NAG FL Interface
e04jcf (bounds_bobyqa_func)
1
Purpose
e04jcf is an easy-to-use algorithm that uses methods of quadratic approximation to find a minimum of an objective function over , subject to fixed lower and upper bounds on the independent variables . Derivatives of are not required.
The routine is intended for functions that are continuous and that have continuous first and second derivatives (although it will usually work even if the derivatives have occasional discontinuities). Efficiency is maintained for large .
2
Specification
Fortran Interface
Subroutine e04jcf ( |
objfun, n, npt, x, bl, bu, rhobeg, rhoend, monfun, maxcal, f, nf, iuser, ruser, ifail) |
Integer, Intent (In) |
:: |
n, npt, maxcal |
Integer, Intent (Inout) |
:: |
iuser(*), ifail |
Integer, Intent (Out) |
:: |
nf |
Real (Kind=nag_wp), Intent (In) |
:: |
bl(n), bu(n), rhobeg, rhoend |
Real (Kind=nag_wp), Intent (Inout) |
:: |
x(n), ruser(*) |
Real (Kind=nag_wp), Intent (Out) |
:: |
f |
External |
:: |
objfun, monfun |
|
C Header Interface
#include <nag.h>
void |
e04jcf_ ( void (NAG_CALL *objfun)(const Integer *n, const double x[], double *f, Integer iuser[], double ruser[], Integer *inform), const Integer *n, const Integer *npt, double x[], const double bl[], const double bu[], const double *rhobeg, const double *rhoend, void (NAG_CALL *monfun)(const Integer *n, const Integer *nf, const double x[], const double *f, const double *rho, Integer iuser[], double ruser[], Integer *inform), const Integer *maxcal, double *f, Integer *nf, Integer iuser[], double ruser[], Integer *ifail) |
|
C++ Header Interface
#include <nag.h> extern "C" {
void |
e04jcf_ ( void (NAG_CALL *objfun)(const Integer &n, const double x[], double &f, Integer iuser[], double ruser[], Integer &inform), const Integer &n, const Integer &npt, double x[], const double bl[], const double bu[], const double &rhobeg, const double &rhoend, void (NAG_CALL *monfun)(const Integer &n, const Integer &nf, const double x[], const double &f, const double &rho, Integer iuser[], double ruser[], Integer &inform), const Integer &maxcal, double &f, Integer &nf, Integer iuser[], double ruser[], Integer &ifail) |
}
|
The routine may be called by the names e04jcf or nagf_opt_bounds_bobyqa_func.
3
Description
e04jcf is applicable to problems of the form:
where
is a nonlinear scalar function whose derivatives may be unavailable, and where the bound vectors are elements of
. Relational operators between vectors are interpreted elementwise.
Fixing variables (that is, setting for some ) is allowed in e04jcf.
You must supply a subroutine to calculate the value of at any given point .
The method used by
e04jcf is based on BOBYQA, the method of Bound Optimization BY Quadratic Approximation described in
Powell (2009). In particular, each iteration of
e04jcf generates a quadratic approximation
to
that agrees with
at
automatically chosen interpolation points. The value of
is a constant prescribed by you. Updates to the independent variables mostly occur from approximate solutions to trust region subproblems, using the current quadratic model.
4
References
Powell M J D (2009) The BOBYQA algorithm for bound constrained optimization without derivatives
Report DAMTP 2009/NA06 University of Cambridge
https://www.damtp.cam.ac.uk/user/na/NA_papers/NA2009_06.pdf
5
Arguments
-
1:
– Subroutine, supplied by the user.
External Procedure
-
objfun must evaluate the objective function
at a specified vector
.
The specification of
objfun is:
Fortran Interface
Integer, Intent (In) |
:: |
n |
Integer, Intent (Inout) |
:: |
iuser(*) |
Integer, Intent (Out) |
:: |
inform |
Real (Kind=nag_wp), Intent (In) |
:: |
x(n) |
Real (Kind=nag_wp), Intent (Inout) |
:: |
ruser(*) |
Real (Kind=nag_wp), Intent (Out) |
:: |
f |
|
C Header Interface
void |
objfun_ (const Integer *n, const double x[], double *f, Integer iuser[], double ruser[], Integer *inform) |
|
C++ Header Interface
#include <nag.h> extern "C" {
void |
objfun_ (const Integer &n, const double x[], double &f, Integer iuser[], double ruser[], Integer &inform) |
}
|
-
1:
– Integer
Input
-
On entry: , the number of independent variables.
