NAG CL Interface
e04jcc (bounds_bobyqa_func)
1
Purpose
e04jcc 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 function 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
void |
e04jcc (
Integer n,
Integer npt,
double x[],
const double bl[],
const double bu[],
double rhobeg,
double rhoend,
Integer maxcal,
double *f,
Integer *nf,
Nag_Comm *comm,
NagError *fail) |
|
The function may be called by the names: e04jcc, nag_opt_bounds_bobyqa_func or nag_opt_bounds_qa_no_deriv.
3
Description
e04jcc 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 e04jcc.
You must supply a function to calculate the value of at any given point .
The method used by
e04jcc is based on BOBYQA, the method of Bound Optimization BY Quadratic Approximation described in
Powell (2009). In particular, each iteration of
e04jcc 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:
– function, supplied by the user
External Function
-
objfun must evaluate the objective function
at a specified vector
.
The specification of
objfun is:
void |
objfun (Integer n,
const double x[],
double *f,
Nag_Comm *comm, Integer *inform)
|
|
-
1:
– Integer
Input
-
On entry: , the number of independent variables.
-
2:
– const double
Input
-
On entry: , the vector at which the objective function is to be evaluated.
-
3:
– double *
Output
-
On exit: must be set to the value of the objective function at .
-
4:
– Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to
objfun.
- user – double *
- iuser – Integer *
- p – Pointer
The type Pointer will be
void *. Before calling
e04jcc you may allocate memory and initialize these pointers with various quantities for use by
objfun when called from
e04jcc (see
Section 3.1.1 in the Introduction to the NAG Library CL Interface).
-
5:
– 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.
Note: objfun should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by
e04jcc. If your code inadvertently
does return any NaNs or infinities,
e04jcc 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:
– double
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
NE_NOERROR on exit,
x is the position of the minimum.
-
5:
– const double
Input
-
6:
– const double
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
X02ALC).
Constraints:
- if is to be fixed at , then;
- otherwise , for .
-
7:
– double
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:
– double
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:
– function, supplied by the user
External Function
-
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 specified as NULLFN.
The specification of
monfun is:
void |
monfun (Integer n,
Integer nf,
const double x[],
double f,
double rho,
Nag_Comm *comm, 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:
– const double
Input
-
On entry: the current best point.
-
4:
– double
Input
-
On entry: the value of
objfun at
x.
-
5:
– double
Input
-
On entry: a lower bound on the current trust region radius.
-
6:
– Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to
monfun.
- user – double *
- iuser – Integer *
- p – Pointer
The type Pointer will be
void *. Before calling
e04jcc you may allocate memory and initialize these pointers with various quantities for use by
monfun when called from
e04jcc (see
Section 3.1.1 in the Introduction to the NAG Library CL Interface).
-
7:
– 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.
-
10:
– Integer
Input
-
On entry: the maximum permitted number of calls to
objfun.
Constraint:
.
-
11:
– double *
Output
-
On exit: the function value at the lowest point found (
x).
-
12:
– Integer *
Output
-
On exit: unless
NE_RESCUE_FAILED,
NE_TOO_MANY_FEVALS,
NE_TR_STEP_FAILED or
NE_USER_STOP on exit, the total number of calls made to
objfun.
-
13:
– Nag_Comm *
-
The NAG communication argument (see
Section 3.1.1 in the Introduction to the NAG Library CL Interface).
-
14:
– NagError *
Input/Output
-
The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
e04jcc returns with
NE_NOERROR if the final trust region radius has reached its lower bound
rhoend.
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_BOUND
-
On entry, , , and .
Constraint: if in coordinate , .
- NE_INT
-
On entry, .
Constraint: .
There were unequal bounds.
Constraint: .
There were unequal bounds and on entry.
Constraint: .
- NE_INTERNAL_ERROR
-
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
NAG for assistance.
See
Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library CL Interface for further information.
- NE_REAL
-
On entry, .
Constraint: .
On entry, .
Constraint: , where , the machine precision.
- NE_REAL_2
-
On entry, and .
Constraint: .
- NE_RESCUE_FAILED
-
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.
- NE_TOO_MANY_FEVALS
-
The function evaluations limit was reached:
objfun has been called
maxcal times.
- NE_TR_STEP_FAILED
-
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.
- NE_USER_STOP
-
User-supplied monitoring function requested termination.
User-supplied objective function requested termination.
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
e04jcc 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 function. Please also consult the
Users' Note for your implementation for any additional implementation-specific information.
For each invocation of
e04jcc, local workspace arrays of fixed length are allocated internally. The total size of these arrays amounts to
double 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