RBFOpt

https://rbfopt.readthedocs.io/en/latest/
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RBFOpt is a Python library for black-box optimization of computationally expensive functions using...
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### Initialization point logging

Source: https://rbfopt.readthedocs.io/en/latest/_modules/rbfopt_algorithm.html

Calculates the gap and logs initialization points during the algorithm setup.

```python
        gap = ru.compute_gap(self.l_settings, self.fbest +
                             self.all_node_err_bounds[self.fbest_index, 1])
        self.current_step = self.first_step
        self.cyclecount += 1
        # Print the initialization points
        for (i, val) in enumerate(self.node_val):
            min_dist = ru.get_min_distance(self.node_pos[i],
                                           np.vstack((self.node_pos[:i],
                                                      self.node_pos[(i+1):])))
            self.update_log('Initialization', self.node_is_noisy[i], val, gap)
```

--------------------------------

### Genetic Algorithm Optimization Setup

Source: https://rbfopt.readthedocs.io/en/latest/_modules/rbfopt_aux_problems.html

Initializes parameters for a simple genetic algorithm used to minimize a subproblem.

```python
    assert(isinstance(var_lower, np.ndarray))
    assert(isinstance(var_upper, np.ndarray))
    assert(isinstance(integer_vars, np.ndarray))
    assert(len(var_lower) == n)
    assert(len(var_upper) == n)
    assert(isinstance(settings, RbfoptSettings))

    # Define parameters here, for now. Will move them to
    # rbfopt_settings later if it seems that the user should be able
    # to change their value.
    population_size = settings.ga_base_population_size + 20 * n//5
    mutation_rate = 0.1

    # Derived parameters. Since the best individual will always remain
    # and mutate, there is a -1 in the count for new individuals.
    num_surviving = population_size//4
    num_new = population_size - 2*num_surviving - 1
```

--------------------------------

### Integer variable model setup

Source: https://rbfopt.readthedocs.io/en/latest/_modules/rbfopt_degree1_models.html

Defines parameters and variables for integer variables within a Pyomo model.

```python
ni = len(integer_vars)
    
    # Number of integer variables
    model.ni = Param(initialize=ni)
    model.NI = RangeSet(0, model.ni-1)

    # Parameter: list of indices of integer variables
    int_var_param = {}
    for i in range(ni):
        int_var_param[i] = integer_vars[i]
    model.integer_vars = Param(model.NI, initialize=int_var_param)

    # Variable: the point in the space
    model.y = Var(model.NI, domain=Integers, bounds=_y_bounds)

    # Constraints: every integer variable is simply equal to one
    # of the existing variables
    model.IntConstraint = Constraint(model.NI, rule=_int_constraint_rule)
```

--------------------------------

### Initialize instance variables function

Source: https://rbfopt.readthedocs.io/en/latest/_modules/rbfopt_aux_problems.html

Initializes the variables of a problem instance to assist local search by starting at a feasible point.

```python
[docs]def initialize_instance_variables(settings, instance, start_point=None):
    """Initialize the variables of a problem instance.

    Initialize the x variables of a problem instance, and set the
    corresponding values for the vectors :math:`(u,\pi)`. This helps
    the local search by starting at a feasible point.

    Parameters
    ----------
    settings : :class:`rbfopt_settings.RbfoptSettings`
        Global and algorithmic settings.

    instance : pyomo.ConcreteModel
        A concrete instance of mathematical optimization model.

```

--------------------------------

### restart(pool=None, remaining_eval=None)

Source: https://rbfopt.readthedocs.io/en/latest/rbfopt_algorithm.html

Perform a complete restart of the optimization. Discards the current RBF model and selects new sample points to start the algorithm from scratch.

```APIDOC
## restart(pool=None, remaining_eval=None)

### Description
Perform a complete restart of the optimization. Discards the current RBF model and selects new sample points to start the algorithm from scratch. Previous point evaluations are ignored but still recorded.

### Method
Function

### Parameters
- **pool** (multiprocessing.Pool()) - Optional - A pool of workers to evaluate the initialization points in parallel. If None, parallel evaluation will not be performed.
- **remaining_eval** (list()) - Optional - A list to which any outstanding initialization point evaluation is appended. If None, these evaluations will be discarded.

### Raises
- **RuntimeError** - If the routine does not find enough good initialization points.
```

