non-processed objective function. Retrieve the name of the objective function. get_observable(name) Retrieve an observable from its name. Return the optimum solution within a given feasibility tolerances. Return theoretical optimal value of Rastrigin function. get_violation_criteria(x_vect) Compute a violation measure associated to an iteration. Return the current values of the design variables after normalization. Check if the problem has equality or inequality constraints. Check if the problem has equality constraints. Check if the problem has inequality constraints. Check if the problem has non-linear constraints. import_hdf(file_path[, x_tolerance]) Import an optimization history from an HDF file. is_point_feasible(out_val[, constraints]) Check if a point is feasible. preprocess_functions([normalize, …]) Pre-process all the functions and eventually the gradien. rastrigin(x_dv) This function computes the order n=2 Rastrigin function. rastrigin_jac(x_dv) This function computes the analytical gradient of 2nd order Rastrigin function. repr_constraint(func, ctype[, value, positive]) Express a constraint as a string expression.
AVAILABLE_PB_TYPES = ['linear', 'non-linear']
COMPLEX_STEP = 'complex_step'
CONSTRAINTS_GROUP = 'constraints'
DESIGN_SPACE_ATTRS = ['u_bounds', 'l_bounds', 'x_0', 'x_names', 'dimension']
DESIGN_SPACE_GROUP = 'design_space'
DESIGN_VAR_NAMES = 'x_names'
DESIGN_VAR_SIZE = 'x_size'
DIFFERENTIATION_METHODS = ['user', 'complex_step', 'finite_differences', 'no_derivatives']
FINITE_DIFFERENCES = 'finite_differences'
FUNCTIONS_ATTRS = ['objective', 'constraints']
GGOBI_FORMAT = 'ggobi'
HDF5_FORMAT = 'hdf5'
LINEAR_PB = 'linear'
NON_LINEAR_PB = 'non-linear'
NO_DERIVATIVES = 'no_derivatives'
OBJECTIVE_GROUP = 'objective'
OPTIM_DESCRIPTION = ['minimize_objective', 'fd_step', 'differentiation_method', 'pb_type', 'ineq_tolerance', 'eq_tolerance']
OPT_DESCR_GROUP = 'opt_description'
SOLUTION_GROUP = 'solution'

Add a callback function after each store operation or new iteration.

Parameters
• callback_func (Callable) – A function to be called after some event.

• each_new_iter (bool) – If True, then callback at every iteration.

• each_store (bool) – If True, then callback at every call to Database.store.

Return type

None

Add a constraint (equality and inequality) to the optimization problem.

Parameters
• cstr_func (MDOFunction) – The constraint.

• value (Optional[value]) – The value for which the constraint is active. If None, this value is 0.

• cstr_type (Optional[str]) – The type of the constraint. Either equality or inequality.

• positive (bool) – If True, then the inequality constraint is positive.

Return type

None

Add an equality constraint to the optimization problem.

Parameters
• cstr_func (gemseo.core.function.MDOFunction) – The constraint.

• value (Optional[float]) – The value for which the constraint is active. If None, this value is 0.

Return type

None

Add an inequality constraint to the optimization problem.

Parameters
• cstr_func (MDOFunction) – The constraint.

• value (Optional[value]) – The value for which the constraint is active. If None, this value is 0.

• positive (bool) – If True, then the inequality constraint is positive.

Return type

None

Add a listener to be called when a new iteration is stored to the database.

Parameters

listener_func (Callable) – The function to be called.

Raises

TypeError – If the argument is not a callable

Return type

None

Add a function to be observed.

Parameters
• obs_func (gemseo.core.function.MDOFunction) – An observable to be observed.

• new_iter (bool) – If True, then the observable will be called at each new iterate.

Return type

None

Add a listener to be called when an item is stored to the database.

Parameters

listener_func (Callable) – The function to be called.

Raises

TypeError – If the argument is not a callable

Return type

None

aggregate_constraint(constr_id, method='max', groups=None, **options)

Aggregates a constraint to generate a reduced dimension constraint.

Parameters
• constr_id (int) – index of the constraint in self.constraints

• method (str or callable, that takes a function and returns a function) – aggregation method, among (‘max’,’KS’, ‘IKS’)

• groups (tuple of ndarray) – if None, a single output constraint is produced otherwise, one output per group is produced.

change_objective_sign()

The OptimizationProblem expresses any optimization problem as a minimization problem. Then, an objective function originally expressed as a performance function to maximize must be converted into a cost function to minimize, by means of this method.

Return type

None

check()

Check if the optimization problem is ready for run.

Raises

ValueError – If the objective function is missing.

Return type

None

static check_format(input_function)

Check that a function is an instance of MDOFunction.

