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scipy.optimize.

# shgo

scipy.optimize.shgo(func, bounds, args=(), constraints=None, n=100, iters=1, callback=None, minimizer_kwargs=None, options=None, sampling_method='simplicial', *, workers=1)

  Finds the global minimum of a function using SHG optimization.

  SHGO stands for “simplicial homology global optimization”.

  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.

    boundssequence or `Bounds`

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

      1. Instance of `Bounds` class.

      2. Sequence of `(min, max)` pairs for each element in x.

    argstuple, optional

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

    constraints{Constraint, dict} or List of {Constraint, dict}, optional

      Constraints definition. Only for COBYLA, COBYQA, SLSQP and trust-constr. See the tutorial [5] for further details on specifying constraints.

      Note

      Only COBYLA, COBYQA, SLSQP, and trust-constr local minimize methods currently support constraint arguments. If the `constraints` sequence used in the local optimization problem is not defined in `minimizer_kwargs` and a constrained method is used then the global `constraints` will be used. (Defining a `constraints` sequence in `minimizer_kwargs` means that `constraints` will not be added so if equality constraints and so forth need to be added then the inequality functions in `constraints` need to be added to `minimizer_kwargs` too). COBYLA only supports inequality constraints.

      Changed in version 1.11.0: `constraints` accepts `NonlinearConstraint`, `LinearConstraint`.

    nint, optional

      Number of sampling points used in the construction of the simplicial complex. For the default `simplicial` sampling method 2**dim + 1 sampling points are generated instead of the default `n=100`. For all other specified values n sampling points are generated. For `sobol`, `halton` and other arbitrary sampling_methods `n=100` or another specified number of sampling points are generated.

    itersint, optional

      Number of iterations used in the construction of the simplicial complex. Default is 1.

    callbackcallable, optional

      Called after each iteration, as `callback(xk)`, where `xk` is the current parameter vector.

    minimizer_kwargsdict, optional

      Extra keyword arguments to be passed to the minimizer `scipy.optimize.minimize`. Some important options could be:

      methodstr

        The minimization method. If not given, chosen to be one of BFGS, L-BFGS-B, SLSQP, depending on whether or not the problem has constraints or bounds.

      argstuple

        Extra arguments passed to the objective function (`func`) and its derivatives (Jacobian, Hessian).

      optionsdict, optional

        Note that by default the tolerance is specified as `{ftol: 1e-12}`

    optionsdict, optional

      A dictionary of solver options. Many of the options specified for the global routine are also passed to the `scipy.optimize.minimize` routine. The options that are also passed to the local routine are marked with “(L)”.

      Stopping criteria, the algorithm will terminate if any of the specified criteria are met. However, the default algorithm does not require any to be specified:

      maxfevint (L)

        Maximum number of function evaluations in the feasible domain. (Note only methods that support this option will terminate the routine at precisely exact specified value. Otherwise the criterion will only terminate during a global iteration)

      f_minfloat

        Specify the minimum objective function value, if it is known.

      f_tolfloat

        Precision goal for the value of f in the stopping criterion. Note that the global routine will also terminate if a sampling point in the global routine is within this tolerance.

      maxiterint

        Maximum number of iterations to perform.

      maxevint

        Maximum number of sampling evaluations to perform (includes searching in infeasible points).

      maxtimefloat

        Maximum processing runtime allowed

      minhgrdint

        Minimum homology group rank differential. The homology group of the objective function is calculated (approximately) during every iteration. The rank of this group has a one-to-one correspondence with the number of locally convex subdomains in the objective function (after adequate sampling points each of these subdomains contain a unique global minimum). If the difference in the hgr is 0 between iterations for `maxhgrd` specified iterations the algorithm will terminate.

      Objective function knowledge:

      symmetrylist or bool

        Specify if the objective function contains symmetric variables. The search space (and therefore performance) is decreased by up to O(n!) times in the fully symmetric case. If True is specified then all variables will be set symmetric to the first variable. Default is set to False.