--------------------------------

### Settings constructor with all default parameters

Source: https://rbfopt.readthedocs.io/en/latest/_modules/rbfopt_settings.html

Defines the __init__ method that sets every configuration parameter with its default value, enabling full customization of the optimization settings.

```python
    def __init__(self,
                 max_iterations=1000,
                 max_evaluations=300,
                 max_noisy_evaluations=200,
                 max_cycles=1000,
                 max_clock_time=1.0e30,
                 num_cpus=1,
                 parallel_wakeup_time=0.1,
                 algorithm='MSRSM',
                 rbf='auto',
                 rbf_shape_parameter=0.1,
                 target_objval=-1.0e10,
                 eps_opt=1.0e-2,
                 eps_zero=1.0e-15,
                 eps_impr=1.0e-4,
                 eps_linear_dependence=1.0e-6,
                 min_dist=1.0e-5,
                 do_infstep=False,
                 do_local_search=True,
                 num_global_searches=5,
                 init_strategy='lhd_maximin',
                 init_include_midpoint=True,
                 init_sample_fraction=-1.0,
                 init_sample_increase_parallel=0.05,
                 max_random_init=50,
                 function_scaling='auto',
                 log_scaling_threshold=1.0e6,
                 domain_scaling='auto',
                 dynamism_clipping='auto',
                 dynamism_threshold=1.0e3,
                 local_search_threshold=0.25,
                 local_search_box_scaling=0.5,
                 max_stalled_iterations=100,
                 discarded_window_size=30,
                 max_fraction_discarded=0.5,
                 max_consecutive_restoration=15,
                 max_cross_validations=50,
                 max_noisy_restarts=2,
                 max_noisy_iterations=200,
                 targetval_clipping=True,
                 global_search_method='genetic',
                 ga_base_population_size=400,
                 ga_num_generations=20,
                 num_samples_aux_problems=1000,
                 modified_msrsm_score=True,
                 max_consecutive_refinement=5,
                 thresh_unlimited_refinement=0.9,
                 thresh_unlimited_refinement_stalled=0.9,
                 refinement_frequency=3,
                 ref_num_integer_candidates=10,
                 ref_acceptable_decrease_shrink=0.2,
                 ref_acceptable_decrease_enlarge=0.6,
                 ref_acceptable_decrease_move=0.1,
                 ref_min_radius=1.0e-3,
                 ref_init_radius_multiplier=2.0,
                 ref_min_grad_norm=1.0e-2,
```

--------------------------------

### RbfoptAlgorithm constructor

Source: https://rbfopt.readthedocs.io/en/latest/rbfopt_algorithm.html

Initializes the optimization algorithm with settings, a black box describing the problem, and optional initial points and values. The do_init_strategy parameter controls whether the algorithm generates an initial set of points.

```APIDOC
## RbfoptAlgorithm(settings, black_box, init_node_pos=None, init_node_val=None, do_init_strategy=True)

### Description
Creates an instance of the RbfoptAlgorithm class, which implements the main optimization algorithm. The algorithm can be warm-started from a given state.

### Parameters
- **settings** (rbfopt_settings.RbfoptSettings) - Required - Global and algorithmic settings.
- **black_box** (rbfopt_black_box.RbfoptBlackBox) - Required - An object derived from class RbfoptBlackBox that describes the problem.
- **init_node_pos** (2D numpy.ndarray[float] or None) - Optional - Coordinates of points at which the function value is known. These points will be added to the points generated by the algorithm. If None, all initial points will be generated by the algorithm.
- **init_node_val** (1D numpy.ndarray[float] or None) - Optional - Function values corresponding to the points given in init_node_pos. Should be None if the previous argument is None. If init_node_pos is not None but init_node_val is None, the points in init_node_pos will be evaluated in the initialization phase.
- **do_init_strategy** (bool) - Optional - Perform initialization strategy. If True, the algorithm will generate an initial set of points on top of any user-provided points. If False, we will rely on user-provided points only. Default True.

### Attributes
- **elapsed_time** (float) - Elapsed CPU time up to the point where state was last saved.
- **best_local_rbf** (string, float) - Best RBF type to construct model for local search, and the corresponding shape parameter.
- **best_global_rbf** (string, float) - Best RBF type to construct model for global search, and the corresponding shape parameter.
- **n** (int) - Dimension of the problem.
- **itercount** (int) - Iteration number.
- **evalcount** (int) - Total number of function evaluations in accurate mode.
- **noisy_evalcount** (int) - Total number of function evaluations in noisy mode.
- **current_step** (int) - Identifier of the current step within the cyclic optimization strategy counter.
- **cyclecount** (int) - Number of cycles.
- **num_cons_refinement** (int) - Current number of consecutive refinement steps.
- **num_stalled_iter** (int) - Number of consecutive iterations without improvement.
- **num_noisy_restarts** (int) - Number of restarts in noisy mode.
- **discarded_iters** (collections.deque) - Rolling window to keep track of discarded iterations.
- **do_init_strategy** (bool) - If True, perform initialization strategy on first cycle.
- **inf_step** (int) - Identifier of the InfStep.
- **local_search_step** (int) - Identifier of the LocalSearchStep.
- **refinement_step** (int) - Identifier of the Step.
- **cycle_length** (int) - Length of an optimization cycle.
- **restoration_step** (int) - Identifier of the RestorationStep.
- **first_step** (int) - Identifier of the first step of an optimization cycle.
- **two_phase_optimization** (bool) - Is the fast but noisy objective function available?
- **eval_mode** (string) - Evaluation mode for the objective function at a given stage. Can be either 'noisy' or 'accurate'.
- **node_pos** (2D numpy.ndarray[float]) - Coordinates of the interpolation nodes (i.e., the points where the objective function has already been evaluated). The coordinates may be in the transformed space. This matrix only includes points since the last restart.
- **node_val** (1D numpy.ndarray[float]) - Objective function value at the points in node_pos. This array only includes points since the last restart.
- **node_is_noisy** (1D numpy.ndarray[bool]) - For each interpolation node in node_pos, was it evaluated in 'noisy' mode?
- **node_err_bounds** (2D numpy.ndarray[float]) - The lower and upper variation of the function value for the nodes in node_pos. The variation is assumed 0 for nodes evaluated in accurate mode.
- **all_node_pos** (2D numpy.ndarray[float]) - Coordinates of the interpolation nodes. This matrix contains all evaluated points in the original space, and is persistent across restarts.
- **all_node_val** (1D numpy.ndarray[float]) - Objective function value at the points in all_node_pos.
```

