gemseo_pymoo / problems / analytical

# chankong_haimes module¶

Chankong and Haimes multi-objective problem.

This module implements the Chankong and Haimes multi-objective problem:

\begin{split}\begin{aligned} \text{minimize the objective function } & f_1(x, y) = 2 + (x - 2)^2 + (y - 1)^2 \\ & f_2(x, y) = 9x - (y - 1)^2 \\ \text{with respect to the design variables }&x,\,y \\ \text{subject to the general constraints } & g_1(x, y) = x^2 + y^2 \leq 225.0\\ & g_2(x, y) = x - 3y + 10 \leq 0.0\\ \text{subject to the bound constraints } & -20.0 \leq x \leq 20.\\ & -20.0 \leq y \leq 20. \end{aligned}\end{split}

Chankong, V., & Haimes, Y. Y. (2008). Multiobjective decision making: theory and methodology. Courier Dover Publications.

class gemseo_pymoo.problems.analytical.chankong_haimes.ChankongHaimes(l_b=- 20.0, u_b=20.0, initial_guess=None)[source]

Chankong and Haimes optimization problem.

The constructor.

Initialize the ChankongHaimes OptimizationProblem by defining the DesignSpace and the objective and constraints functions.

Parameters
• l_b (float) –

The lower bound (common value to all variables).

By default it is set to -20.0.

• u_b (float) –

The upper bound (common value to all variables).

By default it is set to 20.0.

• initial_guess (ndarray | None) –

The initial guess for the optimal solution.

By default it is set to None.

Return type

None

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.

By default it is set to True.

• each_store (bool) –

If True, then callback at every call to Database.store().

By default it is set to False.

Return type

None

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

Parameters
• cstr_func (MDOFunction) – The constraint.

• value (float | None) –

The value for which the constraint is active. If None, this value is 0.

By default it is set to None.

• cstr_type (str | None) –

The type of the constraint. Either equality or inequality.

By default it is set to None.

• positive (bool) –

If True, then the inequality constraint is positive.

By default it is set to False.

Raises
Return type

None

Add an equality constraint to the optimization problem.

Parameters
• cstr_func (MDOFunction) – The constraint.

• value (float | None) –

The value for which the constraint is active. If None, this value is 0.

By default it is set to None.

Return type

None

Add an inequality constraint to the optimization problem.

Parameters
• cstr_func (MDOFunction) – The constraint.

• value (float | None) –

The value for which the constraint is active. If None, this value is 0.

By default it is set to None.

• positive (bool) –

If True, then the inequality constraint is positive.

By default it is set to False.

Return type

None

Add a function to be observed.

When the OptimizationProblem is executed, the observables are called following this sequence:

Parameters
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) – The index of the constraint in constraints.

• method (str | Callable[[Callable], Callable]) –

The aggregation method, e.g. "max", "KS" or "IKS".

By default it is set to max.

• groups (tuple[ndarray] | None) –

The groups for which to produce an output. If None, a single output constraint is produced.

By default it is set to None.

• **options (Any) – The options of the aggregation method.

Raises

ValueError – When the given is index is greater or equal than the number of constraints or when the method is aggregation unknown.

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 (Any) – 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

static compute_constraint_1(design_variables)[source]

Compute the first constraint function.

Parameters

design_variables (numpy.ndarray) – The design variables vector.

Returns

The first constraint’s value.

Return type

numpy.ndarray

static compute_constraint_1_jacobian(design_variables)[source]

Compute the first inequality constraint jacobian.

Parameters

design_variables (numpy.ndarray) – The design variables vector.

Returns

The gradient of the first constraint function wrt the design variables.

Return type

numpy.ndarray

static compute_constraint_2(design_variables)[source]

Compute the second constraint function.

Parameters

design_variables (numpy.ndarray) – The design variables vector.

Returns

The second constraint’s value.

Return type

numpy.ndarray

static compute_constraint_2_jacobian(design_variables)[source]

Compute the second inequality constraint jacobian.

Parameters

design_variables (numpy.ndarray) – The design variables vector.

Returns

The gradient of the second constraint function wrt the design variables.

Return type

numpy.ndarray

static compute_objective(design_variables)[source]

Compute the objectives of the Chankong and Haimes function.

Parameters

design_variables (numpy.ndarray) – The design variables vector.

Returns

The objective function value.

Return type

numpy.ndarray

static compute_objective_jacobian(design_variables)[source]

Compute the gradient of objective function.

