genetic_algorithm_driver.py

Driver for a simple genetic algorithm.

This is the Simple Genetic Algorithm implementation based on 2009 AAE550: MDO Lecture notes of Prof. William A. Crossley.

This basic GA algorithm is compartmentalized into the GeneticAlgorithm class so that it can be used in more complicated driver.

The following reference is only for the automatic population sizing: Williams E.A., Crossley W.A. (1998) Empirically-Derived Population Size and Mutation Rate Guidelines for a Genetic Algorithm with Uniform Crossover. In: Chawdhry P.K., Roy R., Pant R.K. (eds) Soft Computing in Engineering Design and Manufacturing. Springer, London.

The following reference is only for the penalty function: Smith, A. E., Coit, D. W. (1995) Penalty functions. In: Handbook of Evolutionary Computation, 97(1).

The following reference is only for weighted sum multi-objective optimization: Sobieszczanski-Sobieski, J., Morris, A. J., van Tooren, M. J. L. (2015) Multidisciplinary Design Optimization Supported by Knowledge Based Engineering. John Wiley & Sons, Ltd.

class openmdao.drivers.genetic_algorithm_driver.GeneticAlgorithm(objfun, comm=None, model_mpi=None)[source]

Bases: object

Simple Genetic Algorithm.

This is the Simple Genetic Algorithm implementation based on 2009 AAE550: MDO Lecture notes of Prof. William A. Crossley. It can be used standalone or as part of the OpenMDAO Driver.

Attributes

comm (MPI communicator or None) The MPI communicator that will be used objective evaluation for each generation.
elite (bool) Elitism flag.
lchrom (int) Chromosome length.
model_mpi (None or tuple) If the model in objfun is also parallel, then this will contain a tuple with the the total number of population points to evaluate concurrently, and the color of the point to evaluate on this rank.
npop (int) Population size.
objfun (function) Objective function callback.
__init__(objfun, comm=None, model_mpi=None)[source]

Initialize genetic algorithm object.

Parameters:
objfun : function

Objective callback function.

comm : MPI communicator or None

The MPI communicator that will be used objective evaluation for each generation.

model_mpi : None or tuple

If the model in objfun is also parallel, then this will contain a tuple with the the total number of population points to evaluate concurrently, and the color of the point to evaluate on this rank.

crossover(old_gen, Pc)[source]

Apply crossover to the current generation.

Crossover flips two adjacent genes.

Parameters:
old_gen : ndarray

Points in current generation

Pc : float

Probability of crossover.

Returns:
ndarray

Current generation with crossovers applied.

decode(gen, vlb, vub, bits)[source]

Decode from binary array to real value array.

Parameters:
gen : ndarray

Population of points, encoded.

vlb : ndarray

Lower bound array.

vub : ndarray

Upper bound array.

bits : ndarray

Number of bits for decoding.

Returns:
ndarray

Decoded design variable values.

encode(x, vlb, vub, bits)[source]

Encode array of real values to array of binary arrays.

Parameters:
x : ndarray

Design variable values.

vlb : ndarray

Lower bound array.

vub : ndarray

Upper bound array.

bits : int

Number of bits for decoding.

Returns:
ndarray

Population of points, encoded.

execute_ga(x0, vlb, vub, vob, bits, pop_size, max_gen, random_state, Pm=None, Pc=0.5)[source]

Perform the genetic algorithm.

Parameters:
x0 : ndarray

Initial design values

vlb : ndarray

Lower bounds array.

vub : ndarray

Upper bounds array. This includes over-allocation so that every point falls on an integer value.

vob : ndarray

Outer bounds array. This is purely for bounds check.

bits : ndarray

Number of bits to encode the design space for each element of the design vector.

pop_size : int

Number of points in the population.

max_gen : int

Number of generations to run the GA.

random_state : np.random.RandomState, int

Random state (or seed-number) which controls the seed and random draws.

