NAG Library Routine Document
E04LYF
1 Purpose
E04LYF is an easy-to-use modified-Newton algorithm for finding a minimum of a function, Fx1,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
| SUBROUTINE E04LYF ( | N, IBOUND, FUNCT2, HESS2, BL, BU, X, F, G, IW, LIW, W, LW, IUSER, RUSER, IFAIL) |
| INTEGER | N, IBOUND, IW(LIW), LIW, LW, IUSER(*), IFAIL |
| double precision | BL(N), BU(N), X(N), F, G(N), W(LW), RUSER(*) |
| EXTERNAL | FUNCT2, HESS2 |
3 Description
E04LYF is applicable to problems of the form:
when first and second derivatives of
Fx 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 Fx 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 Fx, 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 Parameters
- 1: N – INTEGERInput
-
On entry:
the number n of independent variables.
Constraint:
N≥1.
- 2: IBOUND – INTEGERInput
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 0≤xj.
- IBOUND=3
- If l1=l2=⋯=ln and u1=u2=⋯=un.
Constraint:
0≤IBOUND≤3.
- 3: FUNCT2 – SUBROUTINE, supplied by the user.External Procedure
You must supply this routine to calculate the values of the function
Fx 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:
| SUBROUTINE FUNCT2 ( | N, XC, FC, GC, IUSER, RUSER) |
| INTEGER | N, IUSER(*) |
| double precision | XC(N), FC, GC(N), RUSER(*) |
- 1: N – INTEGERInput
-
On entry: the number n of variables.
- 2: XC(N) – double precision arrayInput
On entry: the point x at which the function and its derivatives are required.
- 3: FC – double precisionOutput
-
On exit: the value of the function F at the current point x.
- 4: GC(N) – double precision arrayOutput
On exit: GCj must be set to the value of the first derivative
∂F
∂xj
at the point x, for j=1,2,…,n.
- 5: IUSER(*) – INTEGER arrayUser Workspace
- 6: RUSER(*) – double precision arrayUser Workspace
FUNCT2 is called from E04LYF with the parameters
IUSER and
RUSER as supplied to E04LYF. You are free to use the arrays
IUSER and
RUSER to supply information to
FUNCT2 as an alternative to using
COMMON.
FUNCT2 must be declared as EXTERNAL in the (sub)program from which E04LYF is called. Parameters denoted as
Input must
not be changed by this procedure.
- 4: HESS2 – SUBROUTINE, supplied by the user.External Procedure
You must supply this routine to evaluate the elements
Hij=
∂2F
∂xi∂xj
of the matrix of second derivatives of
Fx 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:
| SUBROUTINE HESS2 ( | N, XC, HESLC, LH, HESDC, IUSER, RUSER) |
| INTEGER | N, LH, IUSER(*) |
| double precision | XC(N), HESLC(LH), HESDC(N), RUSER(*) |
- 1: N – INTEGERInput
-
On entry: the number n of variables.
- 2: XC(N) – double precision arrayInput
On entry: the point x at which the derivatives are required.
- 3: HESLC(LH) – double precision arrayOutput
On exit:
HESS2 must place the strict lower triangle of the second derivative matrix
H in
HESLC, stored by rows, i.e., set
HESLCi-1i-2/2+j=
∂2F
∂xi∂xj
, for
i=2,3,…,n and
j=1,2,…,i-1. (The upper triangle is not required because the matrix is symmetric.)
- 4: LH – INTEGERInput
-
On entry: the length of the array
HESLC.
- 5: HESDC(N) – double precision arrayOutput
On exit: must contain the diagonal elements of the second derivative matrix, i.e., set HESDCj=
∂2F
∂xj2
, for j=1,2,…,n.
- 6: IUSER(*) – INTEGER arrayUser Workspace
- 7: RUSER(*) – double precision arrayUser Workspace
HESS2 is called from E04LYF with the parameters
IUSER and
RUSER as supplied to E04LYF. You are free to use the arrays
IUSER and
RUSER to supply information to
HESS2 as an alternative to using
COMMON.
HESS2 must be declared as EXTERNAL in the (sub)program from which E04LYF is called. Parameters denoted as
Input must
not be changed by this procedure.
- 5: BL(N) – double precision arrayInput/Output
On entry: the lower bounds
lj.
If
IBOUND is set to
0,
BLj must be set to
lj, for
j=1,2,…,n. (If a lower bound is not specified for any
xj, the corresponding
BLj should be set to
-106.)
If
IBOUND is set to
3, you must set
BL1 to
l1; E04LYF will then set the remaining elements of
BL equal to
BL1.
