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 .
The routine may be called by the names e04jcf or nagf_opt_bounds_bobyqa_func.
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.
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: – IntegerInput
On entry: , the number of independent variables.
and , where denotes the number of non-fixed variables.
3: – IntegerInput
On entry: , the number of interpolation conditions imposed on the quadratic approximation at each iteration.
, where denotes the number of non-fixed variables.
, where denotes the number of non-fixed variables.
4: – Real (Kind=nag_wp) arrayInput/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) arrayInput
6: – Real (Kind=nag_wp) arrayInput
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).
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.
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.
8: – Real (Kind=nag_wp)Input
On entry: a final lower bound on the value of the trust region radius.
rhoend should indicate the absolute accuracy that is required in the final values of the variables.
, 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.
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: – IntegerInput
On entry: the maximum permitted number of calls to objfun.
11: – Real (Kind=nag_wp)Output
On exit: the function value at the lowest point found (x).
12: – IntegerOutput
On exit: unless or on exit, the total number of calls made to objfun.
13: – Integer arrayUser Workspace
14: – Real (Kind=nag_wp) arrayUser 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: – IntegerInput/Output
On entry: ifail must be set to , or to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of means that an error message is printed while a value of means that it is not.
If halting is not appropriate, the value or is recommended. If message printing is undesirable, then the value is recommended. Otherwise, the value is recommended. When the value or 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.
6Error 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, .
On entry, , , and .
Constraint: if in coordinate , .
On entry, .
On entry, and .
On entry, .
Constraint: , where , the machine precision.
There were unequal bounds.
There were unequal bounds and on entry.
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 objective function requested termination.
An unexpected error has been triggered by this routine. Please
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.
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.
8Parallelism 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.