Pm : float or None

Mutation rate

Pc : float

Crossover rate

Returns:
ndarray

Best design point

float

Objective value at best design point.

int

Number of successful function evaluations.

mutate(current_gen, Pm)[source]

Apply mutations to the current generation.

A mutation flips the state of the gene from 0 to 1 or 1 to 0.

Parameters:
current_gen : ndarray

Points in current generation

Pm : float

Probability of mutation.

Returns:
ndarray

Current generation with mutations applied.

shuffle(old_gen)[source]

Shuffle (reorder) the points in the population.

Used in tournament selection.

Parameters:
old_gen : ndarray

Old population.

Returns:
ndarray

New shuffled population.

ndarray(dtype=np.int)

Index array that maps the shuffle from old to new.

tournament(old_gen, fitness)[source]

Apply tournament selection and keep the best points.

Parameters:
old_gen : ndarray

Points in current generation

fitness : ndarray

Objective value of each point.

Returns:
ndarray

New generation with best points.

class openmdao.drivers.genetic_algorithm_driver.SimpleGADriver(**kwargs)[source]

Bases: openmdao.core.driver.Driver

Driver for a simple genetic algorithm.

__init__(**kwargs)[source]

Initialize the SimpleGADriver driver.

Parameters:
**kwargs : dict of keyword arguments

Keyword arguments that will be mapped into the Driver options.

add_recorder(recorder)

Add a recorder to the driver.

Parameters:
recorder : CaseRecorder

A recorder instance.

cleanup()

Clean up resources prior to exit.

get_constraint_values(ctype='all', lintype='all', unscaled=False, filter=None, ignore_indices=False)

Return constraint values.

Parameters:
ctype : string

Default is ‘all’. Optionally return just the inequality constraints with ‘ineq’ or the equality constraints with ‘eq’.

lintype : string

Default is ‘all’. Optionally return just the linear constraints with ‘linear’ or the nonlinear constraints with ‘nonlinear’.

unscaled : bool

Set to True if unscaled (physical) design variables are desired.

filter : list

List of constraint names used by recorders.

ignore_indices : bool

Set to True if the full array is desired, not just those indicated by indices.

Returns:
dict

Dictionary containing values of each constraint.

get_design_var_values(filter=None, unscaled=False, ignore_indices=False)

Return the design variable values.

This is called to gather the initial design variable state.

Parameters:
filter : list

List of desvar names used by recorders.

unscaled : bool

Set to True if unscaled (physical) design variables are desired.

ignore_indices : bool

Set to True if the full array is desired, not just those indicated by indices.

Returns:
dict

Dictionary containing values of each design variable.

get_objective_values(unscaled=False, filter=None, ignore_indices=False)

Return objective values.

Parameters:
unscaled : bool

Set to True if unscaled (physical) design variables are desired.

filter : list

List of objective names used by recorders.

ignore_indices : bool

Set to True if the full array is desired, not just those indicated by indices.

Returns:
dict

Dictionary containing values of each objective.

get_response_values(filter=None)

Return response values.

Parameters:
filter : list

List of response names used by recorders.

Returns:
dict

Dictionary containing values of each response.

objective_callback(x, icase)[source]

Evaluate problem objective at the requested point.

In case of multi-objective optimization, a simple weighted sum method is used:

\[f = (\sum_{k=1}^{N_f} w_k \cdot f_k)^a\]

where \(N_f\) is the number of objectives and \(a>0\) is an exponential weight. Choosing \(a=1\) is equivalent to the conventional weighted sum method.

The weights given in the options are normalized, so:

\[\sum_{k=1}^{N_f} w_k = 1\]

If one of the objectives \(f_k\) is not a scalar, its elements will have the same weights, and it will be normed with length of the vector.

Takes into account constraints with a penalty function.

