scipy.optimize.

differential_evolution

scipy.optimize.differential_evolution(func, bounds, args=(), strategy='best1bin', maxiter=1000, popsize=15, tol=0.01, mutation=(0.5, 1), recombination=0.7, rng=None, callback=None, disp=False, polish=True, init='latinhypercube', atol=0, updating='immediate', workers=1, constraints=(), x0=None, *, integrality=None, vectorized=False)

Finds the global minimum of a multivariate function.

The differential evolution method [1] is stochastic in nature. It does not use gradient methods to find the minimum, and can search large areas of candidate space, but often requires larger numbers of function evaluations than conventional gradient-based techniques.

The algorithm is due to Storn and Price [2].

Parameters:

funccallable

The objective function to be minimized. Must be in the form f(x, *args), where x is the argument in the form of a 1-D array and args is a tuple of any additional fixed parameters needed to completely specify the function. The number of parameters, N, is equal to len(x).

boundssequence or Bounds

Bounds for variables. There are two ways to specify the bounds:

1. Instance of Bounds class.

2. (min, max) pairs for each element in x, defining the finite lower and upper bounds for the optimizing argument of func.

The total number of bounds is used to determine the number of parameters, N. If there are parameters whose bounds are equal the total number of free parameters is N - N_equal.

argstuple, optional

Any additional fixed parameters needed to completely specify the objective function.

strategy{str, callable}, optional

The differential evolution strategy to use. Should be one of:

• ‘best1bin’

• ‘best1exp’

• ‘rand1bin’

• ‘rand1exp’

• ‘rand2bin’

• ‘rand2exp’

• ‘randtobest1bin’

• ‘randtobest1exp’

• ‘currenttobest1bin’

• ‘currenttobest1exp’

• ‘best2exp’

• ‘best2bin’

The default is ‘best1bin’. Strategies that may be implemented are outlined in ‘Notes’. Alternatively the differential evolution strategy can be customized by providing a callable that constructs a trial vector. The callable must have the form strategy(candidate: int, population: np.ndarray, rng=None), where candidate is an integer specifying which entry of the population is being evolved, population is an array of shape (S, N) containing all the population members (where S is the total population size), and rng is the random number generator being used within the solver. candidate will be in the range [0, S). strategy must return a trial vector with shape (N,). The fitness of this trial vector is compared against the fitness of population[candidate].

Changed in version 1.12.0: Customization of evolution strategy via a callable.

maxiterint, optional

The maximum number of generations over which the entire population is evolved. The maximum number of function evaluations (with no polishing) is: (maxiter + 1) * popsize * (N - N_equal)

popsizeint, optional

A multiplier for setting the total population size. The population has popsize * (N - N_equal) individuals. This keyword is overridden if an initial population is supplied via the init keyword. When using init='sobol' the population size is calculated as the next power of 2 after popsize * (N - N_equal).

tolfloat, optional

Relative tolerance for convergence, the solving stops when np.std(population_energies) <= atol + tol * np.abs(np.mean(population_energies)), where and atol and tol are the absolute and relative tolerance respectively.

mutationfloat or tuple(float, float), optional

The mutation constant. In the literature this is also known as differential weight, being denoted by \(F\). If specified as a float it should be in the range [0, 2). If specified as a tuple (min, max) dithering is employed. Dithering randomly changes the mutation constant on a generation by generation basis. The mutation constant for that generation is taken from U[min, max). Dithering can help speed convergence significantly. Increasing the mutation constant increases the search radius, but will slow down convergence.

recombinationfloat, optional

The recombination constant, should be in the range [0, 1]. In the literature this is also known as the crossover probability, being denoted by CR. Increasing this value allows a larger number of mutants to progress into the next generation, but at the risk of population stability.

rng{None, int, numpy.random.Generator}, optional

If rng is passed by keyword, types other than numpy.random.Generator are passed to numpy.random.default_rng to instantiate a Generator. If rng is already a Generator instance, then the provided instance is used. Specify rng for repeatable function behavior.

If this argument is passed by position or seed is passed by keyword, legacy behavior for the argument seed applies:

• If seed is None (or numpy.random), the numpy.random.RandomState singleton is used.

• If seed is an int, a new RandomState instance is used, seeded with seed.

• If seed is already a Generator or RandomState instance then that instance is used.

