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')

Finds the global minimum of a function using SHG optimization.

SHGO stands for “simplicial homology global optimization”.

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

Bounds for variables. (min, max) pairs for each element in x, defining the 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. Use None for one of min or max when there is no bound in that direction. By default bounds are (None, None).

args : tuple, optional

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

constraints : dict or sequence of dict, optional

Constraints definition. Function(s) R**n in the form:

g(x) <= 0 applied as g : R^n -> R^m
h(x) == 0 applied as h : R^n -> R^p

Each constraint is defined in a dictionary with fields:

type : str

Constraint type: ‘eq’ for equality, ‘ineq’ for inequality.

fun : callable

The function defining the constraint.

jac : callable, optional

The Jacobian of fun (only for SLSQP).

args : sequence, optional

Extra arguments to be passed to the function and Jacobian.

Equality constraint means that the constraint function result is to be zero whereas inequality means that it is to be non-negative. Note that COBYLA only supports inequality constraints.

Note

Only the COBYLA and SLSQP 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).

n : int, optional

Number of sampling points used in the construction of the simplicial complex. Note that this argument is only used for sobol and other arbitrary sampling_methods.

iters : int, optional

Number of iterations used in the construction of the simplicial complex.

callback : callable, optional

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

minimizer_kwargs : dict, optional

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

• method : str

  The minimization method (e.g. SLSQP).

• args : tuple

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

• options : dict, optional

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

options : dict, 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:

• maxfev : int (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_min

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

• f_tol : float

  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.

• maxiter : int

  Maximum number of iterations to perform.

• maxev : int

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

• maxtime : float

  Maximum processing runtime allowed

• minhgrd : int

  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:

• symmetry : bool

  Specify True if the objective function contains symmetric variables. The search space (and therefore performance) is decreased by O(n!).

• jac : bool 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.minmize automatically)

• hess, hessp : callable, 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.minmize automatically)

Algorithm settings:

• minimize_every_iter : bool

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

• local_iter : int

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

• infty_constraints: bool

  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:

• disp : bool (L)

  Set to True to print convergence messages.

sampling_method : str or function, optional

Current built in sampling method options are sobol and simplicial. The default simplicial uses less memory and provides the theoretical guarantee of convergence to the global minimum in finite time. The sobol method is faster in terms of sampling point generation at the cost of higher memory resources and 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.

Returns:

res : OptimizeResult

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 optimisations, nit number of iterations performed by the global routine.

Notes

Global optimization using simplicial homology global optimisation [1]. Appropriate for solving general purpose NLP and blackbox optimisation 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 or COBYLA local minimization if inequality constraints are defined for the problem since the other methods do not use constraints.

The sobol method points are generated using the Sobol (1967) [2] sequence. The primitive polynomials and various sets of initial direction numbers for generating Sobol sequences is provided by [3] by Frances Kuo and Stephen Joe. The original program sobol.cc (MIT) is available and described at http://web.maths.unsw.edu.au/~fkuo/sobol/ translated to Python 3 by Carl Sandrock 2016-03-31.

References

[1]	(1, 2, 3) Endres, SC, Sandrock, C, Focke, WW (2018) “A simplicial homology algorithm for lipschitz optimisation”, Journal of Global Optimization.

[2]	(1, 2) Sobol, IM (1967) “The distribution of points in a cube and the approximate evaluation of integrals”, USSR Comput. Math. Math. Phys. 7, 86-112.

[3]	(1, 2) Joe, SW and Kuo, FY (2008) “Constructing Sobol sequences with better two-dimensional projections”, SIAM J. Sci. Comput. 30, 2635-2654.

[4]	(1, 2, 3) Hoch, 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

[5]	(1, 2) Wales, DJ (2015) “Perspective: Insight into reaction coordinates and dynamics from the potential energy landscape”, Journal of Chemical Physics, 142(13), 2015.

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.]), 2.9203923741900809e-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)

>>> 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 two built-in low discrepancy sampling sequences. First we will input 30 initial sampling points of the Sobol sequence:

>>> result = shgo(eggholder, bounds, n=30, sampling_method='sobol')
>>> result.x, result.fun
(array([ 512.        ,  404.23180542]), -959.64066272085051)

shgo also has a return for any other local minima that was found, these

can be called using:

>>> result.xl
array([[ 512.        ,  404.23180542],
       [ 283.07593402, -487.12566542],
       [-294.66820039, -462.01964031],
       [-105.87688985,  423.15324143],
       [-242.97923629,  274.38032063],
       [-506.25823477,    6.3131022 ],
       [-408.71981195, -156.10117154],
       [ 150.23210485,  301.31378508],
       [  91.00922754, -391.28375925],
       [ 202.8966344 , -269.38042147],
       [ 361.66625957, -106.96490692],
       [-219.40615102, -244.06022436],
       [ 151.59603137, -100.61082677]])

>>> 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,
       -202.53912972])

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 [5]).

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 60 and the number of iterations from the default of 1 to 5. This gives us 60 x 5 = 300 initial sampling points.

>>> result_2 = shgo(eggholder, bounds, n=60, iters=5, sampling_method='sobol')
>>> len(result.xl), len(result_2.xl)
(13, 39)

Note the difference between, e.g., n=180, iters=1 and n=60, iters=3. In the first case the promising points contained in the minimiser pool is processed only once. In the latter case it is processed every 60 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) [4]:

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 [4] 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, iters=3, constraints=cons)
>>> res
     fun: 29.894378159142136
    funl: array([29.89437816])
 message: 'Optimization terminated successfully.'
    nfev: 119
     nit: 3
   nlfev: 40
   nlhev: 0
   nljev: 5
 success: True
       x: array([6.35521569e-01, 1.13700270e-13, 3.12701881e-01, 5.17765506e-02])
      xl: array([[6.35521569e-01, 1.13700270e-13, 3.12701881e-01, 5.17765506e-02]])

>>> g1(res.x), g2(res.x), h1(res.x)
(-5.0626169922907138e-14, -2.9594104944408173e-12, 0.0)