--------------------------------

### Get Dimension

Source: https://rbfopt.readthedocs.io/en/latest/_modules/rbfopt_test_functions.html

Returns the total dimension of the function, calculated as the base dimension multiplied by the dimension multiplier.

```python
[docs]
    def get_dimension(self):
        return self._function.dimension * self.dimension_multiplier
```

--------------------------------

### Get most common element in a list

Source: https://rbfopt.readthedocs.io/en/latest/_modules/rbfopt_utils.html

Returns the most frequent element in a list, optionally excluding specific values.

```python
def get_most_common_element(array, to_exclude=[]):
    """Get most common element in a list.

    Parameters
    ----------
    array : List[any]
        The list whose most common element is sought.

    to_exclude : List[any]
        A list of elements to exclude from the count.
    
    Returns
    -------
    Any
        The most common element, None if all elements were excluded.
    """
    counter = collections.Counter(
        val for val in array if val not in to_exclude)
    if (counter.most_common(1)):
        return counter.most_common(1)[0][0]
    return None
```

--------------------------------

### print_init_line()

Source: https://rbfopt.readthedocs.io/en/latest/rbfopt_algorithm.html

Prints the first line of the output, with headers.

```APIDOC
## print_init_line()

### Description
Print first line of the output, with headers.

### Parameters
None

### Returns
None (prints to standard output).
```

--------------------------------

### RbfoptSettings.print

Source: https://rbfopt.readthedocs.io/en/latest/rbfopt_settings.html

Prints the value of all settings to the specified output stream on a single line.

```APIDOC
## RbfoptSettings.print

### Description
Prints the value of all settings to the specified output stream on a single line.

### Method
_instance method_

### Signature
`print(output_stream)`

### Parameters
- **output_stream** (file) - Required - The stream on which messages are printed.
```

--------------------------------

### RbfoptUserBlackBox

Source: https://rbfopt.readthedocs.io/en/latest/_modules/rbfopt_user_black_box.html

A class that implements the necessary methods to describe the black-box function to be minimized, and gets all the required data from the user.

```APIDOC
## RbfoptUserBlackBox

### Description
A class that implements the necessary methods to describe the black-box function to be minimized, and gets all the required data from the user.

### Constructor

#### Parameters
- **dimension** (int) - Required - Dimension of the problem.
- **var_lower** (1D numpy.ndarray[float]) - Required - Lower bounds of the decision variables.
- **var_upper** (1D numpy.ndarray[float]) - Required - Upper bounds of the decision variables.
- **var_type** (1D numpy.ndarray[char]) - Required - An array of length equal to dimension, specifying the type of each variable. Possible types are 'R' for real (continuous) variables, 'I' for integer (discrete, ordered) variables, 'C' for categorical (discrete, unordered) variables. Bounds for categorical variables are interpreted the same way as for integer variables, but categorical variables are handled differently by the optimization algorithm; e.g., a categorical variable with bounds [2, 4] can take the value 2, 3 or 4.
- **obj_funct** (Callable[1D numpy.ndarray[float]]) - Required - The function to optimize. Must take a numpy array as argument, and return a float.
- **obj_funct_noisy** (Callable[1D numpy.ndarray[float]] or None) - Optional - The noisy but fast version of the function to optimize. If given, it must take a numpy array as argument, and return a numpy array with three floats, in the following order: the approximate function value, its lower variation, and its upper variation, where lower <= 0 and upper >= 0 and the true function value is contained between value + lower and value + upper. If it is None, we assume that there is no fast version of the objective function.

### See Also
- :class:`rbfopt_black_box.BlackBox`
```

--------------------------------

### Get Variable Bounds and Types

Source: https://rbfopt.readthedocs.io/en/latest/_modules/rbfopt_test_functions.html

Methods to retrieve the lower bounds, upper bounds, and variable types for the test function.

```python
[docs]
    def get_var_lower(self):        
        return self.var_lower

[docs]
    def get_var_upper(self):
        return self.var_upper

[docs]
    def get_var_type(self):
        return self.var_type
```

