NAG FL Interface
e04lyf (bounds_​mod_​deriv2_​easy)

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1 Purpose

e04lyf is an easy-to-use modified-Newton algorithm for finding a minimum of a function, F(x1,x2,,xn) subject to fixed upper and lower bounds on the independent variables, x1,x2,,xn when first and second derivatives of F are available. It is intended for functions which are continuous and which have continuous first and second derivatives (although it will usually work even if the derivatives have occasional discontinuities).

2 Specification

Fortran Interface
Subroutine e04lyf ( n, ibound, funct2, hess2, bl, bu, x, f, g, iw, liw, w, lw, iuser, ruser, ifail)
Integer, Intent (In) :: n, ibound, liw, lw
Integer, Intent (Inout) :: iuser(*), ifail
Integer, Intent (Out) :: iw(liw)
Real (Kind=nag_wp), Intent (Inout) :: bl(n), bu(n), x(n), ruser(*)
Real (Kind=nag_wp), Intent (Out) :: f, g(n), w(lw)
External :: funct2, hess2
C Header Interface
#include <nag.h>
void  e04lyf_ (const Integer *n, const Integer *ibound,
void (NAG_CALL *funct2)(const Integer *n, const double xc[], double *fc, double gc[], Integer iuser[], double ruser[]),
void (NAG_CALL *hess2)(const Integer *n, const double xc[], double heslc[], const Integer *lh, double hesdc[], Integer iuser[], double ruser[]),
double bl[], double bu[], double x[], double *f, double g[], Integer iw[], const Integer *liw, double w[], const Integer *lw, Integer iuser[], double ruser[], Integer *ifail)
The routine may be called by the names e04lyf or nagf_opt_bounds_mod_deriv2_easy.

3 Description

e04lyf is applicable to problems of the form:
MinimizeF(x1,x2,,xn)  subject to  ljxjuj,  j=1,2,,n  
when first and second derivatives of F(x) are available.
Special provision is made for problems which actually have no bounds on the xj, problems which have only non-negativity bounds and problems in which l1=l2==ln and u1=u2==un. You must supply a subroutine to calculate the values of F(x) and its first derivatives at any point x and a subroutine to calculate the second derivatives.
From a starting point you supplied there is generated, on the basis of estimates of the curvature of F(x), a sequence of feasible points which is intended to converge to a local minimum of the constrained function.

4 References

Gill P E and Murray W (1976) Minimization subject to bounds on the variables NPL Report NAC 72 National Physical Laboratory