-
2:
– Real (Kind=nag_wp) array
Input
-
On entry: , the vector at which the objective function is to be evaluated.
-
3:
– Real (Kind=nag_wp)
Output
-
On exit: must be set to the value of the objective function at .
-
4:
– Integer array
User Workspace
-
5:
– Real (Kind=nag_wp) array
User Workspace
-
objfun is called with the arguments
iuser and
ruser as supplied to
e04jcf. You should use the arrays
iuser and
ruser to supply information to
objfun.
-
6:
– Integer
Output
-
On exit: must be set to a value describing the action to be taken by the solver on return from
objfun. Specifically, if the value is negative the solution of the current problem will terminate immediately; otherwise, computations will continue.
objfun must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which
e04jcf is called. Arguments denoted as
Input must
not be changed by this procedure.
Note: objfun should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by
e04jcf. If your code inadvertently
does return any NaNs or infinities,
e04jcf is likely to produce unexpected results.
-
2:
– Integer
Input
-
On entry: , the number of independent variables.
Constraint:
and , where denotes the number of non-fixed variables.
-
3:
– Integer
Input
-
On entry: , the number of interpolation conditions imposed on the quadratic approximation at each iteration.
Suggested value:
, where denotes the number of non-fixed variables.
Constraint:
, where denotes the number of non-fixed variables.
-
4:
– Real (Kind=nag_wp) array
Input/Output
-
On entry: an estimate of the position of the minimum. If any component is out-of-bounds it is replaced internally by the bound it violates.
On exit: the lowest point found during the calculations. Thus, if
on exit,
x is the position of the minimum.
-
5:
– Real (Kind=nag_wp) array
Input
-
6:
– Real (Kind=nag_wp) array
Input
-
On entry: the fixed vectors of bounds: the lower bounds
and the upper bounds
, respectively. To signify that a variable is unbounded you should choose a large scalar
appropriate to your problem, then set the lower bound on that variable to
and the upper bound to
. For well-scaled problems
may be suitable, where
denotes the largest positive model number (see
x02alf).
Constraints:
- if is to be fixed at , then;
- otherwise , for .
-
7:
– Real (Kind=nag_wp)
Input
-
On entry: an initial lower bound on the value of the trust region radius.
Suggested value:
rhobeg should be about one tenth of the greatest expected overall change to a variable: the initial quadratic model will be constructed by taking steps from the initial
x of length
rhobeg along each coordinate direction.
Constraints:
- ;
- .
-
8:
– Real (Kind=nag_wp)
Input
-
On entry: a final lower bound on the value of the trust region radius.
Suggested value:
rhoend should indicate the absolute accuracy that is required in the final values of the variables.
Constraint:
, where , the machine precision..
-
9:
– Subroutine, supplied by the NAG Library or the user.
External Procedure
-
monfun may be used to monitor the optimization process. It is invoked every time a new trust region radius is chosen.
If no monitoring is required,
monfun may be the dummy monitoring routine
e04jcp supplied by the NAG Library.
The specification of
monfun is:
Fortran Interface
Integer, Intent (In) |
:: |
n, nf |
Integer, Intent (Inout) |
:: |
iuser(*) |
Integer, Intent (Out) |
:: |
inform |
Real (Kind=nag_wp), Intent (In) |
:: |
x(n), f, rho |
Real (Kind=nag_wp), Intent (Inout) |
:: |
ruser(*) |
|
C Header Interface
void |
monfun_ (const Integer *n, const Integer *nf, const double x[], const double *f, const double *rho, Integer iuser[], double ruser[], Integer *inform) |
|
C++ Header Interface
#include <nag.h> extern "C" {
void |
monfun_ (const Integer &n, const Integer &nf, const double x[], const double &f, const double &rho, Integer iuser[], double ruser[], Integer &inform) |
}
|
-
1:
– Integer
Input
-
On entry: , the number of independent variables.