--------------------------------

### rbfopt_utils.initialize_nodes(settings, var_lower, var_upper, integer_vars, categorical_info)

Source: https://rbfopt.readthedocs.io/en/latest/rbfopt_utils.html

Computes the initial sample points for the optimization algorithm based on the provided settings and variable bounds.

```APIDOC
## rbfopt_utils.initialize_nodes(settings, var_lower, var_upper, integer_vars, categorical_info)

### Description
Compute the initial sample points using the initialization strategy indicated in the algorithmic settings.

### Parameters
- **settings** (rbfopt_settings.RbfoptSettings) - Required - Global and algorithmic settings.
- **var_lower** (1D numpy.ndarray[float]) - Required - List of lower bounds of the variables.
- **var_upper** (1D numpy.ndarray[float]) - Required - List of upper bounds of the variables.
- **integer_vars** (1D numpy.ndarray[int]) - Required - A list containing the indices of the integrality constrained variables. If empty, all variables are assumed to be continuous.
- **categorical_info** ((1D numpy.ndarray[int], 1D numpy.ndarray[int], List[(int, 1D numpy.ndarray[int])]) or None) - Required - Information on categorical variables: array of indices of categorical variables in original space, array of indices of noncategorical variables in original space, and expansion of each categorical variable, given as a tuple (original index, indices of expanded variables).

### Returns
- 2D numpy.ndarray[float] - Matrix containing at least n+1 corner points, one for each row, where n is the dimension of the space. The number and position of points depends on the chosen strategy.

### Raises
- RuntimeError - If a set of feasible and linearly independent sample points cannot be computed within the prescribed number of iterations.
```

--------------------------------

### Initialize environment settings

Source: https://rbfopt.readthedocs.io/en/latest/_modules/rbfopt_utils.html

Sets the random seed and suppresses Pyomo logging output based on the provided settings object.

```python
def init_environment(settings):
    """Initialize the random seed and disable Pyomo output.

    Parameters
    ----------
    settings : :class:`rbfopt_settings.RbfoptSettings`
        Global and algorithmic settings.
    """
    assert(isinstance(settings, RbfoptSettings))
    # Numpy's random seed
    np.random.seed(settings.rand_seed)
    # Disable Pyomo's logger if necessary
    if (not settings.debug):
        logging.getLogger('pyomo.core').setLevel(logging.CRITICAL)
        logging.getLogger('pyomo.opt').setLevel(logging.CRITICAL)
        logging.getLogger('pyomo.solvers').setLevel(logging.CRITICAL)
```

--------------------------------

### TestBlackBox

Source: https://rbfopt.readthedocs.io/en/latest/_modules/rbfopt_test_functions.html

A black-box constructed from a known test function. It provides methods to get dimension, bounds, variable types, and evaluate the function.

```APIDOC
## TestBlackBox

### Description
A black-box constructed from a known test function. It provides methods to get dimension, bounds, variable types, and evaluate the function.

### Constructor

#### `__init__(self, name)`
- **name** (string) - Required - The name of the function to be implemented.

### Methods

#### `get_dimension(self)`
Returns the dimension of the black-box function.

#### `get_var_lower(self)`
Returns the lower bounds of the variables.

#### `get_var_upper(self)`
Returns the upper bounds of the variables.

#### `get_var_type(self)`
Returns the variable types (e.g., 'R' for real, 'I' for integer, 'C' for categorical).

#### `evaluate(self, point)`
Evaluates the black-box function at the given point.

#### `evaluate_noisy(self, point)`
Raises `NotImplementedError` because noisy evaluation is not implemented.

#### `has_evaluate_noisy(self)`
Returns `False` indicating noisy evaluation is not available.

```

--------------------------------

### Get index of ready async computation

Source: https://rbfopt.readthedocs.io/en/latest/_modules/rbfopt_utils.html

Returns the index of the last completed asynchronous computation in a list, or the list length if none are ready.

```python
def get_one_ready_index(results):
    """Get index of a single async computation result that is ready.

    Given a list containing results of asynchronous computations
    dispatched to a worker pool, obtain the index of the last
    computation that has concluded. (Last is better to use the pop()
    function in the list.)

    Parameters
    ----------
    results : List[(multiprocessing.pool.AsyncResult, any)]
        A list of tasks, where each task is a list and the first
        element is the output of a call to apply_async. The other
        elements of the list will never be scanned by this function,
        and they could be anything.

    Returns
    -------
    int
        Index of last computation that completed, or len(results) if
        no computation is ready.

    """
    for i in reversed(range(len(results))):
        if results[i][0].ready():
            return i
    return len(results)
```