All constraints are converted to the form of \(g_i(x) \leq 0\) for inequality constraints and \(h_i(x) = 0\) for equality constraints. The constraint vector for inequality constraints is the following:

\[ \begin{align}\begin{aligned}g = [g_1, g_2 \dots g_N], g_i \in R^{N_{g_i}}\\h = [h_1, h_2 \dots h_N], h_i \in R^{N_{h_i}}\end{aligned}\end{align} \]

The number of all constraints:

\[N_g = \sum_{i=1}^N N_{g_i}, N_h = \sum_{i=1}^N N_{h_i}\]

The fitness function is constructed with the penalty parameter \(p\) and the exponent \(\kappa\):

\[\Phi(x) = f(x) + p \cdot \sum_{k=1}^{N^g}(\delta_k \cdot g_k)^{\kappa} + p \cdot \sum_{k=1}^{N^h}|h_k|^{\kappa}\]

where \(\delta_k = 0\) if \(g_k\) is satisfied, 1 otherwise

Note

The values of \(\kappa\) and \(p\) can be defined as driver options.

Parameters:
x : ndarray

Value of design variables.

icase : int

Case number, used for identification when run in parallel.

Returns:
float

Objective value

bool

Success flag, True if successful

int

Case number, used for identification when run in parallel.

record_iteration()

Record an iteration of the current Driver.

run()[source]

Execute the genetic algorithm.

Returns:
boolean

Failure flag; True if failed to converge, False is successful.

set_design_var(name, value)

Set the value of a design variable.

Parameters:
name : str

Global pathname of the design variable.

value : float or ndarray

Value for the design variable.

set_simul_deriv_color(simul_info)

Set the coloring (and possibly the sub-jac sparsity) for simultaneous total derivatives.

Parameters:
simul_info : str or dict
# Information about simultaneous coloring for design vars and responses.  If a
# string, then simul_info is assumed to be the name of a file that contains the
# coloring information in JSON format.  If a dict, the structure looks like this:

{
"fwd": [
    # First, a list of column index lists, each index list representing columns
    # having the same color, except for the very first index list, which contains
    # indices of all columns that are not colored.
    [
        [i1, i2, i3, ...]    # list of non-colored columns
        [ia, ib, ...]    # list of columns in first color
        [ic, id, ...]    # list of columns in second color
           ...           # remaining color lists, one list of columns per color
    ],

    # Next is a list of lists, one for each column, containing the nonzero rows for
    # that column.  If a column is not colored, then it will have a None entry
    # instead of a list.
    [
        [r1, rn, ...]   # list of nonzero rows for column 0
        None,           # column 1 is not colored
        [ra, rb, ...]   # list of nonzero rows for column 2
            ...
    ],
],
# This example is not a bidirectional coloring, so the opposite direction, "rev"
# in this case, has an empty row index list.  It could also be removed entirely.
"rev": [[[]], []],
"sparsity":
    # The sparsity entry can be absent, indicating that no sparsity structure is
    # specified, or it can be a nested dictionary where the outer keys are response
    # names, the inner keys are design variable names, and the value is a tuple of
    # the form (row_list, col_list, shape).
    {
        resp1_name: {
            dv1_name: (rows, cols, shape),  # for sub-jac d_resp1/d_dv1
            dv2_name: (rows, cols, shape),
              ...
        },
        resp2_name: {
            ...
        }
        ...
    }
}
set_total_jac_sparsity(sparsity)

Set the sparsity of sub-jacobians of the total jacobian.

Note: This currently will have no effect if you are not using the pyOptSparseDriver.

Parameters:
sparsity : str or dict
# Sparsity is a nested dictionary where the outer keys are response
# names, the inner keys are design variable names, and the value is a tuple of
# the form (row_list, col_list, shape).
{
    resp1: {
        dv1: (rows, cols, shape),  # for sub-jac d_resp1/d_dv1
        dv2: (rows, cols, shape),
          ...
    },
    resp2: {
        ...
    }
    ...
}