# NAG Library Routine Document

## 1Purpose

e04lyf is an easy-to-use modified-Newton algorithm for finding a minimum of a function, $F\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)$ subject to fixed upper and lower bounds on the independent variables, ${x}_{1},{x}_{2},\dots ,{x}_{n}$ 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).

## 2Specification

Fortran Interface
 Subroutine e04lyf ( n, bl, bu, x, f, g, iw, liw, w, lw,
 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
#include nagmk26.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)

## 3Description

e04lyf is applicable to problems of the form:
 $Minimize⁡Fx1,x2,…,xn subject to lj≤xj≤uj, j=1,2,…,n$
when first and second derivatives of $F\left(x\right)$ are available.
Special provision is made for problems which actually have no bounds on the ${x}_{j}$, problems which have only non-negativity bounds and problems in which ${l}_{1}={l}_{2}=\cdots ={l}_{n}$ and ${u}_{1}={u}_{2}=\cdots ={u}_{n}$. You must supply a subroutine to calculate the values of $F\left(x\right)$ 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 $F\left(x\right)$, a sequence of feasible points which is intended to converge to a local minimum of the constrained function.
Gill P E and Murray W (1976) Minimization subject to bounds on the variables NPL Report NAC 72 National Physical Laboratory

## 5Arguments

1:     $\mathbf{n}$ – IntegerInput
On entry: the number $n$ of independent variables.
Constraint: ${\mathbf{n}}\ge 1$.
2:     $\mathbf{ibound}$ – 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:
${\mathbf{ibound}}=0$
If you are supplying all the ${l}_{j}$ and ${u}_{j}$ individually.
${\mathbf{ibound}}=1$
If there are no bounds on any ${x}_{j}$.
${\mathbf{ibound}}=2$
If all the bounds are of the form $0\le {x}_{j}$.
${\mathbf{ibound}}=3$
If ${l}_{1}={l}_{2}=\cdots ={l}_{n}$ and ${u}_{1}={u}_{2}=\cdots ={u}_{n}$.
Constraint: $0\le {\mathbf{ibound}}\le 3$.
3:     $\mathbf{funct2}$ – Subroutine, supplied by the user.External Procedure
You must supply this routine to calculate the values of the function $F\left(x\right)$ and its first derivatives $\frac{\partial F}{\partial {x}_{j}}$ 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,
 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)
#include nagmk26.h
 void funct2 (const Integer *n, const double xc[], double *fc, double gc[], Integer iuser[], double ruser[])
1:     $\mathbf{n}$ – IntegerInput
On entry: the number $n$ of variables.
2:     $\mathbf{xc}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the point $x$ at which the function and its derivatives are required.
3:     $\mathbf{fc}$ – Real (Kind=nag_wp)Output
On exit: the value of the function $F$ at the current point $x$.
4:     $\mathbf{gc}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{gc}}\left(\mathit{j}\right)$ must be set to the value of the first derivative $\frac{\partial F}{\partial {x}_{\mathit{j}}}$ at the point $x$, for $\mathit{j}=1,2,\dots ,n$.
5:     $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace
6:     $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser 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:     $\mathbf{hess2}$ – Subroutine, supplied by the user.External Procedure
You must supply this routine to evaluate the elements ${H}_{ij}=\frac{{\partial }^{2}F}{\partial {x}_{i}\partial {x}_{j}}$ of the matrix of second derivatives of $F\left(x\right)$ 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, lh,
 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)
#include nagmk26.h
 void hess2 (const Integer *n, const double xc[], double heslc[], const Integer *lh, double hesdc[], Integer iuser[], double ruser[])
1:     $\mathbf{n}$ – IntegerInput
On entry: the number $n$ of variables.
2:     $\mathbf{xc}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the point $x$ at which the derivatives are required.
3:     $\mathbf{heslc}\left({\mathbf{lh}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: hess2 must place the strict lower triangle of the second derivative matrix $H$ in heslc, stored by rows, i.e., set ${\mathbf{heslc}}\left(\left(\mathit{i}-1\right)\left(\mathit{i}-2\right)/2+\mathit{j}\right)=\frac{{\partial }^{2}F}{\partial {x}_{\mathit{i}}\partial {x}_{\mathit{j}}}$, for $\mathit{i}=2,3,\dots ,n$ and $\mathit{j}=1,2,\dots ,\mathit{i}-1$. (The upper triangle is not required because the matrix is symmetric.)
4:     $\mathbf{lh}$ – IntegerInput
On entry: the length of the array heslc.
5:     $\mathbf{hesdc}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: must contain the diagonal elements of the second derivative matrix, i.e., set ${\mathbf{hesdc}}\left(\mathit{j}\right)=\frac{{\partial }^{2}F}{\partial {x}_{\mathit{j}}^{2}}$, for $\mathit{j}=1,2,\dots ,n$.
6:     $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace
7:     $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser 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:     $\mathbf{bl}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: the lower bounds ${l}_{j}$.
If ibound is set to $0$, ${\mathbf{bl}}\left(\mathit{j}\right)$ must be set to ${l}_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,n$. (If a lower bound is not specified for any ${x}_{j}$, the corresponding ${\mathbf{bl}}\left(j\right)$ should be set to $-{10}^{6}$.)
If ibound is set to $3$, you must set ${\mathbf{bl}}\left(1\right)$ to ${l}_{1}$; e04lyf will then set the remaining elements of bl equal to ${\mathbf{bl}}\left(1\right)$.
On exit: the lower bounds actually used by e04lyf.
6:     $\mathbf{bu}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: the upper bounds ${u}_{j}$.