--------------------------------

### initialize_nodes

Source: https://rbfopt.readthedocs.io/en/latest/_modules/rbfopt_utils.html

Computes the initial sample points using the initialization strategy indicated in the algorithmic settings.

```APIDOC
## initialize_nodes

### Description
Computes the initial sample points using the initialization strategy indicated in the algorithmic settings.

### Parameters
- **settings** (RbfoptSettings) - Required - Global and algorithmic settings.
- **var_lower** (1D numpy.ndarray[float]) - Required - List of lower bounds of the variables.
- **var_upper** (1D numpy.ndarray[float]) - Required - List of upper bounds of the variables.
- **integer_vars** (1D numpy.ndarray[int]) - Required - A List containing the indices of the integrality constrained variables. If empty, all variables are assumed to be continuous.
- **categorical_info** (tuple or None) - Required - Information on categorical variables: array of indices of categorical variables in original space, array of indices of noncategorical variables in original space, and expansion of each categorical variable, given as a tuple (original index, indices of expanded variables).

### Returns
- **2D numpy.ndarray[float]** - Matrix containing at least n+1 corner points, one for each row, where n is the dimension of the space. The number and position of points depends on the chosen strategy.

### Raises
- **RuntimeError** - If a set of feasible and linearly independent sample points cannot be computed within the prescribed number of iterations.

```

--------------------------------

### Algorithm Parameter Assertions and Initialization

Source: https://rbfopt.readthedocs.io/en/latest/_modules/rbfopt_algorithm.html

Validates input dimensions and initialization strategies, then sets up RBF model references and environment settings.

```python
assert(len(var_lower) == dimension)
        assert(len(var_upper) == dimension)
        assert((len(integer_vars) == 0) or (max(integer_vars) < dimension))
        assert(init_node_pos is None or init_node_val is None or
               len(init_node_pos) == len(init_node_val))
        assert(do_init_strategy or
               (init_node_pos is not None and len(init_node_pos) >=
                round((dimension + 1)*settings.init_sample_fraction)))

        # Set the value of 'auto' parameters if necessary
        l_settings = settings.set_auto_parameters(dimension, var_lower,
                                                  var_upper, integer_vars)
        # Save references
        self.l_settings = l_settings

        # Local and global RBF models are initially the same
        self.best_local_rbf = (l_settings.rbf, l_settings.rbf_shape_parameter)
        self.best_global_rbf = (l_settings.rbf, l_settings.rbf_shape_parameter)
        self.best_local_rbf_list = list()
        self.best_global_rbf_list = list()
    
        # We use n to denote the dimension of the problem, same notation
        # of the paper. This is redundant but it simplifies our life.
        self.n = dimension

        # Set random seed and external loggers. We always use numpy's
        # random generator. This is taken care of by an appropriate
        # function (useful for parallelization).
        ru.init_environment(l_settings)
```

--------------------------------

### get_integer_candidate(settings, n, k, h, start_point, ref_radius, candidate, integer_vars, categorical_info)

Source: https://rbfopt.readthedocs.io/en/latest/rbfopt_refinement.html

Gets an integer candidate point from a fractional point.

```APIDOC
## get_integer_candidate(settings, n, k, h, start_point, ref_radius, candidate, integer_vars, categorical_info)

### Description
Gets an integer candidate point from a fractional point. It looks for integer points around the given fractional point, trying to find one with a good value of the linear model.

### Parameters
- **settings** (rbfopt_settings.RbfoptSettings) - Required - Global and algorithmic settings.
- **n** (int) - Required - Dimension of the problem.
- **k** (int) - Required - Number of interpolation nodes.
- **h** (1D numpy.ndarray[float]) - Required - Linear coefficients of the model.
- **start_point** (1D numpy.ndarray[float]) - Required - Starting point for the descent.
- **ref_radius** (float) - Required - Radius of the local search.
- **candidate** (1D numpy.ndarray[float]) - Required - Fractional point to begin the search.
- **integer_vars** (1D numpy.ndarray[int]) - Required - Indices of the integer variables.
- **categorical_info** ((1D numpy.ndarray[int], 1D numpy.ndarray[int], List[(int, 1D numpy.ndarray[int])]) or None) - Required - Information on categorical variables: array of indices of categorical variables in original space, array of indices of noncategorical variables in original space, and expansion of each categorical variable, given as a tuple (original index, indices of expanded variables).

### Returns
- (1D numpy.ndarray[float], float) - Next candidate point for the search, and the corresponding change in model value compared to the given point.
```

--------------------------------

### get_num_init_samples

Source: https://rbfopt.readthedocs.io/en/latest/_modules/rbfopt_utils.html

Computes the number of initial sample points based on problem dimension and settings.

```APIDOC
## get_num_init_samples

### Description
Computes the number of initial sample points based on problem dimension and settings.

### Parameters
- **settings** (RbfoptSettings) - Required - Global and algorithmic settings.
- **n** (int) - Required - Dimension of the problem, i.e. number of variables.

### Returns
- **int** - Number of samples, depending on problem dimension and settings.