5 Arguments

1: n Integer Input
On entry: the number n of independent variables.
Constraint: n1.
2: ibound Integer Input
On entry: indicates whether the facility for dealing with bounds of special forms is to be used. It must be set to one of the following values:
ibound=0
If you are supplying all the lj and uj individually.
ibound=1
If there are no bounds on any xj.
ibound=2
If all the bounds are of the form 0xj.
ibound=3
If l1=l2==ln and u1=u2==un.
Constraint: 0ibound3.
3: funct2 Subroutine, supplied by the user. External Procedure
You must supply this routine to calculate the values of the function F(x) and its first derivatives F xj at any point x. It should be tested separately before being used in conjunction with e04lyf (see the E04 Chapter Introduction).
The specification of funct2 is:
Fortran Interface
Subroutine funct2 ( n, xc, fc, gc, iuser, ruser)
Integer, Intent (In) :: n
Integer, Intent (Inout) :: iuser(*)
Real (Kind=nag_wp), Intent (In) :: xc(n)
Real (Kind=nag_wp), Intent (Inout) :: ruser(*)
Real (Kind=nag_wp), Intent (Out) :: fc, gc(n)
C Header Interface
void  funct2 (const Integer *n, const double xc[], double *fc, double gc[], Integer iuser[], double ruser[])
1: n Integer Input
On entry: the number n of variables.
2: xc(n) Real (Kind=nag_wp) array Input
On entry: the point x at which the function and its derivatives are required.
3: fc Real (Kind=nag_wp) Output
On exit: the value of the function F at the current point x.
4: gc(n) Real (Kind=nag_wp) array Output
On exit: gc(j) must be set to the value of the first derivative F xj at the point x, for j=1,2,,n.
5: iuser(*) Integer array User Workspace
6: ruser(*) Real (Kind=nag_wp) array User Workspace
funct2 is called with the arguments iuser and ruser as supplied to e04lyf. You should use the arrays iuser and ruser to supply information to funct2.
funct2 must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which e04lyf is called. Arguments denoted as Input must not be changed by this procedure.
Note: funct2 should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by e04lyf. If your code inadvertently does return any NaNs or infinities, e04lyf is likely to produce unexpected results.
4: hess2 Subroutine, supplied by the user. External Procedure
You must supply this routine to evaluate the elements Hij= 2F xixj of the matrix of second derivatives of F(x) at any point x. It should be tested separately before being used in conjunction with e04lyf (see the E04 Chapter Introduction).
The specification of hess2 is:
Fortran Interface
Subroutine hess2 ( n, xc, heslc, lh, hesdc, iuser, ruser)
Integer, Intent (In) :: n, lh
Integer, Intent (Inout) :: iuser(*)
Real (Kind=nag_wp), Intent (In) :: xc(n)
Real (Kind=nag_wp), Intent (Inout) :: ruser(*)
Real (Kind=nag_wp), Intent (Out) :: heslc(lh), hesdc(n)
C Header Interface
void  hess2 (const Integer *n, const double xc[], double heslc[], const Integer *lh, double hesdc[], Integer iuser[], double ruser[])
1: n Integer Input
On entry: the number n of variables.
2: xc(n) Real (Kind=nag_wp) array Input
On entry: the point x at which the derivatives are required.
3: heslc(lh) Real (Kind=nag_wp) array Output
On exit: hess2 must place the strict lower triangle of the second derivative matrix H in heslc, stored by rows, i.e., set heslc((i-1)(i-2)/2+j)= 2F xixj , for i=2,3,,n and j=1,2,,i-1. (The upper triangle is not required because the matrix is symmetric.)
4: lh Integer Input
On entry: the length of the array heslc.
5: hesdc(n) Real (Kind=nag_wp) array Output
On exit: must contain the diagonal elements of the second derivative matrix, i.e., set hesdc(j)= 2F xj2 , for j=1,2,,n.
6: iuser(*) Integer array User Workspace
7: ruser(*) Real (Kind=nag_wp) array User Workspace
hess2 is called with the arguments iuser and ruser as supplied to e04lyf. You should use the arrays iuser and ruser to supply information to hess2.
hess2 must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which e04lyf is called. Arguments denoted as Input must not be changed by this procedure.
Note: hess2 should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by e04lyf. If your code inadvertently does return any NaNs or infinities, e04lyf is likely to produce unexpected results.
5: bl(n) Real (Kind=nag_wp) array Input/Output
On entry: the lower bounds lj.
If ibound is set to 0, bl(j) must be set to lj, for j=1,2,,n. (If a lower bound is not specified for any xj, the corresponding bl(j) should be set to -106.)
If ibound is set to 3, you must set bl(1) to l1; e04lyf will then set the remaining elements of bl equal to bl(1).
On exit: the lower bounds actually used by e04lyf.
6: bu(n) Real (Kind=nag_wp) array Input/Output
On entry: the upper bounds uj.
If ibound is set to 0, bu(j) must be set to uj, for j=1,2,,n. (If an upper bound is not specified for any xj the corresponding bu(j) should be set to 106.)
If ibound is set to 3, you must set bu(1) to u1; e04lyf will then set the remaining elements of bu equal to bu(1).
On exit: the upper bounds actually used by e04lyf.
7: x(n) Real (Kind=nag_wp) array Input/Output
On entry: x(j) must be set to a guess at the jth component of the position of the minimum, for j=1,2,,n. The routine checks the gradient and the Hessian matrix at the starting point, and is more likely to detect any error in your programming if the initial x(j) are nonzero and mutually distinct.
On exit: the lowest point found during the calculations. Thus, if ifail=0 on exit, x(j) is the jth component of the position of the minimum.
8: f Real (Kind=nag_wp) Output
On exit: the value of F(x) corresponding to the final point stored in x.
9: g(n) Real (Kind=nag_wp) array Output
On exit: the value of F xj corresponding to the final point stored in x, for j=1,2,,n; the value of g(j) for variables not on a bound should normally be close to zero.
10: iw(liw) Integer array Workspace
11: liw Integer Input
On entry: the dimension of the array iw as declared in the (sub)program from which e04lyf is called.
Constraint: liwn+2.
12: w(lw) Real (Kind=nag_wp) array Workspace
13: lw Integer Input
On entry: the dimension of the array w as declared in the (sub)program from which e04lyf is called.
Constraint: lwmax(n×(n+7),10).
14: iuser(*) Integer array User Workspace
15: ruser(*) Real (Kind=nag_wp) array User Workspace
iuser and ruser are not used by e04lyf, but are passed directly to funct2 and hess2 and may be used to pass information to these routines.
16: ifail Integer Input/Output
On entry: ifail must be set to 0, -1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of -1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value -1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value -1 is recommended since useful values can be provided in some output arguments even when ifail0 on exit. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry ifail=0 or -1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
Note: in some cases e04lyf may return useful information.
If you are not satisfied with the result (e.g., because ifail=5, 6, 7 or 8), it is worth restarting the calculations from a different starting point (not the point at which the failure occurred) in order to avoid the region which caused the failure.
ifail=1
On entry, ibound=value.
Constraint: 0ibound3.
On entry, ibound=0 and bl(j)>bu(j) for some j.
On entry, ibound=3 and bl(1)>bu(1).
On entry, liw=value.
Constraint: liwvalue.
On entry, lw=value.
Constraint: lwvalue.
On entry, n=value.
Constraint: n1.
ifail=2
There have been 50×n function evaluations.
The algorithm does not seem to be converging. The calculations can be restarted from the final point held in x. The error may also indicate that F(x) has no minimum.
ifail=3
The conditions for a minimum have not all been satisfied, but a lower point could not be found. See Section 7 for further information.
ifail=5
It is probable that a local minimum has been found, but it cannot be guaranteed.
ifail=6
It is possible that a local minimum has been found, but it cannot be guaranteed.
ifail=7
It is unlikely that a local minimum has been found.
ifail=8
It is very unlikely that a local minimum has been found.
ifail=9
The modulus of a variable has become very large. There may be a mistake in your supplied routines, your problem has no finite solution, or the problem needs rescaling.
ifail=10
It is very likely that you have made an error forming the gradient.
ifail=11
It is very likely that you have made an error forming the 2nd derivatives.
ifail=-99
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.
ifail=-399
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.
ifail=-999
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