-
2:
– Integer
Input
-
On entry: the cumulative number of calls made to
objfun.
-
3:
– Real (Kind=nag_wp) array
Input
-
On entry: the current best point.
-
4:
– Real (Kind=nag_wp)
Input
-
On entry: the value of
objfun at
x.
-
5:
– Real (Kind=nag_wp)
Input
-
On entry: a lower bound on the current trust region radius.
-
6:
– Integer array
User Workspace
-
7:
– Real (Kind=nag_wp) array
User Workspace
-
monfun is called with the arguments
iuser and
ruser as supplied to
e04jcf. You should use the arrays
iuser and
ruser to supply information to
monfun.
-
8:
– Integer
Output
-
On exit: must be set to a value describing the action to be taken by the solver on return from
monfun. Specifically, if the value is negative the solution of the current problem will terminate immediately; otherwise, computations will continue.
monfun must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which
e04jcf is called. Arguments denoted as
Input must
not be changed by this procedure.
-
10:
– Integer
Input
-
On entry: the maximum permitted number of calls to
objfun.
Constraint:
.
-
11:
– Real (Kind=nag_wp)
Output
-
On exit: the function value at the lowest point found (
x).
-
12:
– Integer
Output
-
On exit: unless
or
on exit, the total number of calls made to
objfun.
-
13:
– Integer array
User Workspace
-
14:
– Real (Kind=nag_wp) array
User Workspace
-
iuser and
ruser are not used by
e04jcf, but are passed directly to
objfun and
monfun and may be used to pass information to these routines.
-
15:
– Integer
Input/Output
-
On entry:
ifail must be set to
,
. If you are unfamiliar with this argument you should refer to
Section 4 in the Introduction to the NAG Library FL Interface for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
is recommended. If the output of error messages is undesirable, then the value
is recommended. Otherwise, if you are not familiar with this argument, the recommended value is
.
When the value is used it is essential to test the value of ifail on exit.
On exit:
unless the routine detects an error or a warning has been flagged (see
Section 6).
e04jcf returns with
if the final trust region radius has reached its lower bound
rhoend.
6
Error Indicators and Warnings
If on entry
or
, explanatory error messages are output on the current error message unit (as defined by
x04aaf).
Errors or warnings detected by the routine:
-
On entry, .
Constraint: .
On entry, , , and .
Constraint: if in coordinate , .
On entry, .
Constraint: .
On entry, and .
Constraint: .
On entry, .
Constraint: , where , the machine precision.
There were unequal bounds.
Constraint: .
There were unequal bounds and on entry.
Constraint: .
-
The function evaluations limit was reached:
objfun has been called
maxcal times.
-
The predicted reduction in a trust region step was non-positive. Check your specification of
objfun and whether the function needs rescaling. Try a different initial
x.
-
A rescue procedure has been called in order to correct damage from rounding errors when computing an update to a quadratic approximation of
, but no further progess could be made. Check your specification of
objfun and whether the function needs rescaling. Try a different initial
x.
-
User-supplied monitoring routine requested termination.
User-supplied objective function requested termination.
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See
Section 9 in the Introduction to the NAG Library FL Interface for further information.
7
Accuracy
Experience shows that, in many cases, on successful termination the
-norm distance from the best point
to a local minimum of
is less than
, unless
rhoend is so small that such accuracy is unattainable.
8
Parallelism and Performance
e04jcf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the
Users' Note for your implementation for any additional implementation-specific information.
For each invocation of
e04jcf, local workspace arrays of fixed length are allocated internally. The total size of these arrays amounts to
real elements and
integer elements, where
denotes the number of non-fixed variables; that is, the total size is
. If you follow the recommendation for the choice of
npt on entry, this total size reduces to
.
Usually the total number of function evaluations (
nf) is substantially less than
, and often, if
on entry,
nf is only of magnitude
or less.
10
Example
This example involves the minimization of
subject to
starting from the initial guess
.
10.1
Program Text
10.2
Program Data
None.
10.3
Program Results