```

--------------------------------

### global_step(settings, n, k, var_lower, var_upper, integer_vars, categorical_info, node_pos, rbf_lambda, rbf_h, tfv, Amatinv, fmin_index, current_step)

Source: https://rbfopt.readthedocs.io/en/latest/rbfopt_algorithm.html

Perform a global search step, with a different methodology depending on the algorithm chosen.

```APIDOC
## global_step(settings, n, k, var_lower, var_upper, integer_vars, categorical_info, node_pos, rbf_lambda, rbf_h, tfv, Amatinv, fmin_index, current_step)

### Description
Perform a global search step, with a different methodology depending on the algorithm chosen.

### Method
Function

### Parameters
- **settings** (rbfopt_settings.RbfoptSettings) - Required - Global and algorithmic settings.
- **n** (int) - Required - The dimension of the problem, i.e. size of the space.
- **k** (int) - Required - Number of nodes, i.e. interpolation points.
- **var_lower** (1D numpy.ndarray[float]) - Required - Vector of variable lower bounds.
- **var_upper** (1D numpy.ndarray[float]) - Required - Vector of variable upper bounds.
- **integer_vars** (1D numpy.ndarray[int]) - Required - A list containing the indices of the integrality constrained variables. If empty list, all variables are assumed to be continuous.
- **categorical_info** ((1D numpy.ndarray[int], 1D numpy.ndarray[int], List[(int, 1D numpy.ndarray[int])])) - Required - Information on categorical variables: array of indices of categorical variables in original space, array of indices of noncategorical variables in original space, and expansion of each categorical variable, given as a tuple (original index, indices of expanded variables).
- **node_pos** (2D numpy.ndarray[float]) - Required - List of coordinates of the nodes.
- **rbf_lambda** (1D numpy.ndarray[float]) - Required - The lambda coefficients of the RBF interpolant, corresponding to the radial basis functions. List of dimension k.
- **rbf_h** (1D numpy.ndarray[float]) - Required - The h coefficients of the RBF interpolant, corresponding to the polynomial. List of dimension n+1.
- **tfv** ((List[float], float, float, List[(float, float)])) - Required - Transformed function values: scaled node values, scaled minimum, scaled maximum, and node error bounds.
- **Amatinv** (2D numpy.ndarray[float]) - Required - Inverse of the RBF matrix.
- **fmin_index** (int) - Required - Index of the node with the minimum function value.
- **current_step** (int) - Required - Current step number.
```

--------------------------------

### Compute RBF Interpolant

Source: https://rbfopt.readthedocs.io/en/latest/_modules/rbfopt_utils.html

Example of computing the RBF interpolant by transforming function values and selecting the appropriate coefficient calculation method based on the number of nodes.

```python
tfv = transform_function_values(settings, node_val, fmin, fmax,
                                node_err_bounds)
(scaled_node_val, scaled_fmin, scaled_fmax, 
 scaled_err_bounds, rescale_function) = tfv

Amat = get_rbf_matrix(settings, n, k, node_pos)

# Compute RBF interpolant at current stage
if (k < n+1):
    # Underdetermined RBF
    rbf_l, rbf_h = get_rbf_coefficients_underdet(
        settings, n, k, Amat, scaled_node_val,
        categorical_info)
else:
    # Fully accurate RBF
    rbf_l, rbf_h = get_rbf_coefficients(
        settings, n, k, Amat, scaled_node_val,
        categorical_info)
return bulk_evaluate_rbf(settings, points, n, k, node_pos,
                         rbf_l, rbf_h)
```

--------------------------------

### Initialize MSRSM auxiliary variables

Source: https://rbfopt.readthedocs.io/en/latest/_modules/rbfopt_aux_problems.html

Initializes the rbfval and mindist variables of a problem instance using existing x and u_pi values to facilitate local search starting at a feasible point.

```python
def initialize_msrsm_aux_variables(settings, instance):
    """Initialize auxiliary variables for the MSRSM model.

    Initialize the rbfval and mindist variables of a problem
    instance, using the values for for x and u_pi already given. This
    helps the local search by starting at a feasible point.

    Parameters
    ----------
    settings : :class:`rbfopt_settings.RbfoptSettings`
        Global and algorithmic settings.

    instance : pyomo.ConcreteModel
        A concrete instance of mathematical optimization model.
    """
    assert(isinstance(settings, RbfoptSettings))

    dist = min(sum((instance.x[j].value - instance.node[i, j])**2
                   for j in instance.N) for i in instance.K)
    instance.mindistsq = dist
```

--------------------------------

### Initialize and Solve RBF Coefficients

Source: https://rbfopt.readthedocs.io/en/latest/_modules/rbfopt_aux_problems.html