        E.g. f(x) = (x_1 + x_2 + x_3) + (x_4)**2 + (x_5)**2 + (x_6)**2

        In this equation x_2 and x_3 are symmetric to x_1, while x_5 and x_6 are symmetric to x_4, this can be specified to the solver as:

            symmetry = [0,  # Variable 1
                        0,  # symmetric to variable 1
                        0,  # symmetric to variable 1
                        3,  # Variable 4
                        3,  # symmetric to variable 4
                        3,  # symmetric to variable 4
                        ]

      jacbool or callable, optional

        Jacobian (gradient) of objective function. Only for CG, BFGS, Newton-CG, L-BFGS-B, TNC, SLSQP, dogleg, trust-ncg. If `jac` is a boolean and is True, `fun` is assumed to return the gradient along with the objective function. If False, the gradient will be estimated numerically. `jac` can also be a callable returning the gradient of the objective. In this case, it must accept the same arguments as `fun`. (Passed to `scipy.optimize.minimize` automatically)

      hess, hesspcallable, optional

        Hessian (matrix of second-order derivatives) of objective function or Hessian of objective function times an arbitrary vector p. Only for Newton-CG, dogleg, trust-ncg. Only one of `hessp` or `hess` needs to be given. If `hess` is provided, then `hessp` will be ignored. If neither `hess` nor `hessp` is provided, then the Hessian product will be approximated using finite differences on `jac`. `hessp` must compute the Hessian times an arbitrary vector. (Passed to `scipy.optimize.minimize` automatically)

      Algorithm settings:

      minimize_every_iterbool

        If True then promising global sampling points will be passed to a local minimization routine every iteration. If True then only the final minimizer pool will be run. Defaults to True.

      local_iterint

        Only evaluate a few of the best minimizer pool candidates every iteration. If False all potential points are passed to the local minimization routine.

      infty_constraintsbool

        If True then any sampling points generated which are outside will the feasible domain will be saved and given an objective function value of `inf`. If False then these points will be discarded. Using this functionality could lead to higher performance with respect to function evaluations before the global minimum is found, specifying False will use less memory at the cost of a slight decrease in performance. Defaults to True.

      Feedback:

      dispbool (L)

        Set to True to print convergence messages.

    sampling_methodstr or function, optional

      Current built in sampling method options are `halton`, `sobol` and `simplicial`. The default `simplicial` provides the theoretical guarantee of convergence to the global minimum in finite time. `halton` and `sobol` method are faster in terms of sampling point generation at the cost of the loss of guaranteed convergence. It is more appropriate for most “easier” problems where the convergence is relatively fast. User defined sampling functions must accept two arguments of `n` sampling points of dimension `dim` per call and output an array of sampling points with shape n x dim.

    workersint or map-like callable, optional

      Sample and run the local serial minimizations in parallel. Supply -1 to use all available CPU cores, or an int to use that many Processes (uses `multiprocessing.Pool`).

      Alternatively supply a map-like callable, such as multiprocessing.Pool.map for parallel evaluation. This evaluation is carried out as `workers(func, iterable)`. Requires that func be pickleable.

      Added in version 1.11.0.

  Returns:
    resOptimizeResult

      The optimization result represented as a `OptimizeResult` object. Important attributes are: `x` the solution array corresponding to the global minimum, `fun` the function output at the global solution, `xl` an ordered list of local minima solutions, `funl` the function output at the corresponding local solutions, `success` a Boolean flag indicating if the optimizer exited successfully, `message` which describes the cause of the termination, `nfev` the total number of objective function evaluations including the sampling calls, `nlfev` the total number of objective function evaluations culminating from all local search optimizations, `nit` number of iterations performed by the global routine.

  Notes

  Global optimization using simplicial homology global optimization [1]. Appropriate for solving general purpose NLP and blackbox optimization problems to global optimality (low-dimensional problems).

  In general, the optimization problems are of the form:

      minimize f(x) subject to

      g_i(x) >= 0,  i = 1,...,m
      h_j(x)  = 0,  j = 1,...,p

  where x is a vector of one or more variables. `f(x)` is the objective function `R^n -> R`, `g_i(x)` are the inequality constraints, and `h_j(x)` are the equality constraints.

