5.6. pyopus.problems.madsprob — Mesh adaptive direct search test problems

Optimization test functions from papers on MADS. (PyOPUS subsystem name: MADSPROB)

All test functions in this module are maps from \(R^n\) to \(R\).

The nc nonlinear constraints are of the form

\[cl \leq c(x) \leq ch\]

Beside linear constraints a problem can also have bounds of the form

\[xl \leq x \leq xh\]

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 and the mgh module.

The STYRENE and the MDO problem depend on the _mads module. HS114 and MAD6 depend on the _lvns module.

References:

mads

Audet C., Dennis J.E. Jr., Mesh Adaptive Direct Search Algorithms for Constrained Optimization. SIAM Journal on Optimization, vol. 17, pp. 188-217, 2006.

madspsd(1,2,3)

Audet C., Dennis Jr J.E., Le Digabel S., Parallel Space Decomposition of the Mesh Adaptive Direct Search Algorithm. SIAM Journal on Optimization, vol. 19, pp. 1150-1170, 2008.

madsort(1,2)

Abramson M.A., Audet C., Dennis J.E. Jr., Le Digabel S., ORTHO-MADS: A Deterministic MADS Instance with Orthogonal Directions. SIAM Journal on Optimization, vol. 20, pp. 948-966, 2009.

madspb(1,2,3)

Audet C., Dennis J.E. Jr., A Progressive Barrier for Derivative-Free Nonlinear Programming. SIAM Journal on Optimization, vol. 20, pp. 445-472, 2009.

madsqm

Conn A.R., Le Digabel S., Use of Quadratic Models with Mesh Adaptive Direct Search for Constrained Black Box Optimization. Optimization Methods and Software, vol. 28, pp. 139-158, 2013.

madsvns(1,2)

Audet C., Bechard V., Le Digabel S., Nonsmooth Optimization Through Mesh Adaptive Direct Search and Variable Neighbourhood Search. Journal of Global Optimization, vol 41, pp. 299-318, 2008.

ufo(1,2)

Lukšan L, et. al., UFO 2011 interactive system for univeral functional optimization. TR 1151 ICS, AS CR, 2011.

class pyopus.problems.madsprob.B250

B250 test function (n=60, nc=1).

Mentioned as problem B in section 5.2 of [madspsd].

Rewritten from C++ source obtained from the NOMAD test problems collection.

See the MADSPROB class for more details.

class pyopus.problems.madsprob.B500

B500 test function (n=60, nc=1).

Mentioned as problem B in section 5.2 of [madspsd].

Rewritten from C++ source obtained from the NOMAD test problems collection.

See the MADSPROB class for more details.

class pyopus.problems.madsprob.CRESCENT(n=10)

CRESCENT test function (n=arbitrary, nc=2).

Published as the first problem in section 4.3 of [madspb].

See the MADSPROB class for more details.

class pyopus.problems.madsprob.DIFF2

DIFF2 test function (n=2, nc=0).

Published as the problem (4.1) in [madsqm].

See the MADSPROB class for more details.

class pyopus.problems.madsprob.DISK(n=10)

DISK test function (n=arbitrary, nc=1). Also referred to a the Hypersphere problem.

Published as the first problem in section 4.2 of [madspb] or as the first problem in section 5.3 of [mads].

See the MADSPROB class for more details.

class pyopus.problems.madsprob.G2(n=10)

G2 test function (n=arbitrary, nc=1).

Published as problem A in section 5.2 of [madspsd].

See the MADSPROB class for more details.

class pyopus.problems.madsprob.HS114_mod

Modification of the HS114 problem from the lvns module.

The HS114 problem has 10 variables. One linear equality constraint links x1, x4, and x5. This modification eliminates x5 resulting in a problem with 9 variables and no equality constraint.

Published in [madsort].

cpi()

Returns the common problem interface.

xmin, f, and c are not available. Only fc is available.

See the CPI class for more information.

fc(x)

Returns the value of the function and the constraints at x.

setup()

Initializes the binary implementation of the function. After this function is called no other function from the same test set may be created or initialized because that will change the internal variables and break the function. Returns an info structure.

class pyopus.problems.madsprob.MAD6_mod

Modification of the MAD6 problem from the lvns module.

The MAD6 problem has 7 variables. One of them has an equality constraint imposed (x7). There is also a linear equality constraint involving x4 and x6. This modification eliminates x6 and x7 resulting in a problem with 5 variables and no equality constraint.

Published in [madsort].

cpi()

Returns the common problem interface.

xmin, f, and c are not available. Only fc is available.