Changed in version 1.15.0: As part of the SPEC-007 transition from use of numpy.random.RandomState to numpy.random.Generator, this keyword was changed from seed to rng. For an interim period, both keywords will continue to work, although only one may be specified at a time. After the interim period, function calls using the seed keyword will emit warnings. The behavior of both seed and rng are outlined above, but only the rng keyword should be used in new code.

dispbool, optional

Prints the evaluated func at every iteration.

callbackcallable, optional

A callable called after each iteration. Has the signature:

callback(intermediate_result: OptimizeResult)

where intermediate_result is a keyword parameter containing an OptimizeResult with attributes x and fun, the best solution found so far and the objective function. Note that the name of the parameter must be intermediate_result for the callback to be passed an OptimizeResult.

The callback also supports a signature like:

callback(x, convergence: float=val)

val represents the fractional value of the population convergence. When val is greater than 1.0, the function halts.

Introspection is used to determine which of the signatures is invoked.

Global minimization will halt if the callback raises StopIteration or returns True; any polishing is still carried out.

Changed in version 1.12.0: callback accepts the intermediate_result keyword.

polishbool, optional

If True (default), then scipy.optimize.minimize with the L-BFGS-B method is used to polish the best population member at the end, which can improve the minimization slightly. If a constrained problem is being studied then the trust-constr method is used instead. For large problems with many constraints, polishing can take a long time due to the Jacobian computations.

Changed in version 1.15.0: If workers is specified then the map-like callable that wraps func is supplied to minimize instead of it using func directly. This allows the caller to control how and where the invocations actually run.

initstr or array-like, optional

Specify which type of population initialization is performed. Should be one of:

• ‘latinhypercube’

• ‘sobol’

• ‘halton’

• ‘random’

• array specifying the initial population. The array should have shape (S, N), where S is the total population size and N is the number of parameters.

init is clipped to bounds before use.

The default is ‘latinhypercube’. Latin Hypercube sampling tries to maximize coverage of the available parameter space.

‘sobol’ and ‘halton’ are superior alternatives and maximize even more the parameter space. ‘sobol’ will enforce an initial population size which is calculated as the next power of 2 after popsize * (N - N_equal). ‘halton’ has no requirements but is a bit less efficient. See scipy.stats.qmc for more details.

‘random’ initializes the population randomly - this has the drawback that clustering can occur, preventing the whole of parameter space being covered. Use of an array to specify a population could be used, for example, to create a tight bunch of initial guesses in an location where the solution is known to exist, thereby reducing time for convergence.

atolfloat, optional

Absolute tolerance for convergence, the solving stops when np.std(pop) <= atol + tol * np.abs(np.mean(population_energies)), where and atol and tol are the absolute and relative tolerance respectively.

updating{‘immediate’, ‘deferred’}, optional

If 'immediate', the best solution vector is continuously updated within a single generation [4]. This can lead to faster convergence as trial vectors can take advantage of continuous improvements in the best solution. With 'deferred', the best solution vector is updated once per generation. Only 'deferred' is compatible with parallelization or vectorization, and the workers and vectorized keywords can over-ride this option.

Added in version 1.2.0.

workersint or map-like callable, optional

If workers is an int the population is subdivided into workers sections and evaluated in parallel (uses multiprocessing.Pool). Supply -1 to use all available CPU cores. Alternatively supply a map-like callable, such as multiprocessing.Pool.map for evaluating the population in parallel. This evaluation is carried out as workers(func, iterable). This option will override the updating keyword to updating='deferred' if workers != 1. This option overrides the vectorized keyword if workers != 1. Requires that func be pickleable.

Added in version 1.2.0.

constraints{NonLinearConstraint, LinearConstraint, Bounds}

Constraints on the solver, over and above those applied by the bounds kwd. Uses the approach by Lampinen [5].

Added in version 1.4.0.

x0None or array-like, optional

Provides an initial guess to the minimization. Once the population has been initialized this vector replaces the first (best) member. This replacement is done even if init is given an initial population. x0.shape == (N,).

Added in version 1.7.0.

integrality1-D array, optional

For each decision variable, a boolean value indicating whether the decision variable is constrained to integer values. The array is broadcast to (N,). If any decision variables are constrained to be integral, they will not be changed during polishing. Only integer values lying between the lower and upper bounds are used. If there are no integer values lying between the bounds then a ValueError is raised.

