# 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, seed=None, callback=None, disp=False, polish=True, init='latinhypercube', atol=0, updating='immediate', workers=1, constraints=(), x0=None)

  Finds the global minimum of a multivariate function.

  Differential Evolution is stochastic in nature (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 [1].

  Parameters
    func callable

      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.

    bounds sequence 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. It is required to have `len(bounds) == len(x)`. `len(bounds)` is used to determine the number of parameters in `x`.

    args tuple, optional

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

    strategy str, optional

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

      > - ‘best1bin’

      > - ‘best1exp’

      > - ‘rand1exp’

      > - ‘randtobest1exp’

      > - ‘currenttobest1exp’

      > - ‘best2exp’

      > - ‘rand2exp’

      > - ‘randtobest1bin’

      > - ‘currenttobest1bin’

      > - ‘best2bin’

      > - ‘rand2bin’

      > - ‘rand1bin’

      The default is ‘best1bin’.

    maxiter int, 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 * len(x)`

    popsize int, optional

      A multiplier for setting the total population size. The population has `popsize * len(x)` 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 * len(x)`.

    tol float, optional

      Relative 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.

    mutation float 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.

    recombination float, 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.

    seed{None, int, `numpy.random.Generator`,

      > `numpy.random.RandomState`}, optional

      If seed is None (or np.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. Specify seed for repeatable minimizations.

    disp bool, optional

      Prints the evaluated func at every iteration.

    callback callable, callback(xk, convergence=val), optional

      A function to follow the progress of the minimization. `xk` is the current value of `x0`. `val` represents the fractional value of the population convergence. When `val` is greater than one the function halts. If callback returns True, then the minimization is halted (any polishing is still carried out).

    polish bool, 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.

    init str 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 `(M, len(x))`, where M is the total population size and len(x) 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 * len(x)`. ‘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.

    atol float, 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, and the workers keyword can over-ride this option.

      New in version 1.2.0.

    workers int 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`. Requires that func be pickleable.

      New 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].

      New in version 1.4.0.

    x0 None 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.

      New in version 1.7.0.

  Returns
    res OptimizeResult

      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 and `message` which describes the cause of the termination. 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 [2] 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’), \(b_0\), so far:

  \[b' = b_0 + mutation * (population[rand0] - population[rand1])\]

  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. 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'`.

  New in version 0.15.0.

  References

  1

    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.

  2

    http://www1.icsi.berkeley.edu/~storn/code.html

  3

    http://en.wikipedia.org/wiki/Differential_evolution

  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.

  Examples

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

      >>> 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.

      >>> bounds = [(0,2), (0, 2), (0, 2), (0, 2), (0, 2)]
      >>> result = differential_evolution(rosen, bounds, updating='deferred',
      ...                                 workers=2)
      >>> result.x, result.fun
      (array([1., 1., 1., 1., 1.]), 1.9216496320061384e-19)

  Let’s try and do a constrained minimization

      >>> from scipy.optimize import NonlinearConstraint, Bounds
      >>> def constr_f(x):
      ...     return np.array(x[0] + x[1])
      >>>
      >>> # the sum of x[0] and x[1] must be less than 1.9
      >>> nlc = NonlinearConstraint(constr_f, -np.inf, 1.9)
      >>> # specify limits using a `Bounds` object.
      >>> bounds = Bounds([0., 0.], [2., 2.])
      >>> result = differential_evolution(rosen, bounds, constraints=(nlc),
      ...                                 seed=1)
      >>> result.x, result.fun
      (array([0.96633867, 0.93363577]), 0.0011361355854792312)

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

      >>> from scipy.optimize import differential_evolution
      >>> import numpy as np
      >>> 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)
      >>> result.x, result.fun
      (array([ 0.,  0.]), 4.4408920985006262e-16)

scipy.optimize.brute scipy.optimize.shgo