If ibound is set to $0$, ${\mathbf{bu}}\left(\mathit{j}\right)$ must be set to ${u}_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,n$. (If an upper bound is not specified for any ${x}_{j}$ the corresponding ${\mathbf{bu}}\left(j\right)$ should be set to ${10}^{6}$.)
If ibound is set to $3$, you must set ${\mathbf{bu}}\left(1\right)$ to ${u}_{1}$; e04lyf will then set the remaining elements of bu equal to ${\mathbf{bu}}\left(1\right)$.
On exit: the upper bounds actually used by e04lyf.
7:     $\mathbf{x}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: ${\mathbf{x}}\left(\mathit{j}\right)$ must be set to a guess at the $\mathit{j}$th component of the position of the minimum, for $\mathit{j}=1,2,\dots ,n$. 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 ${\mathbf{x}}\left(j\right)$ are nonzero and mutually distinct.
On exit: the lowest point found during the calculations. Thus, if ${\mathbf{ifail}}={\mathbf{0}}$ on exit, ${\mathbf{x}}\left(j\right)$ is the $j$th component of the position of the minimum.
8:     $\mathbf{f}$ – Real (Kind=nag_wp)Output
On exit: the value of $F\left(x\right)$ corresponding to the final point stored in x.
9:     $\mathbf{g}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the value of $\frac{\partial F}{\partial {x}_{\mathit{j}}}$ corresponding to the final point stored in x, for $\mathit{j}=1,2,\dots ,n$; the value of ${\mathbf{g}}\left(j\right)$ for variables not on a bound should normally be close to zero.
10:   $\mathbf{iw}\left({\mathbf{liw}}\right)$ – Integer arrayWorkspace
11:   $\mathbf{liw}$ – IntegerInput
On entry: the dimension of the array iw as declared in the (sub)program from which e04lyf is called.
Constraint: ${\mathbf{liw}}\ge {\mathbf{n}}+2$.
12:   $\mathbf{w}\left({\mathbf{lw}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
13:   $\mathbf{lw}$ – IntegerInput
On entry: the dimension of the array w as declared in the (sub)program from which e04lyf is called.
Constraint: ${\mathbf{lw}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{n}}×\left({\mathbf{n}}+7\right),10\right)$.
14:   $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace
15:   $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser 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:   $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ 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 arguments may be useful even if ${\mathbf{ifail}}\ne {\mathbf{0}}$ on exit, the recommended value is $-1$. When the value $-\mathbf{1}\text{​ or ​}1$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ 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:
${\mathbf{ifail}}=1$
 On entry, ${\mathbf{n}}<1$, or ${\mathbf{ibound}}<0$, or ${\mathbf{ibound}}>3$, or ${\mathbf{ibound}}=0$ and ${\mathbf{bl}}\left(j\right)>{\mathbf{bu}}\left(j\right)$ for some $j$, or ${\mathbf{ibound}}=3$ and ${\mathbf{bl}}\left(1\right)>{\mathbf{bu}}\left(1\right)$, or ${\mathbf{liw}}<{\mathbf{n}}+2$, or ${\mathbf{lw}}<\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(10,{\mathbf{n}}×\left({\mathbf{n}}+7\right)\right)$.
${\mathbf{ifail}}=2$
There have been $50×{\mathbf{n}}$ 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 $F\left(x\right)$ has no minimum.
${\mathbf{ifail}}=3$
The conditions for a minimum have not all been met but a lower point could not be found and the algorithm has failed.
${\mathbf{ifail}}=4$
Not used. (This value of the argument is included so as to make the significance of ${\mathbf{ifail}}={\mathbf{5}}$ etc. consistent in the easy-to-use routines.)
${\mathbf{ifail}}=5$
${\mathbf{ifail}}=6$
${\mathbf{ifail}}=7$
${\mathbf{ifail}}=8$
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 ${\mathbf{ifail}}={\mathbf{5}}$ it is probable that the final $x$ gives a good estimate of the position of a minimum, but when ${\mathbf{ifail}}={\mathbf{8}}$ it is very unlikely that the routine has found a minimum.
${\mathbf{ifail}}=9$
In the search for a minimum, the modulus of one of the variables has become very large $\left(\sim {10}^{6}\right)$. 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 9).
${\mathbf{ifail}}=10$
It is very likely that you have made an error in forming the gradient.
${\mathbf{ifail}}=11$
It is very likely that you have made an error in forming the second derivatives.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.
If you are dissatisfied with the result (e.g., because ${\mathbf{ifail}}={\mathbf{5}}$, ${\mathbf{6}}$, ${\mathbf{7}}$ or ${\mathbf{8}}$), 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.

## 7Accuracy

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.

## 8Parallelism 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.

The number of iterations required depends on the number of variables, the behaviour of $F\left(x\right)$ and the distance of the starting point from the solution. The number of operations performed in an iteration of e04lyf is roughly proportional to ${n}^{3}+\mathit{O}\left({n}^{2}\right)$. In addition, each iteration makes one call of hess2 and at least one call of funct2. So, unless $F\left(x\right)$, 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 $F\left(x\right)$ and the corresponding values of ${x}_{1},{x}_{2},\dots {x}_{n}$ are each in the range $\left(-1,+1\right)$, 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.

## 10Example

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 $\left(3,-1,0,1\right)$. (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.)

### 10.1Program Text

Program Text (e04lyfe.f90)

None.

### 10.3Program Results

Program Results (e04lyfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017