Parameters

input_function – The function to be tested.

Raises

TypeError – If the function is not a MDOFunction.

Return type

None

clear_listeners()

Clear all the listeners.

Return type

None

property dimension

The dimension of the design space.

evaluate_functions(x_vect=None, eval_jac=False, eval_obj=True, normalize=True, no_db_no_norm=False)

Compute the objective and the constraints.

Some optimization libraries require the number of constraints as an input parameter which is unknown by the formulation or the scenario. Evaluation of initial point allows to get this mandatory information. This is also used for design of experiments to evaluate samples.

Parameters
• x_vect (Optional[numpy.ndarray]) – The input vector at which the functions must be evaluated; if None, x_0 is used.

• eval_jac (bool) – If True, then the Jacobian is evaluated

• eval_obj (bool) – If True, then the objective function is evaluated

• normalize (bool) – If True, then input vector is considered normalized

• no_db_no_norm (bool) – If True, then do not use the pre-processed functions, so we have no database, nor normalization.

Returns

The functions values and/or the Jacobian values according to the passed arguments.

Raises

ValueError – If both no_db_no_norm and normalize are True.

Return type

Tuple[Dict[str, Union[float, numpy.ndarray]], Dict[str, numpy.ndarray]]

export_hdf(file_path, append=False)

Export the optimization problem to an HDF file.

Parameters
• file_path (str) – The file to store the data.

• append (bool) – If True, then the data are appended to the file if not empty.

Return type

None

Export the database of the optimization problem to a Dataset.

The variables can be classified into groups, separating the design variables and functions (objective function and constraints). This classification can use either an optimization naming, with Database.DESIGN_GROUP and Database.FUNCTION_GROUP or an input-output naming, with Database.INPUT_GROUP and Database.OUTPUT_GROUP

Parameters
• name (str) – A name to be given to the dataset.

• by_group (bool) – If True, then store the data by group. Otherwise, store them by variables.

• categorize (bool) – If True, then distinguish between the different groups of variables.

• opt_naming (bool) – If True, then use an optimization naming.

• export_gradients (bool) – If True, then export also the gradients of the functions (objective function, constraints and observables) if the latter are available in the database of the optimization problem.

Returns

A dataset built from the database of the optimization problem.

Return type

gemseo.core.dataset.Dataset

get_active_ineq_constraints(x_vect, tol=1e-06)

For each constraint, indicate if its different components are active.

Parameters
• x_vect (numpy.ndarray) – The vector of design variables.

• tol (float) – The tolerance for deciding whether a constraint is active.

Returns

For each constraint, a boolean indicator of activation of its different components.

Return type

Dict[str, numpy.ndarray]

get_all_functions()

Retrieve all the functions of the optimization problem.

These functions are the constraints, the objective function and the observables.

Returns

All the functions of the optimization problem.

Return type
get_all_functions_names()

Retrieve the names of all the function of the optimization problem.

These functions are the constraints, the objective function and the observables.

Returns

The names of all the functions of the optimization problem.

Return type

List[str]

get_best_infeasible_point()

Retrieve the best infeasible point within a given tolerance.

Returns

The best infeasible point expressed as the design variables values, the objective function value, the feasibility of the point and the functions values.

Return type

Tuple[Optional[numpy.ndarray], Optional[numpy.ndarray], bool, Dict[str, numpy.ndarray]]

get_constraints_names()

Retrieve the names of the constraints.

Returns

The names of the constraints.

Return type

List[str]

get_constraints_number()

Retrieve the number of constraints.

Returns

The number of constraints.

Return type

int

get_design_variable_names()

Retrieve the names of the design variables.

Returns

The names of the design variables.

Return type

List[str]

get_dimension()

Retrieve the total number of design variables.

Returns

The dimension of the design space.

Return type

int

get_eq_constraints()

Retrieve all the equality constraints.

Returns

The equality constraints.

Return type
get_eq_constraints_number()

Retrieve the number of equality constraints.

Returns

The number of equality constraints.

Return type

int

get_eq_cstr_total_dim()

Retrieve the total dimension of the equality constraints.

This dimension is the sum of all the outputs dimensions of all the equality constraints.

Returns

The total dimension of the equality constraints.

Return type

int

get_feasible_points()

Retrieve the feasible points within a given tolerance.

This tolerance is defined by OptimizationProblem.eq_tolerance for equality constraints and OptimizationProblem.ineq_tolerance for inequality ones.

Returns

The values of the design variables and objective function for the feasible points.

Return type

Tuple[List[numpy.ndarray], List[Dict[str, Union[float, List[int]]]]]

get_ineq_constraints()

Retrieve all the inequality constraints.

