scipy.optimize.shgo¶
- scipy.optimize.shgo(func, bounds, args=(), constraints=None, n=None, iters=1, callback=None, minimizer_kwargs=None, options=None, sampling_method='simplicial')[source]¶
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)
, wherex
is the argument in the form of a 1-D array andargs
is a tuple of any additional fixed parameters needed to completely specify the function.- boundssequence
Bounds for variables.
(min, max)
pairs for each element inx
, defining the lower and upper bounds for the optimizing argument of func. It is required to havelen(bounds) == len(x)
.len(bounds)
is used to determine the number of parameters inx
. UseNone
for one of min or max when there is no bound in that direction. By default bounds are(None, None)
.- argstuple, optional
Any additional fixed parameters needed to completely specify the objective function.
- constraintsdict 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:
- typestr
Constraint type: ‘eq’ for equality, ‘ineq’ for inequality.
- funcallable
The function defining the constraint.
- jaccallable, optional
The Jacobian of fun (only for SLSQP).
- argssequence, 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 inminimizer_kwargs
and a constrained method is used then the globalconstraints
will be used. (Defining aconstraints
sequence inminimizer_kwargs
means thatconstraints
will not be added so if equality constraints and so forth need to be added then the inequality functions inconstraints
need to be added tominimizer_kwargs
too).- nint, 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. In case ofsobol
, it must be a power of 2:n=2**m
, and the argument will automatically be converted to the next higher power of 2. Default is 100 forsampling_method='simplicial'
and 128 forsampling_method='sobol'
.- 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)
, wherexk
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, the default is
SLSQP
.
- 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_min
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:
- symmetrybool
Specify True if the objective function contains symmetric variables. The search space (and therefore performance) is decreased by O(n!).
- 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 asfun
. (Passed to scipy.optimize.minmize 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
orhess
needs to be given. Ifhess
is provided, thenhessp
will be ignored. If neitherhess
norhessp
is provided, then the Hessian product will be approximated using finite differences onjac
.hessp
must compute the Hessian times an arbitrary vector. (Passed to scipy.optimize.minmize automatically)
Algorithm settings:
- minimize_every_iterbool
If True then promising global sampling points will be passed to a local minimization routine every iteration. If False then only the final minimizer pool will be run. Defaults to False.
- 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_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:
- dispbool (L)
Set to True to print convergence messages.
- sampling_methodstr or function, optional
Current built in sampling method options are
halton
,sobol
andsimplicial
. The defaultsimplicial
provides the theoretical guarantee of convergence to the global minimum in finite time.halton
andsobol
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 ofn
sampling points of dimensiondim
per call and output an array of sampling points with shape n x dim.
- 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 functionR^n -> R
,g_i(x)
are the inequality constraints, andh_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 wheref(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 toscipy.optimize.minimize
. By default, theSLSQP
method is used. In general, it is recommended to use theSLSQP
orCOBYLA
local minimization if inequality constraints are defined for the problem since the other methods do not use constraints.The
halton
andsobol
method points are generated usingscipy.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)
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
- 4
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.]), 2.920392374190081e-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 likenp.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 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, 20)
Note the difference between, e.g.,
n=192, iters=1
andn=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, iters=3, constraints=cons) >>> res fun: 29.894378159142136 funl: array([29.89437816]) message: 'Optimization terminated successfully.' nfev: 114 nit: 3 nlfev: 35 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) (-5.062616992290714e-14, -2.9594104944408173e-12, 0.0)