  Optionally, the lower and upper bounds for each element in x can also be specified using the bounds argument.

  While most of the theoretical advantages of SHGO are only proven for when `f(x)` is a Lipschitz smooth function, the algorithm is also proven to converge to the global optimum for the more general case where `f(x)` is non-continuous, non-convex and non-smooth, if the default sampling method is used [1].

  The local search method may be specified using the `minimizer_kwargs` parameter which is passed on to `scipy.optimize.minimize`. By default, the `SLSQP` method is used. In general, it is recommended to use the `SLSQP`, `COBYLA`, or `COBYQA` local minimization if inequality constraints are defined for the problem since the other methods do not use constraints.

  The `halton` and `sobol` method points are generated using `scipy.stats.qmc`. Any other QMC method could be used.

  References

  [1] (1,2)

  Endres, SC, Sandrock, C, Focke, WW (2018) “A simplicial homology algorithm for lipschitz optimisation”, Journal of Global Optimization.

  [2]

  Joe, SW and Kuo, FY (2008) “Constructing Sobol’ sequences with better two-dimensional projections”, SIAM J. Sci. Comput. 30, 2635-2654.

  [3] (1,2)

  Hock, W and Schittkowski, K (1981) “Test examples for nonlinear programming codes”, Lecture Notes in Economics and Mathematical Systems, 187. Springer-Verlag, New York. http://www.ai7.uni-bayreuth.de/test_problem_coll.pdf

  [4]

  Wales, DJ (2015) “Perspective: Insight into reaction coordinates and dynamics from the potential energy landscape”, Journal of Chemical Physics, 142(13), 2015.

  [5]

  https://docs.scipy.org/doc/scipy/tutorial/optimize.html#constrained-minimization-of-multivariate-scalar-functions-minimize

  Examples

  First consider the problem of minimizing the Rosenbrock function, `rosen`:

      >>> from scipy.optimize import rosen, shgo
      >>> bounds = [(0,2), (0, 2), (0, 2), (0, 2), (0, 2)]
      >>> result = shgo(rosen, bounds)
      >>> result.x, result.fun
      (array([1., 1., 1., 1., 1.]), 2.920392374190081e-18)

  Note that bounds determine the dimensionality of the objective function and is therefore a required input, however you can specify empty bounds using `None` or objects like `np.inf` which will be converted to large float numbers.

      >>> bounds = [(None, None), ]*4
      >>> result = shgo(rosen, bounds)
      >>> result.x
      array([0.99999851, 0.99999704, 0.99999411, 0.9999882 ])

  Next, we consider the Eggholder function, a problem with several local minima and one global minimum. We will demonstrate the use of arguments and the capabilities of `shgo`. (https://en.wikipedia.org/wiki/Test_functions_for_optimization)

      >>> import numpy as np
      >>> def eggholder(x):
      ...     return (-(x[1] + 47.0)
      ...             * np.sin(np.sqrt(abs(x[0]/2.0 + (x[1] + 47.0))))
      ...             - x[0] * np.sin(np.sqrt(abs(x[0] - (x[1] + 47.0))))
      ...             )
      ...
      >>> bounds = [(-512, 512), (-512, 512)]

  `shgo` has built-in low discrepancy sampling sequences. First, we will input 64 initial sampling points of the Sobol’ sequence:

      >>> result = shgo(eggholder, bounds, n=64, sampling_method='sobol')
      >>> result.x, result.fun
      (array([512.        , 404.23180824]), -959.6406627208397)

  `shgo` also has a return for any other local minima that was found, these can be called using:

      >>> result.xl
      array([[ 512.        ,  404.23180824],
             [ 283.0759062 , -487.12565635],
             [-294.66820039, -462.01964031],
             [-105.87688911,  423.15323845],
             [-242.97926   ,  274.38030925],
             [-506.25823477,    6.3131022 ],
             [-408.71980731, -156.10116949],
             [ 150.23207937,  301.31376595],
             [  91.00920901, -391.283763  ],
             [ 202.89662724, -269.38043241],
             [ 361.66623976, -106.96493868],
             [-219.40612786, -244.06020508]])

