5.4. `pyopus.problems.mgh` — More-Garbow-Hillstrom set of test functions¶

Inheritance diagram of pyopus.problems.mgh

More-Garbow-Hillstrom test functions with first derivatives (PyOPUS subsystem name: MGH)

Translated from MATLAB implementation (with bugs omitted).

All test functions in this module are maps from $R$ to $R^n$ . Every function is comprised of m auxiliary functions

$f_1(x) ... f_m(x)$

where x is a n-dimensional vector.

The actual test function is then constructed as

$f(x) = \sum_{i=1}^m \left( f_i(x) \right)^2$

The i-th component of the function’s gradient can be expressed as

$(\nabla f(x))_i = 2 \sum_{k=1}^{m} J_{ki}(x) f_{k}(x)$

where J(x) is the Jacobian matrix of the auxiliary functions at x where the first index corresponds to the auxiliary function and the second index corresponds to the component of auxiliary function’s gradient.

An exception is the McKinnon function which is not part of the original test suite but is included here because it is a well known counterexample for the Nelder-Mead simplex algorithm. Another exception is the Gao-Han almost quadratic function.

The functions were first published as a test suite in [mgh]. Later bounds were added to most test functions in [gay].

If a function’s documentation does not say anything about bounds, the bounds are defined for all allowed values of n.

This module is independent of PyOPUS, meaning that it can be taken as is and used as a module in some other package. It depends only on the cpi module.

[mgh]

More J.J, Garbow B. S., Hillstrom K. E.: Testing Unconstrained Optimization Software. ACM Transactions on Mathematical Software, vol. 7, pp. 17-41, 1981.

[gay]

Gay D. M., A trust-region approach to linearly constrained optimization. Numerical Analysis (Griffiths, D.F., ed.), Lecture Notes in Mathematics 1066, pp. 72-105, Springer, Berlin, 1984.

class pyopus.problems.mgh.MGH(n=2, m=2)¶

Base class for test functions

The initial point can be obtained from the initial member. The full name of the problem is in the name member.

The xl and xh members specify the lower and the upper bound given by D. M. Gay in his paper. If they are None no known lower or upper bound is available in the literature.

Objects of this class are callable. The calling convention is

object(x, gradients)

where x is the input values vector and gradients is a boolean flag specifying whether the Jacobian should be evaluated. The values of the auxiliary functions and the jacobian are stored in the fi and J members.

To create an instance of the extended Rosenbrock function with n=m=4 and evaluate the function and the gradient at the initial point one should:

from pyopus.optimizer.mgh import ExtendedRosenbrock
rb=ExtendedRosenbrock(n=4, m=4)
(f, g)=rb.fg(rb.initial)

cpi(bounds=False)¶

Returns the common problem interface.

If bounds is True the bounds defined by Gay are included. Best known minimum information is missing for some problems.

The info member of the returned dictionary is itself a dictionary with the following members:

m - m parameter of the MGH problem

See the CPI class for more information.

f(x)¶: Returns the value of the test function at x.

fg(x)¶: Returns a tuple of the form (f, g) where f is the function value at x and g is the corresponding value of the function’s gradient.

g(x)¶: Returns the value of the gradient at x.

name = None¶: The name of the test function

pyopus.problems.mgh.MGHsuite = [<class 'pyopus.problems.mgh.Rosenbrock'>, <class 'pyopus.problems.mgh.FreudensteinAndRoth'>, <class 'pyopus.problems.mgh.PowellBadlyScaled'>, <class 'pyopus.problems.mgh.BrownBadlyScaled'>, <class 'pyopus.problems.mgh.Beale'>, <class 'pyopus.problems.mgh.JennrichAndSampson'>, <class 'pyopus.problems.mgh.HelicalValley'>, <class 'pyopus.problems.mgh.Bard'>, <class 'pyopus.problems.mgh.Gaussian'>, <class 'pyopus.problems.mgh.Meyer'>, <class 'pyopus.problems.mgh.GulfResearchAndDevelopement'>, <class 'pyopus.problems.mgh.Box3D'>, <class 'pyopus.problems.mgh.PowellSingular'>, <class 'pyopus.problems.mgh.Wood'>, <class 'pyopus.problems.mgh.KowalikAndOsborne'>, <class 'pyopus.problems.mgh.BrownAndDennis'>, <class 'pyopus.problems.mgh.Osborne1'>, <class 'pyopus.problems.mgh.BiggsEXP6'>, <class 'pyopus.problems.mgh.Osborne2'>, <class 'pyopus.problems.mgh.Watson'>, <class 'pyopus.problems.mgh.ExtendedRosenbrock'>, <class 'pyopus.problems.mgh.ExtendedPowellSingular'>, <class 'pyopus.problems.mgh.PenaltyI'>, <class 'pyopus.problems.mgh.PenaltyII'>, <class 'pyopus.problems.mgh.VariablyDimensioned'>, <class 'pyopus.problems.mgh.Trigonometric'>, <class 'pyopus.problems.mgh.BrownAlmostLinear'>, <class 'pyopus.problems.mgh.DiscreteBoundaryValue'>, <class 'pyopus.problems.mgh.DiscreteIntegralEquation'>, <class 'pyopus.problems.mgh.BroydenTridiagonal'>, <class 'pyopus.problems.mgh.BroydenBanded'>, <class 'pyopus.problems.mgh.LinearFullRank'>, <class 'pyopus.problems.mgh.LinearRank1'>, <class 'pyopus.problems.mgh.LinearRank1ZeroColumnsAndRows'>, <class 'pyopus.problems.mgh.Chebyquad'>, <class 'pyopus.problems.mgh.Quadratic'>, <class 'pyopus.problems.mgh.McKinnon'>, <class 'pyopus.problems.mgh.GaoHanAlmostQuadratic'>, <class 'pyopus.problems.mgh.Dixon'>, <class 'pyopus.problems.mgh.OrenPower'>]¶: A list holding references to all function classes in this module.

class pyopus.problems.mgh.Rosenbrock(n=2, m=2)¶

Rosenbrock test function (n=m=2).