Added in version 1.9.0.

vectorizedbool, optional

If vectorized is True, func is sent an x array with x.shape == (N, S), and is expected to return an array of shape (S,), where S is the number of solution vectors to be calculated. If constraints are applied, each of the functions used to construct a Constraint object should accept an x array with x.shape == (N, S), and return an array of shape (M, S), where M is the number of constraint components. This option is an alternative to the parallelization offered by workers, and may help in optimization speed by reducing interpreter overhead from multiple function calls. This keyword is ignored if workers != 1. This option will override the updating keyword to updating='deferred'. See the notes section for further discussion on when to use 'vectorized', and when to use 'workers'.

Added in version 1.9.0.

Returns:

resOptimizeResult

The optimization result represented as a OptimizeResult object. Important attributes are: x the solution array, success a Boolean flag indicating if the optimizer exited successfully, message which describes the cause of the termination, population the solution vectors present in the population, and population_energies the value of the objective function for each entry in population. See OptimizeResult for a description of other attributes. If polish was employed, and a lower minimum was obtained by the polishing, then OptimizeResult also contains the jac attribute. If the eventual solution does not satisfy the applied constraints success will be False.

Notes

Differential evolution is a stochastic population based method that is useful for global optimization problems. At each pass through the population the algorithm mutates each candidate solution by mixing with other candidate solutions to create a trial candidate. There are several strategies [3] for creating trial candidates, which suit some problems more than others. The ‘best1bin’ strategy is a good starting point for many systems. In this strategy two members of the population are randomly chosen. Their difference is used to mutate the best member (the ‘best’ in ‘best1bin’), \(x_0\), so far:

\[b' = x_0 + F \cdot (x_{r_0} - x_{r_1})\]

where \(F\) is the mutation parameter. A trial vector is then constructed. Starting with a randomly chosen ith parameter the trial is sequentially filled (in modulo) with parameters from b' or the original candidate. The choice of whether to use b' or the original candidate is made with a binomial distribution (the ‘bin’ in ‘best1bin’) - a random number in [0, 1) is generated. If this number is less than the recombination constant then the parameter is loaded from b', otherwise it is loaded from the original candidate. The final parameter is always loaded from b'. Once the trial candidate is built its fitness is assessed. If the trial is better than the original candidate then it takes its place. If it is also better than the best overall candidate it also replaces that.

The other strategies available are outlined in Qiang and Mitchell (2014) [3].

• rand1 : \(b' = x_{r_0} + F \cdot (x_{r_1} - x_{r_2})\)

• rand2 : \(b' = x_{r_0} + F \cdot (x_{r_1} + x_{r_2} - x_{r_3} - x_{r_4})\)

• best1 : \(b' = x_0 + F \cdot (x_{r_0} - x_{r_1})\)

• best2 : \(b' = x_0 + F \cdot (x_{r_0} + x_{r_1} - x_{r_2} - x_{r_3})\)

• currenttobest1 : \(b' = x_i + F \cdot (x_0 - x_i + x_{r_0} - x_{r_1})\)

• randtobest1 : \(b' = x_{r_0} + F \cdot (x_0 - x_{r_0} + x_{r_1} - x_{r_2})\)

where the integers \(r_0, r_1, r_2, r_3, r_4\) are chosen randomly from the interval [0, NP) with NP being the total population size and the original candidate having index i. The user can fully customize the generation of the trial candidates by supplying a callable to strategy.

To improve your chances of finding a global minimum use higher popsize values, with higher mutation and (dithering), but lower recombination values. This has the effect of widening the search radius, but slowing convergence.

By default the best solution vector is updated continuously within a single iteration (updating='immediate'). This is a modification [4] of the original differential evolution algorithm which can lead to faster convergence as trial vectors can immediately benefit from improved solutions. To use the original Storn and Price behaviour, updating the best solution once per iteration, set updating='deferred'. The 'deferred' approach is compatible with both parallelization and vectorization ('workers' and 'vectorized' keywords). These may improve minimization speed by using computer resources more efficiently. The 'workers' distribute calculations over multiple processors. By default the Python multiprocessing module is used, but other approaches are also possible, such as the Message Passing Interface (MPI) used on clusters [6] [7]. The overhead from these approaches (creating new Processes, etc) may be significant, meaning that computational speed doesn’t necessarily scale with the number of processors used. Parallelization is best suited to computationally expensive objective functions. If the objective function is less expensive, then 'vectorized' may aid by only calling the objective function once per iteration, rather than multiple times for all the population members; the interpreter overhead is reduced.

Added in version 0.15.0.