      >>> result.funl
      array([-959.64066272, -718.16745962, -704.80659592, -565.99778097,
             -559.78685655, -557.36868733, -507.87385942, -493.9605115 ,
             -426.48799655, -421.15571437, -419.31194957, -410.98477763])

  These results are useful in applications where there are many global minima and the values of other global minima are desired or where the local minima can provide insight into the system (for example morphologies in physical chemistry [4]).

  If we want to find a larger number of local minima, we can increase the number of sampling points or the number of iterations. We’ll increase the number of sampling points to 64 and the number of iterations from the default of 1 to 3. Using `simplicial` this would have given us 64 x 3 = 192 initial sampling points.

      >>> result_2 = shgo(eggholder,
      ...                 bounds, n=64, iters=3, sampling_method='sobol')
      >>> len(result.xl), len(result_2.xl)
      (12, 23)

  Note the difference between, e.g., `n=192, iters=1` and `n=64, iters=3`. In the first case the promising points contained in the minimiser pool are processed only once. In the latter case it is processed every 64 sampling points for a total of 3 times.

  To demonstrate solving problems with non-linear constraints consider the following example from Hock and Schittkowski problem 73 (cattle-feed) [3]:

      minimize: f = 24.55 * x_1 + 26.75 * x_2 + 39 * x_3 + 40.50 * x_4

      subject to: 2.3 * x_1 + 5.6 * x_2 + 11.1 * x_3 + 1.3 * x_4 - 5    >= 0,

                  12 * x_1 + 11.9 * x_2 + 41.8 * x_3 + 52.1 * x_4 - 21
                      -1.645 * sqrt(0.28 * x_1**2 + 0.19 * x_2**2 +
                                    20.5 * x_3**2 + 0.62 * x_4**2)      >= 0,

                  x_1 + x_2 + x_3 + x_4 - 1                             == 0,

                  1 >= x_i >= 0 for all i

  The approximate answer given in [3] is:

      f([0.6355216, -0.12e-11, 0.3127019, 0.05177655]) = 29.894378

      >>> def f(x):  # (cattle-feed)
      ...     return 24.55*x[0] + 26.75*x[1] + 39*x[2] + 40.50*x[3]
      ...
      >>> def g1(x):
      ...     return 2.3*x[0] + 5.6*x[1] + 11.1*x[2] + 1.3*x[3] - 5  # >=0
      ...
      >>> def g2(x):
      ...     return (12*x[0] + 11.9*x[1] +41.8*x[2] + 52.1*x[3] - 21
      ...             - 1.645 * np.sqrt(0.28*x[0]**2 + 0.19*x[1]**2
      ...                             + 20.5*x[2]**2 + 0.62*x[3]**2)
      ...             ) # >=0
      ...
      >>> def h1(x):
      ...     return x[0] + x[1] + x[2] + x[3] - 1  # == 0
      ...
      >>> cons = ({'type': 'ineq', 'fun': g1},
      ...         {'type': 'ineq', 'fun': g2},
      ...         {'type': 'eq', 'fun': h1})
      >>> bounds = [(0, 1.0),]*4
      >>> res = shgo(f, bounds, n=150, constraints=cons)
      >>> res
       message: Optimization terminated successfully.
       success: True
           fun: 29.894378159142136
          funl: [ 2.989e+01]
             x: [ 6.355e-01  1.137e-13  3.127e-01  5.178e-02] # may vary
            xl: [[ 6.355e-01  1.137e-13  3.127e-01  5.178e-02]] # may vary
           nit: 1
          nfev: 142 # may vary
         nlfev: 35 # may vary
         nljev: 5
         nlhev: 0

      >>> g1(res.x), g2(res.x), h1(res.x)
      (-5.062616992290714e-14, -2.9594104944408173e-12, 0.0)