When a successful exit is made then, for a computer with a mantissa of t decimals, one would expect to get about t/2-1 decimals accuracy in x, and about t-1 decimals accuracy in F, provided the problem is reasonably well scaled.

8 Parallelism and Performance

e04lyf 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.

9 Further Comments

The number of iterations required depends on the number of variables, the behaviour of F(x) and the distance of the starting point from the solution. The number of operations performed in an iteration of e04lyf is roughly proportional to n3+O(n2). In addition, each iteration makes one call of hess2 and at least one call of funct2. So, unless F(x), the gradient vector and the matrix of second derivatives can be evaluated very quickly, the run time will be dominated by the time spent in user-supplied subroutines funct2 and hess2.
Ideally the problem should be scaled so that at the solution the value of F(x) and the corresponding values of x1,x2,xn are each in the range (-1,+1), and so that at points a unit distance away from the solution, F is approximately a unit value greater than at the minimum. It is unlikely that you will be able to follow these recommendations very closely, but it is worth trying (by guesswork), as sensible scaling will reduce the difficulty of the minimization problem, so that e04lyf will take less computer time.

10 Example

A program to minimize
F= (x1+10x2) 2+5 (x3-x4) 2+ (x2-2x3) 4+10 (x1-x4) 4  
subject to
1 x1 3 -2 x2 0 1 x4 3  
starting from the initial guess (3,-1,0,1) . (In practice, it is worth trying to make user-supplied subroutines funct2 and hess2 as efficient as possible. This has not been done in the example program for reasons of clarity.)

10.1 Program Text

Program Text (e04lyfe.f90)

10.2 Program Data

None.

10.3 Program Results

Program Results (e04lyfe.r)