References

[1]

Differential evolution, Wikipedia, http://en.wikipedia.org/wiki/Differential_evolution

[2]

Storn, R and Price, K, Differential Evolution - a Simple and Efficient Heuristic for Global Optimization over Continuous Spaces, Journal of Global Optimization, 1997, 11, 341 - 359.

[3] (1,2)

Qiang, J., Mitchell, C., A Unified Differential Evolution Algorithm for Global Optimization, 2014, https://www.osti.gov/servlets/purl/1163659

[4] (1,2)

Wormington, M., Panaccione, C., Matney, K. M., Bowen, D. K., - Characterization of structures from X-ray scattering data using genetic algorithms, Phil. Trans. R. Soc. Lond. A, 1999, 357, 2827-2848

[5]

Lampinen, J., A constraint handling approach for the differential evolution algorithm. Proceedings of the 2002 Congress on Evolutionary Computation. CEC’02 (Cat. No. 02TH8600). Vol. 2. IEEE, 2002.

[6]

https://mpi4py.readthedocs.io/en/stable/

[7]

https://schwimmbad.readthedocs.io/en/latest/

Examples

Try it in your browser!

Let us consider the problem of minimizing the Rosenbrock function. This function is implemented in rosen in scipy.optimize.

>>> import numpy as np
>>> from scipy.optimize import rosen, differential_evolution
>>> bounds = [(0,2), (0, 2), (0, 2), (0, 2), (0, 2)]
>>> result = differential_evolution(rosen, bounds)
>>> result.x, result.fun
(array([1., 1., 1., 1., 1.]), 1.9216496320061384e-19)

Now repeat, but with parallelization.

>>> result = differential_evolution(rosen, bounds, updating='deferred',
...                                 workers=2)
>>> result.x, result.fun
(array([1., 1., 1., 1., 1.]), 1.9216496320061384e-19)

Let’s do a constrained minimization.

>>> from scipy.optimize import LinearConstraint, Bounds

We add the constraint that the sum of x[0] and x[1] must be less than or equal to 1.9. This is a linear constraint, which may be written A @ x <= 1.9, where A = array([[1, 1]]). This can be encoded as a LinearConstraint instance:

>>> lc = LinearConstraint([[1, 1]], -np.inf, 1.9)

Specify limits using a Bounds object.

>>> bounds = Bounds([0., 0.], [2., 2.])
>>> result = differential_evolution(rosen, bounds, constraints=lc,
...                                 rng=1)
>>> result.x, result.fun
(array([0.96632622, 0.93367155]), 0.0011352416852625719)

Next find the minimum of the Ackley function (https://en.wikipedia.org/wiki/Test_functions_for_optimization).

>>> def ackley(x):
...     arg1 = -0.2 * np.sqrt(0.5 * (x[0] ** 2 + x[1] ** 2))
...     arg2 = 0.5 * (np.cos(2. * np.pi * x[0]) + np.cos(2. * np.pi * x[1]))
...     return -20. * np.exp(arg1) - np.exp(arg2) + 20. + np.e
>>> bounds = [(-5, 5), (-5, 5)]
>>> result = differential_evolution(ackley, bounds, rng=1)
>>> result.x, result.fun
(array([0., 0.]), 4.440892098500626e-16)

The Ackley function is written in a vectorized manner, so the 'vectorized' keyword can be employed. Note the reduced number of function evaluations.

>>> result = differential_evolution(
...     ackley, bounds, vectorized=True, updating='deferred', rng=1
... )
>>> result.x, result.fun
(array([0., 0.]), 4.440892098500626e-16)

The following custom strategy function mimics ‘best1bin’:

>>> def custom_strategy_fn(candidate, population, rng=None):
...     parameter_count = population.shape(-1)
...     mutation, recombination = 0.7, 0.9
...     trial = np.copy(population[candidate])
...     fill_point = rng.choice(parameter_count)
...
...     pool = np.arange(len(population))
...     rng.shuffle(pool)
...
...     # two unique random numbers that aren't the same, and
...     # aren't equal to candidate.
...     idxs = []
...     while len(idxs) < 2 and len(pool) > 0:
...         idx = pool[0]
...         pool = pool[1:]
...         if idx != candidate:
...             idxs.append(idx)
...
...     r0, r1 = idxs[:2]
...
...     bprime = (population[0] + mutation *
...               (population[r0] - population[r1]))
...
...     crossovers = rng.uniform(size=parameter_count)
...     crossovers = crossovers < recombination
...     crossovers[fill_point] = True
...     trial = np.where(crossovers, bprime, trial)
...     return trial

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