Initializes RBF coefficients from provided values to ensure feasibility, then solves the optimization problem and loads the results.

```python
    # Initialize instance variables with the solution provided (if
    # available). This should avoid any infeasibility.
    if (init_rbf_lambda is not None and init_rbf_h is not None):
        for i in range(len(init_rbf_lambda)):
            instance.rbf_lambda[i] = init_rbf_lambda[i]
        for i in range(len(init_rbf_h)):
            instance.rbf_h[i] = init_rbf_h[i]

    # Solve and load results
    try:
        results = opt.solve(instance, keepfiles=False,
                            tee=settings.print_solver_output)
        if ((results.solver.status == pyomo.opt.SolverStatus.ok) and
            (results.solver.termination_condition ==
             pyomo.opt.TerminationCondition.optimal)):
            # this is feasible and optimal
            instance.solutions.load_from(results)
            rbf_lambda = np.array([instance.rbf_lambda[i].value 
                                   for i in instance.K])
            if (ru.get_size_P_matrix(settings, n) > 0):
                rbf_h = np.array([instance.rbf_h[i].value for i in instance.P])
            else:
                rbf_h = np.array([])
        else:
            # If we have initialization information, return it. It is
            # a feasible solution. Otherwise, this will be None.
            rbf_lambda = init_rbf_lambda
            rbf_h = init_rbf_h
    except:
        # If we have initialization information, return it. It is
        # a feasible solution. Otherwise, this will be None.
        rbf_lambda = init_rbf_lambda
        rbf_h = init_rbf_h

    return (rbf_lambda, rbf_h)
```

--------------------------------

### Build Pyomo model for minimum-distance subproblem (rbfopt_degreem1_models)

Source: https://rbfopt.readthedocs.io/en/latest/_modules/rbfopt_degreem1_models.html

The first block builds a Pyomo ConcreteModel for the minimum-distance subproblem: it sets node coordinates, variable bounds, the decision variable x, the auxiliary mindistsq variable, the objective, and the distance constraint, then adds integrality and categorical constraints if needed. The second block contains the assertion checks and the initial model setup (dimension and node count parameters) that precede the model construction. Both blocks are part of the same function body in the source, so they are grouped as one snippet.

```python
    # Coordinates of the nodes
    node_param = {}
    for i in range(k):
        for j in range(n):
            node_param[i, j] = float(node_pos[i][j])
    model.node = Param(model.K, model.N, initialize=node_param)

    # Variable bounds
    var_lower_param = {}
    var_upper_param = {}
    for i in range(n):
        var_lower_param[i] = float(var_lower[i])
        var_upper_param[i] = float(var_upper[i])
    model.var_lower = Param(model.N, initialize=var_lower_param)
    model.var_upper = Param(model.N, initialize=var_upper_param)

    # Variable: the point in the space
    model.x = Var(model.N, domain=Reals, bounds=_x_bounds)

    # Auxiliary variable: value of the minimum distance to the nodes
    model.mindistsq = Var(domain=NonNegativeReals,
                          bounds=(settings.min_dist,float('inf')))

    # Objective function.
    model.OBJ = Objective(expr=(model.mindistsq), sense=maximize)
    model.MdistdefConstraint = Constraint(model.K,
                                          rule=_mdistdef_constraint_rule)

    # Add integer variables if necessary
    if (len(integer_vars)):
        add_integrality_constraints(model, integer_vars)

    # Add categorical variables if necessary
    if (categorical_info is not None and categorical_info[2]):
        add_categorical_constraints(model, *categorical_info)

    return model


# -- end function
```

```python
    assert(isinstance(var_lower, np.ndarray))
    assert(isinstance(var_upper, np.ndarray))
    assert(isinstance(integer_vars, np.ndarray))
    assert(isinstance(node_pos, np.ndarray))
    assert(isinstance(rbf_lambda, np.ndarray))
    assert(isinstance(rbf_h, np.ndarray))
    assert(len(var_lower) == n)
    assert(len(var_upper) == n)
    assert(len(rbf_lambda) == k)
    assert(len(rbf_h) == 0)
    assert(len(node_pos) == k)
    assert(isinstance(settings, RbfoptSettings))
    assert(ru.get_degree_polynomial(settings) == -1)
    assert(0 <= dist_weight <= 1)
    assert(dist_max >= dist_min >= 0)

    model = ConcreteModel()

    # Dimension of the space
    model.n = Param(initialize=n)
    model.N = RangeSet(0, model.n - 1)
        
    # Number of interpolation nodes
    model.k = Param(initialize=k)
    model.K = RangeSet(0, model.k - 1)
```

--------------------------------

### Define schoen_6_2_int configuration

Source: https://rbfopt.readthedocs.io/en/latest/_modules/rbfopt_test_functions.html