See the CPI class for more information.

fc(x)

Returns the value of the function and the constraints at x.

setup()

Initializes the binary implementation of the function. After this function is called no other function from the same test set may be created or initialized because that will change the internal variables and break the function. Returns an info structure.

class pyopus.problems.madsprob.MADSPROB(n, m=1, nc=0)

Base class for the test functions

The fesible initial point can be obtained from the initial member. The infeasible initial point is in the initialinf member. If any of these two points is not given, it is set to None.

The full name of the problem is in the name member. The n member holds the dimension of the problem. nc is the number of nonlinear constraints.

The best known minimum value of the function is in the fmin member. If this value is unknown it is set to None.

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 gradients should be evaluated. The values of the auxiliary functions and their gradients are stored in the fi and J members. The values of the cobnstraints and the corresponding Jacobian are stored in the ci and J members.

This creates an instance of the CRESCENT function and evaluates the function and the constraints along with the gradient and the constraint Jacobian at the feasible initial point:

from pyopus.optimizer.madsprob import CRESCENT
cr=CRESCENT(n=10)
(f, g, c, Jc)=cr.fgcjc(cr.initial)

c(x)

Returns the value of the constraints at x.

cjc(x)

Returns the constraint Jacobian at x.

cpi()

Returns the common problem interface.

xmin is not available.

See the CPI class for more information.

f(x)

Returns the value of the test function at x.

fc(x)

Returns the value of teh function and constraints at x.

g(x)

Returns the value of the gradient at x.

name = None

The name of the test function

pyopus.problems.madsprob.MADSPROBsuite = [<class 'pyopus.problems.madsprob.SNAKE'>, <class 'pyopus.problems.madsprob.DISK'>, <class 'pyopus.problems.madsprob.CRESCENT'>, <class 'pyopus.problems.madsprob.G2'>, <class 'pyopus.problems.madsprob.B250'>, <class 'pyopus.problems.madsprob.B500'>, <class 'pyopus.problems.madsprob.DIFF2'>, <class 'pyopus.problems.madsprob.UFO7_26'>, <class 'pyopus.problems.madsprob.UFO7_29'>, <class 'pyopus.problems.madsprob.MDO'>, <class 'pyopus.problems.madsprob.STYRENE'>, <class 'pyopus.problems.madsprob.MAD6_mod'>, <class 'pyopus.problems.madsprob.HS114_mod'>]

A list holding references to all function classes in this module.

class pyopus.problems.madsprob.MDO(eps=1e-12, maxiter=100)

A multidisciplinary design optimization problem - maximization of aircraft range.

Warning - possible memory leaks and crashes the c++ code is not cleaned up.

Published in [madsvns].

cpi()

Returns the common problem interface.

xmin, f, and c are not available. Only fc is available.

See the CPI class for more information.

fc(x)

Returns the value of the function and the constraints at x.

class pyopus.problems.madsprob.SNAKE

SNAKE test function (n=2, nc=2).

Published as the first problem in section 4.1 of [madspb].

See the MADSPROB class for more details.

class pyopus.problems.madsprob.STYRENE

An engineering optimization problem - styrene process optimization

Warning - possible memory leaks and crashes the c++ code is not cleaned up.

Published in [madsvns].

cpi()

Returns the common problem interface.

xmin, f, and c are not available. Only fc is available.

See the CPI class for more information.

fc(x)

Returns the value of the function and the constraints at x.

class pyopus.problems.madsprob.UFO7_26(n=10)

Problem from UFO manual (n, nc=int(3*(n-2)/2)).

Published as the first problem in section 7.26 of [ufo].

See the MADSPROB class for more details.

cpi()

Returns the common problem interface.

xmin is not available.

See the CPI class for more information.

class pyopus.problems.madsprob.UFO7_29(n=10)

Problem from UFO manual (n, nc=n-2).

Published as the first problem in section 7.29 of [ufo].

See the MADSPROB class for more details.

cpi()

Returns the common problem interface.

xmin is not available.

See the CPI class for more information.

Example file madsprob.py in folder demo/problems/

# Problems from papers on MADS

from pyopus.problems.madsprob import MADSPROBsuite

if __name__=='__main__':
	print("MADS problems")
	for ii in range(len(MADSPROBsuite)):
		prob=MADSPROBsuite[ii]()
		fx, cx = prob.fc(prob.initial)
		print("%2d: %20s n=%2d: f0=%e" % (ii, prob.name, prob.n, fx))
	print()