On exit: the lower bounds actually used by E04LYF.
- 6: BU(N) – double precision arrayInput/Output
On entry: the upper bounds
uj.
If
IBOUND is set to
0,
BUj must be set to
uj, for
j=1,2,…,n. (If an upper bound is not specified for any
xj the corresponding
BUj should be set to
106.)
If
IBOUND is set to
3, you must set
BU1 to
u1; E04LYF will then set the remaining elements of
BU equal to
BU1.
On exit: the upper bounds actually used by E04LYF.
- 7: X(N) – double precision arrayInput/Output
On entry: Xj 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 Xj are nonzero and mutually distinct.
On exit: the lowest point found during the calculations. Thus, if IFAIL=0 on exit, Xj is the jth component of the position of the minimum.
- 8: F – double precisionOutput
-
On exit: the value of
Fx corresponding to the final point stored in
X.
- 9: G(N) – double precision arrayOutput
-
On exit: the value of
∂F
∂xj
corresponding to the final point stored in
X, for
j=1,2,…,n; the value of
Gj for variables not on a bound should normally be close to zero.
- 10: IW(LIW) – INTEGER arrayWorkspace
- 11: LIW – INTEGERInput
On entry: the dimension of the array
IW as declared in the (sub)program from which E04LYF is called.
Constraint:
LIW≥N+2.
- 12: W(LW) – double precision arrayWorkspace
- 13: LW – INTEGERInput
On entry: the dimension of the array
W as declared in the (sub)program from which E04LYF is called.
Constraint:
LW≥maxN×N+7,10.
- 14: IUSER(*) – INTEGER arrayUser Workspace
Note: the dimension of the array
IUSER
must be at least
1.
IUSER is not used by E04LYF, but is passed directly to
FUNCT2 and
HESS2 and may be used to pass information to those routines.
- 15: RUSER(*) – double precision arrayUser Workspace
Note: the dimension of the array
RUSER
must be at least
1.
RUSER is not used by E04LYF, but is passed directly to
FUNCT2 and
HESS2 and may be used to pass information to those routines.
- 16: IFAIL – INTEGERInput/Output
-
On entry:
IFAIL must be set to
0,
-1 or 1. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction for details.
On exit:
IFAIL=0 unless the routine detects an error (see
Section 6).
For environments where it might be inappropriate to halt program execution when an error is detected, the value
-1 or 1 is recommended. If the output of error messages is undesirable, then the value
1 is recommended. Otherwise, because for this routine the values of the output parameters may be useful even if
IFAIL≠0 on exit, the recommended value is
-1.
When the value -1 or 1 is used it is essential to test the value of IFAIL on exit.
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).
Note: E04LYF may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the routine:
- IFAIL=1
-
| On entry, | N<1, |
| or | IBOUND<0, |
| or | IBOUND>3, |
| or | IBOUND=0 and BLj>BUj for some j, |
| or | IBOUND=3 and BL1>BU1, |
| or | LIW<N+2, |
| or | LW<max10,N×N+7. |
- IFAIL=2
-
There have been
50×N function evaluations, yet 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
Fx has no minimum.
- IFAIL=3
The conditions for a minimum have not all been met but a lower point could not be found and the algorithm has failed.
- IFAIL=4
-
Not used. (This value of the parameter is included so as to make the significance of IFAIL=5 etc. consistent in the easy-to-use routines.)
- IFAIL=5
- IFAIL=6
- IFAIL=7
- IFAIL=8
-
There is some doubt about whether the point
x found by E04LYF is a minimum. The degree of confidence in the result decreases as
IFAIL increases. Thus, when
IFAIL=5 it is probable that the final
x gives a good estimate of the position of a minimum, but when
IFAIL=8 it is very unlikely that the routine has found a minimum.
- IFAIL=9
-
In the search for a minimum, the modulus of one of the variables has become very large
∼106. This indicates that there is a mistake in user-supplied subroutines
FUNCT2 or
HESS2, that your problem has no finite solution, or that the problem needs rescaling (see
Section 8).
- IFAIL=10
It is very likely that you have made an error in forming the gradient.
- IFAIL=11
-
It is very likely that you have made an error in forming the second derivatives.
If you are dissatisfied 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.
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 Further Comments
The number of iterations required depends on the number of variables, the behaviour of
Fx 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+On2. In addition, each iteration makes one call of
HESS2 and at least one call of
FUNCT2. So, unless
Fx, 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 Fx 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.
9 Example
A program to minimize
subject to
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.)
9.1 Program Text
Program Text (e04lyfe.f)
9.2 Program Data
None.
9.3 Program Results
Program Results (e04lyfe.r)