Parameters

design_variables (numpy.ndarray) – The design variables vector.

Returns

The gradient of the objective functions wrt the design variables.

Return type

numpy.ndarray

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

Compute the functions of interest, and possibly their derivatives.

These functions of interest are the constraints, and possibly the objective.

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 (ndarray) –

The input vector at which the functions must be evaluated; if None, the initial point x_0 is used.

By default it is set to None.

• eval_jac (bool) –

Whether to compute the Jacobian matrices of the functions of interest.

By default it is set to False.

• eval_obj (bool) –

Whether to consider the objective function as a function of interest.

By default it is set to True.

• normalize (bool) –

Whether to consider the input vector x_vect normalized.

By default it is set to True.

• no_db_no_norm (bool) –

If True, then do not use the pre-processed functions, so we have no database, nor normalization.

By default it is set to False.

• eval_observables (bool) –

By default it is set to False.

Returns

The output values of the functions of interest, as well as their Jacobian matrices if eval_jac is True.

Return type

tuple[dict[str, float | ndarray], dict[str, ndarray]]

execute_observables_callback(last_x)

The callback function to be passed to the database.

Call all the observables with the last design variables values as argument.

Parameters

last_x (numpy.ndarray) – The design variables values from the last evaluation.

Return type

None

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.

By default it is set to False.

Return type

None

export_to_dataset(name=None, by_group=True, categorize=True, opt_naming=True, export_gradients=False, input_values=None)

Export the database of the optimization problem to a Dataset.

The variables can be classified into groups: Dataset.DESIGN_GROUP or Dataset.INPUT_GROUP for the design variables and Dataset.FUNCTION_GROUP or Dataset.OUTPUT_GROUP for the functions (objective, constraints and observables).

Parameters
Returns

A dataset built from the database of the optimization problem.

Return type

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.

By default it is set to 1e-06.

Returns

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

Return type
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_data_by_names(names, as_dict=True, filter_non_feasible=False)

Return the data for specific names of variables.

Parameters
• names (str | Iterable[str]) – The names of the variables.

• as_dict (bool) –

If True, return values as dictionary.

By default it is set to True.

• filter_non_feasible (bool) –

If True, remove the non-feasible points from the data.

By default it is set to False.

Returns

The data related to the variables.

Return type

ndarray | dict[str, ndarray]

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[ndarray], list[dict[str, float | list[int]]]]

get_function_dimension(name)

Return the dimension of a function of the problem (e.g. a constraint).

Parameters

name (str) – The name of the function.

Returns

The dimension of the function.

Raises
• ValueError – If the function name is unknown to the problem.

• RuntimeError – If the function dimension is not unavailable.

Return type

int

get_function_names(names)

Return the names of the functions stored in the database.

Parameters

names (Iterable[str]) – The names of the outputs or constraints specified by the user.

Returns

The names of the constraints stored in the database.

Return type

list[str]

get_functions_dimensions(names=None)

Return the dimensions of the outputs of the problem functions.

Parameters

names (Iterable[str] | None) –

The names of the functions. If None, then the objective and all the constraints are considered.

By default it is set to None.

Returns

The dimensions of the outputs of the problem functions. The dictionary keys are the functions names and the values are the functions dimensions.

Return type

dict[str, 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.mdofunctions.mdo_function.MDOFunction

get_number_of_unsatisfied_constraints(design_variables)

Return the number of scalar constraints not satisfied by design variables.

Parameters

design_variables (numpy.ndarray) – The design variables.

Returns

The number of unsatisfied scalar constraints.

Return type

int

get_objective_name(standardize=True)

Retrieve the name of the objective function.

Parameters

standardize (bool) –

Whether to use the name of the objective expressed as a cost, e.g. "-f" when the user seeks to maximize "f".

By default it is set to True.

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.mdofunctions.mdo_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]]

get_scalar_constraints_names()

Return the names of the scalar constraints.

Returns

The names of the scalar constraints.

Return type

list[str]

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
get_x0_normalized(cast_to_real=False)

Return the current values of the design variables after normalization.

Parameters

cast_to_real (bool) –

Whether to cast the return value to real.

By default it is set to False.

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.

By default it is set to 0.0.

Returns

Return type

gemseo.algos.opt_problem.OptimizationProblem

is_max_iter_reached()

Check if the maximum amount of iterations has been reached.

Returns

Whether the maximum amount of iterations has been reached.

Return type

bool

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, ndarray]) – The values of the objective function, and eventually constraints.

