naginterfaces.library.opt.uncon_simplex¶
- naginterfaces.library.opt.uncon_simplex(x, tolf, tolx, funct, maxcal, monit=None, data=None)[source]¶
uncon_simplex
minimizes a general function of independent variables by the Nelder and Mead simplex method (see Nelder and Mead (1965)). Derivatives of the function need not be supplied.For full information please refer to the NAG Library document for e04cb
https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/e04/e04cbf.html
- Parameters
- xfloat, array-like, shape
A guess at the position of the minimum. Note that the problem should be scaled so that the values of the are of order unity.
- tolffloat
The error tolerable in the function values, in the following sense. If , for , are the individual function values at the vertices of the current simplex, and if is the mean of these values, then you can request that
uncon_simplex
should terminate ifYou may specify if you wish to use only the termination criterion (2) on the spatial values: see the description of .
- tolxfloat
The error tolerable in the spatial values, in the following sense. If denotes the ‘linearized’ volume of the current simplex, and if denotes the ‘linearized’ volume of the initial simplex, then you can request that
uncon_simplex
should terminate ifYou may specify if you wish to use only the termination criterion (1) on function values: see the description of .
- functcallable fc = funct(xc, data=None)
must evaluate the function at a specified point.
It should be tested separately before being used in conjunction with
uncon_simplex
.- Parameters
- xcfloat, ndarray, shape
The point at which the function value is required.
- dataarbitrary, optional, modifiable in place
User-communication data for callback functions.
- Returns
- fcfloat
The value of the function at the current point .
- maxcalint
The maximum number of function evaluations to be allowed.
- monitNone or callable monit(fmin, fmax, sim, ncall, serror, vratio, data=None), optional
Note: if this argument is None then a NAG-supplied facility will be used.
may be used to monitor the optimization process.
It is invoked once every iteration.
If no monitoring is required, may be None.
- Parameters
- fminfloat
The smallest function value in the current simplex.
- fmaxfloat
The largest function value in the current simplex.
- simfloat, ndarray, shape
The position vectors of the current simplex.
- ncallint
The number of times that has been called so far.
- serrorfloat
The current value of the standard deviation in function values used in termination test (1).
- vratiofloat
The current value of the linearized volume ratio used in termination test (2).
- dataarbitrary, optional, modifiable in place
User-communication data for callback functions.
- dataarbitrary, optional
User-communication data for callback functions.
- Returns
- xfloat, ndarray, shape
The value of corresponding to the function value in .
- ffloat
The lowest function value found.
- Raises
- NagValueError
- (errno )
On entry, .
Constraint: .
- (errno )
On entry, and .
Constraint: if then is greater than or equal to the machine precision.
- (errno )
On entry, and .
Constraint: if then is greater than or equal to the machine precision.
- (errno )
On entry, and .
Constraint: if and then both should be greater than or equal to the machine precision.
- (errno )
On entry, .
Constraint: .
- Warns
- NagAlgorithmicMajorWarning
- (errno )
function evaluations have been completed without any other termination test passing. Check the coding of before increasing the value of .
- Notes
uncon_simplex
finds an approximation to a minimum of a function of variables. You must supply a function to calculate the value of for any set of values of the variables.The method is iterative. A simplex of points is set up in the -dimensional space of the variables (for example, in dimensions the simplex is a triangle) under the assumption that the problem has been scaled so that the values of the independent variables at the minimum are of order unity. The starting point you have provided is the first vertex of the simplex, the remaining vertices are generated by
uncon_simplex
. The vertex of the simplex with the largest function value is reflected in the centre of gravity of the remaining vertices and the function value at this new point is compared with the remaining function values. Depending on the outcome of this test the new point is accepted or rejected, a further expansion move may be made, or a contraction may be carried out. See Nelder and Mead (1965) and Parkinson and Hutchinson (1972) for more details. When no further progress can be made the sides of the simplex are reduced in length and the method is repeated.The method can be slow, but computational bottlenecks have been reduced following Singer and Singer (2004). However,
uncon_simplex
is robust, and, therefore, very useful for functions that are subject to inaccuracies.There are the following options for successful termination of the method: based only on the function values at the vertices of the current simplex (see (1)); based only on a volume ratio between the current simplex and the initial one (see (2)); or based on which one of the previous two tests passes first. The volume test may be useful if is discontinuous, while the function-value test should be sufficient on its own if is continuous.
- References
Nelder, J A and Mead, R, 1965, A simplex method for function minimization, Comput. J. (7), 308–313
Parkinson, J M and Hutchinson, D, 1972, An investigation into the efficiency of variants of the simplex method, Numerical Methods for Nonlinear Optimization, (ed F A Lootsma), Academic Press
Singer, S and Singer, S, 2004, Efficient implementation of the Nelder–Mead search algorithm, Appl. Num. Anal. Comp. Math. (1(3)), 524–534