Sets up the data arrays, bounds, and variable types for the schoen_6_2_int function.

```python
z = np.array(
    [[0.298854, 0.181010, 0.984817, 0.125272, 0.548396, 0.894658],
     [0.800371, 0.817380, 0.398577, 0.652349, 0.250843, 0.130235],
     [0.268631, 0.929778, 0.640422, 0.462004, 0.492930, 0.434955],
     [0.257863, 0.729198, 0.210810, 0.364378, 0.228216, 0.947432],
     [0.767627, 0.592150, 0.103788, 0.696895, 0.472449, 0.244504],
     [0.369630, 0.110889, 0.072344, 0.515753, 0.068087, 0.103057],
     [0.425457, 0.807081, 0.491209, 0.449497, 0.065690, 0.592775],
     [0.544229, 0.619841, 0.704609, 0.573098, 0.044844, 0.305800],
     [0.164031, 0.722884, 0.670496, 0.517915, 0.176386, 0.921565],
     [0.153788, 0.703577, 0.899129, 0.406134, 0.941356, 0.538215],
     [0.984781, 0.510479, 0.573361, 0.884599, 0.399472, 0.712935],
     [0.488416, 0.403997, 0.888823, 0.048434, 0.265197, 0.478025],
     [0.047985, 0.280071, 0.709960, 0.278919, 0.035737, 0.037699],
     [0.656172, 0.498412, 0.458622, 0.982970, 0.041234, 0.921127],
     [0.590802, 0.359690, 0.396516, 0.338153, 0.320793, 0.847369],
     [0.649160, 0.846974, 0.451818, 0.064864, 0.818545, 0.955844],
     [0.583716, 0.669610, 0.463098, 0.492710, 0.989690, 0.002397],
     [0.097300, 0.112389, 0.128759, 0.182995, 0.262808, 0.701887],
     [0.487363, 0.892520, 0.269056, 0.116046, 0.905416, 0.808013],
     [0.908316, 0.023997, 0.670399, 0.985859, 0.178548, 0.450410],
     [0.230409, 0.381732, 0.613667, 0.697260, 0.016950, 0.736507],
     [0.132544, 0.526349, 0.650042, 0.084086, 0.979257, 0.771499],
     [0.872978, 0.008826, 0.587481, 0.624637, 0.623175, 0.939539],
     [0.447828, 0.836386, 0.223285, 0.422756, 0.344488, 0.555953],
     [0.546839, 0.153934, 0.953017, 0.640891, 0.666774, 0.647583],
     [0.762237, 0.608920, 0.401447, 0.056202, 0.203535, 0.890609],
     [0.655150, 0.444544, 0.495582, 0.247926, 0.155128, 0.188004],
     [0.481813, 0.387178, 0.597276, 0.634671, 0.285404, 0.714793],
     [0.976385, 0.018854, 0.262585, 0.640434, 0.086314, 0.669879],
     [0.120164, 0.882300, 0.057626, 0.695111, 0.735135, 0.004711],
     [0.414644, 0.715618, 0.642033, 0.770645, 0.407019, 0.502945],
     [0.257475, 0.620029, 0.840603, 0.638546, 0.636521, 0.883558],
     [0.788980, 0.374926, 0.448016, 0.081941, 0.225763, 0.944905],
     [0.661591, 0.178832, 0.790349, 0.141653, 0.424235, 0.571960],
     [0.546361, 0.624907, 0.190470, 0.412713, 0.124748, 0.662788],
     [0.226384, 0.065829, 0.960836, 0.767766, 0.089695, 0.441792],
     [0.303675, 0.370047, 0.973692, 0.830432, 0.424719, 0.173571],
     [0.548375, 0.823234, 0.334253, 0.078398, 0.097269, 0.195120],
     [0.646225, 0.100478, 0.723833, 0.891035, 0.386094, 0.360272],
     [0.362757, 0.114700, 0.731020, 0.783785, 0.250399, 0.244399],
     [0.904335, 0.869074, 0.479004, 0.525872, 0.359411, 0.338333],
     [0.563175, 0.245903, 0.694417, 0.833524, 0.205055, 0.132535],
     [0.401356, 0.920963, 0.401902, 0.120625, 0.765834, 0.381552],
     [0.769562, 0.279591, 0.567598, 0.017192, 0.697366, 0.813451],
     [0.738572, 0.984740, 0.007616, 0.005382, 0.592976, 0.771773],
     [0.683721, 0.824097, 0.731623, 0.936945, 0.182420, 0.393537],
     [0.375859, 0.541929, 0.974640, 0.377459, 0.754060, 0.019335],
     [0.4, 0.6, 0.1, 0.4, 0.637412, 0.204038],
     [0.5, 0.4, 0.4, 0.0, 0.198525, 0.074668],
     [0.7, 0.1, 0.3, 0.5, 0.143614, 0.961610]])
f = np.array(
    [672.2, 861.4, 520.9, 121.0, 11.5, 48.2, 702.4, 536.2,
     457.7, 801.3, 787.7, 768.6, 292.4, 960.0, 573.1, 303.7,
     283.3, 474.1, 216.9, 462.2, 853.6, 677.1, 464.6, 830.6,
     831.8, 109.6, 967.6, 122.9, 896.2, 490.2, 710.4, 81.1,
     802.9, 999.8, 945.5, 672.3, 712.9, 235.8, 266.5, 772.4,
     326.6, 585.5, 16.9, 135.9, 224.2, 382.1, 614.6, -1000,
     -1000, -1000])

dimension = 6
var_lower = np.array([0 for i in range(6)])
var_upper = np.array([10 for i in range(6)])
optimum_point = np.array([04., 06., 01., 04., 06.37412, 02.04038])
optimum_value = -1000
var_type = np.array(['I'] * 4 + ['R'] * 2)
```
Tokens:
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