• constraints (Iterable[MDOFunction] | None) –

The constraints whose values are to be tested. If None, then take all constraints of the problem.

By default it is set to None.

Returns

The feasibility of the point.

Return type

bool

preprocess_functions(is_function_input_normalized=True, use_database=True, round_ints=True, eval_obs_jac=False)

Pre-process all the functions and eventually the gradient.

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

Parameters
• is_function_input_normalized (bool) –

Whether to consider the function input as normalized and unnormalize it before the evaluation takes place.

By default it is set to True.

• use_database (bool) –

Whether to wrap the functions in the database.

By default it is set to True.

• round_ints (bool) –

Whether to round the integer variables.

By default it is set to True.

• eval_obs_jac (bool) –

Whether to evaluate the Jacobian of the observables.

By default it is set to False.

Return type

None

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

Express a constraint as a string expression.

Parameters
• func (MDOFunction) – The constraint function.

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

• value (float | None) –

The value for which the constraint is active. If None, this value is 0.

By default it is set to None.

• positive (bool) –

If True, then the inequality constraint is positive.

By default it is set to False.

Returns

A string representation of the constraint.

Return type

str

reset(database=True, current_iter=True, design_space=True, function_calls=True, preprocessing=True)

Partially or fully reset the optimization problem.

Parameters
• database (bool) –

Whether to clear the database.

By default it is set to True.

• current_iter (bool) –

Whether to reset the current iteration OptimizationProblem.current_iter.

By default it is set to True.

• design_space (bool) –

Whether to reset the current point of the OptimizationProblem.design_space to its initial value (possibly none).

By default it is set to True.

• function_calls (bool) –

Whether to reset the number of calls of the functions.

By default it is set to True.

• preprocessing (bool) –

Whether to turn the pre-processing of functions to False.

By default it is set to True.

Return type

None

AVAILABLE_PB_TYPES: ClassVar[str] = ['linear', 'non-linear']
COMPLEX_STEP: Final[str] = 'complex_step'
CONSTRAINTS_GROUP: Final[str] = 'constraints'
DESIGN_SPACE_ATTRS: Final[str] = ['u_bounds', 'l_bounds', 'x_0', 'x_names', 'dimension']
DESIGN_SPACE_GROUP: Final[str] = 'design_space'
DESIGN_VAR_NAMES: Final[str] = 'x_names'
DESIGN_VAR_SIZE: Final[str] = 'x_size'
DIFFERENTIATION_METHODS: ClassVar[str] = ['user', 'complex_step', 'finite_differences', 'no_derivatives']
FINITE_DIFFERENCES: Final[str] = 'finite_differences'
FUNCTIONS_ATTRS: ClassVar[str] = ['objective', 'constraints']
GGOBI_FORMAT: Final[str] = 'ggobi'
HDF5_FORMAT: Final[str] = 'hdf5'
LINEAR_PB: Final[str] = 'linear'
NON_LINEAR_PB: Final[str] = 'non-linear'
NO_DERIVATIVES: Final[str] = 'no_derivatives'
OBJECTIVE_GROUP: Final[str] = 'objective'
OBSERVABLES_GROUP: Final[str] = 'observables'
OPTIM_DESCRIPTION: ClassVar[str] = ['minimize_objective', 'fd_step', 'differentiation_method', 'pb_type', 'ineq_tolerance', 'eq_tolerance']
OPT_DESCR_GROUP: Final[str] = 'opt_description'
SOLUTION_GROUP: Final[str] = 'solution'
activate_bound_check: ClassVar[bool] = True

Whether to check if a point is in the design space before calling functions.

constraint_names: dict[str, list[str]]

The standardized constraint names bound to the original ones.

constraints: list[MDOFunction]

The constraints.

database: Database

The database to store the optimization problem data.

design_space: DesignSpace

The design space on which the optimization problem is solved.

property differentiation_method: str

The differentiation method.

property dimension: int

The dimension of the design space.

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.

property is_mono_objective: bool

Whether the optimization problem is mono-objective.

minimize_objective: bool

Whether to maximize the objective.

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.

property objective: gemseo.core.mdofunctions.mdo_function.MDOFunction

The objective function.

observables: list[MDOFunction]

The observables.

property parallel_differentiation: bool

Whether to approximate the derivatives in parallel.

property parallel_differentiation_options: bool

The options to approximate the derivatives in parallel.

pb_type: str

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 DriverLib works with this standardized objective and the Database stores its values. However, for convenience, it may be more relevant to log the expression and the values of the original objective.