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 , * , integrality = None , vectorized = False ) [source] #
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 : 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. The number of parameters, N, is equal to len(x) .
bounds sequence or Bounds
Bounds for variables. There are two ways to specify the bounds:
- Instance of Bounds class.
- (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 .
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’
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 * (N — N_equal)
popsize int, 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) .
tol float, optional
Relative tolerance for convergence, the solving stops when np.std(pop) 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.
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 best solution found so far. 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. For large problems with many constraints, polishing can take a long time due to the Jacobian computations.
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 (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.
atol float, optional
Absolute tolerance for convergence, the solving stops when np.std(pop) atol and tol are the absolute and relative tolerance respectively.
updating , 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.
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.
Constraints on the solver, over and above those applied by the bounds kwd. Uses the approach by Lampinen [5].
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,) .
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.
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’ .
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.
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’), \(b_0\) , so far:
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’ . 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.