Returns

The inequality constraints.

Return type
get_ineq_constraints_number()

Retrieve the number of inequality constraints.

Returns

The number of inequality constraints.

Return type

int

get_ineq_cstr_total_dim()

Retrieve the total dimension of the inequality constraints.

This dimension is the sum of all the outputs dimensions of all the inequality constraints.

Returns

The total dimension of the inequality constraints.

Return type

int

get_nonproc_constraints()

Retrieve the non-processed constraints.

Returns

The non-processed constraints.

Return type
get_nonproc_objective()

Retrieve the non-processed objective function.

Return type

gemseo.core.function.MDOFunction

get_objective_name()

Retrieve the name of the objective function.

Returns

The name of the objective function.

Return type

str

get_observable(name)

Retrieve an observable from its name.

Parameters

name (str) – The name of the observable.

Returns

The observable.

Raises

ValueError – If the observable cannot be found.

Return type

gemseo.core.function.MDOFunction

get_optimum()

Return the optimum solution within a given feasibility tolerances.

Returns

The optimum result, defined by:

• the value of the objective function,

• the value of the design variables,

• the indicator of feasibility of the optimal solution,

• the value of the constraints,

• the value of the gradients of the constraints.

Return type

Tuple[numpy.ndarray, numpy.ndarray, bool, Dict[str, numpy.ndarray], Dict[str, numpy.ndarray]]

static get_solution()[source]

Return theoretical optimal value of Rastrigin function.

Returns

design variables values of optimized values, function value at optimum

Return type

numpy array

get_violation_criteria(x_vect)

Compute a violation measure associated to an iteration.

For each constraint, when it is violated, add the absolute distance to zero, in L2 norm.

If 0, all constraints are satisfied

Parameters

x_vect (numpy.ndarray) – The vector of the design variables values.

Returns

The feasibility of the point and the violation measure.

Return type

Tuple[bool, float]

get_x0_normalized()

Return the current values of the design variables after normalization.

Returns

The current values of the design variables normalized between 0 and 1 from their lower and upper bounds.

Return type

numpy.ndarray

has_constraints()

Check if the problem has equality or inequality constraints.

Returns

True if the problem has equality or inequality constraints.

has_eq_constraints()

Check if the problem has equality constraints.

Returns

True if the problem has equality constraints.

Return type

bool

has_ineq_constraints()

Check if the problem has inequality constraints.

Returns

True if the problem has inequality constraints.

Return type

bool

has_nonlinear_constraints()

Check if the problem has non-linear constraints.

Returns

True if the problem has equality or inequality constraints.

Return type

bool

classmethod import_hdf(file_path, x_tolerance=0.0)

Import an optimization history from an HDF file.

Parameters
• file_path (str) – The file containing the optimization history.

• x_tolerance (float) – The tolerance on the design variables when reading the file.

Returns

Return type

gemseo.algos.opt_problem.OptimizationProblem

is_point_feasible(out_val, constraints=None)

Check if a point is feasible.

Note

If the value of a constraint is absent from this point, then this constraint will be considered satisfied.

Parameters
• out_val (Dict[str, numpy.ndarray]) – The values of the objective function, and eventually constraints.

• constraints (Optional[Iterable[gemseo.core.function.MDOFunction]]) – The constraints whose values are to be tested. If None, then take all constraints of the problem.

Returns

The feasibility of the point.

Return type

bool

property objective

The objective function.

preprocess_functions(normalize=True, use_database=True, round_ints=True)

Pre-process all the functions and eventually the gradien.

Required to wrap the objective function and constraints with the database and eventually the gradients by complex step or finite differences.

Parameters
• normalize (bool) – If True, then the functions are normalized.

• use_database (bool) – If True, then the functions are wrapped in the database.

• round_ints (bool) – If True, then round the integer variables.

Return type

None

static rastrigin(x_dv)[source]

This function computes the order n=2 Rastrigin function.

Parameters

x_dv – design variable vector of size 2

Returns

result of Rastrigin function evaluation

static rastrigin_jac(x_dv)[source]

This function computes the analytical gradient of 2nd order Rastrigin function.

Parameters

x_dv (numpy array) – design variable vector

Returns

analytical gradient vector of Rastrigin function

Return type

numpy array

static repr_constraint(func, ctype, value=None, positive=False)

Express a constraint as a string expression.

Parameters
• func (gemseo.core.function.MDOFunction) – The constraint function.

• ctype (str) – The type of the constraint. Either equality or inequality.

• value (Optional[float]) – The value for which the constraint is active. If None, this value is 0.

• positive (bool) – If True, then the inequality constraint is positive.

Returns

A string representation of the constraint.

Return type

str