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NAG Toolbox: nag_opt_bounds_mod_deriv2_easy (e04ly)
Purpose
nag_opt_bounds_mod_deriv2_easy (e04ly) is an easytouse modifiedNewton 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).
Syntax
[
bl,
bu,
x,
f,
g,
user,
ifail] = e04ly(
ibound,
funct2,
hess2,
bl,
bu,
x, 'n',
n, 'user',
user)
[
bl,
bu,
x,
f,
g,
user,
ifail] = nag_opt_bounds_mod_deriv2_easy(
ibound,
funct2,
hess2,
bl,
bu,
x, 'n',
n, 'user',
user)
Description
nag_opt_bounds_mod_deriv2_easy (e04ly) is applicable to problems of the form:
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 nonnegativity bounds and problems in which ${l}_{1}={l}_{2}=\cdots ={l}_{n}$ and ${u}_{1}={u}_{2}=\cdots ={u}_{n}$. You must supply a function to calculate the values of $F\left(x\right)$ and its first derivatives at any point $x$ and a function 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.
References
Gill P E and Murray W (1976) Minimization subject to bounds on the variables NPL Report NAC 72 National Physical Laboratory
Parameters
Compulsory Input Parameters
 1:
$\mathrm{ibound}$ – int64int32nag_int scalar

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:
$0\le {\mathbf{ibound}}\le 3$.
 2:
$\mathrm{funct2}$ – function handle or string containing name of mfile

You must supply this function 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
nag_opt_bounds_mod_deriv2_easy (e04ly) (see the
E04 Chapter Introduction).
[fc, gc, user] = funct2(n, xc, user)
Input Parameters
 1:
$\mathrm{n}$ – int64int32nag_int scalar

The number $n$ of variables.
 2:
$\mathrm{xc}\left({\mathbf{n}}\right)$ – double array

The point $x$ at which the function and its derivatives are required.
 3:
$\mathrm{user}$ – Any MATLAB object
funct2 is called from
nag_opt_bounds_mod_deriv2_easy (e04ly) with the object supplied to
nag_opt_bounds_mod_deriv2_easy (e04ly).
Output Parameters
 1:
$\mathrm{fc}$ – double scalar

The value of the function $F$ at the current point $x$.
 2:
$\mathrm{gc}\left({\mathbf{n}}\right)$ – double array

${\mathbf{gc}}\left(\mathit{j}\right)$ must be set to the value of the first derivative $\frac{\partial F}{\partial {x}_{\mathit{j}}}$ at the point $x$, for $\mathit{j}=1,2,\dots ,n$.
 3:
$\mathrm{user}$ – Any MATLAB object
 3:
$\mathrm{hess2}$ – function handle or string containing name of mfile

You must supply this function 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
nag_opt_bounds_mod_deriv2_easy (e04ly) (see the
E04 Chapter Introduction).
[heslc, hesdc, user] = hess2(n, xc, lh, user)
Input Parameters
 1:
$\mathrm{n}$ – int64int32nag_int scalar

The number $n$ of variables.
 2:
$\mathrm{xc}\left({\mathbf{n}}\right)$ – double array

The point $x$ at which the derivatives are required.
 3:
$\mathrm{lh}$ – int64int32nag_int scalar

The length of the array
heslc.
 4:
$\mathrm{user}$ – Any MATLAB object
hess2 is called from
nag_opt_bounds_mod_deriv2_easy (e04ly) with the object supplied to
nag_opt_bounds_mod_deriv2_easy (e04ly).
Output Parameters
 1:
$\mathrm{heslc}\left({\mathbf{lh}}\right)$ – double array

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.)
 2:
$\mathrm{hesdc}\left({\mathbf{n}}\right)$ – double array

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 $\mathit{j}=1,2,\dots ,n$.
 3:
$\mathrm{user}$ – Any MATLAB object
 4:
$\mathrm{bl}\left({\mathbf{n}}\right)$ – double array

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
${l}_{1}$;
nag_opt_bounds_mod_deriv2_easy (e04ly) will then set the remaining elements of
bl equal to
${\mathbf{bl}}\left(1\right)$.
 5:
$\mathrm{bu}\left({\mathbf{n}}\right)$ – double array

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
${u}_{1}$;
nag_opt_bounds_mod_deriv2_easy (e04ly) will then set the remaining elements of
bu equal to
${\mathbf{bu}}\left(1\right)$.
 6:
$\mathrm{x}\left({\mathbf{n}}\right)$ – double array

${\mathbf{x}}\left(\mathit{j}\right)$ 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 function 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.
Optional Input Parameters
 1:
$\mathrm{n}$ – int64int32nag_int scalar

Default:
the dimension of the arrays
bl,
bu,
x. (An error is raised if these dimensions are not equal.)
The number $n$ of independent variables.
Constraint:
${\mathbf{n}}\ge 1$.
 2:
$\mathrm{user}$ – Any MATLAB object
user is not used by
nag_opt_bounds_mod_deriv2_easy (e04ly), but is passed to
funct2 and
hess2. Note that for large objects it may be more efficient to use a global variable which is accessible from the mfiles than to use
user.
Output Parameters
 1:
$\mathrm{bl}\left({\mathbf{n}}\right)$ – double array

The lower bounds actually used by nag_opt_bounds_mod_deriv2_easy (e04ly).
 2:
$\mathrm{bu}\left({\mathbf{n}}\right)$ – double array

The upper bounds actually used by nag_opt_bounds_mod_deriv2_easy (e04ly).
 3:
$\mathrm{x}\left({\mathbf{n}}\right)$ – double array

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.
 4:
$\mathrm{f}$ – double scalar

The value of
$F\left(x\right)$ corresponding to the final point stored in
x.
 5:
$\mathrm{g}\left({\mathbf{n}}\right)$ – double array

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.
 6:
$\mathrm{user}$ – Any MATLAB object
 7:
$\mathrm{ifail}$ – int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see
Error Indicators and Warnings).
Error Indicators and Warnings
Note: nag_opt_bounds_mod_deriv2_easy (e04ly) may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the function:
Cases prefixed with W are classified as warnings and
do not generate an error of type NAG:error_n. See nag_issue_warnings.
 ${\mathbf{ifail}}=1$

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  $\mathit{liw}<{\mathbf{n}}+2$, 
or  $\mathit{lw}<\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(10,{\mathbf{n}}\times \left({\mathbf{n}}+7\right)\right)$. 
 ${\mathbf{ifail}}=2$

There have been
$50\times {\mathbf{n}}$ 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.
 W ${\mathbf{ifail}}=3$
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 easytouse functions.)
 W ${\mathbf{ifail}}=5$
 W ${\mathbf{ifail}}=6$
 W ${\mathbf{ifail}}=7$
 W ${\mathbf{ifail}}=8$

There is some doubt about whether the point
$x$ found by
nag_opt_bounds_mod_deriv2_easy (e04ly) 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 function 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 usersupplied functions
funct2 or
hess2, that your problem has no finite solution, or that the problem needs rescaling (see
Further Comments).
 ${\mathbf{ifail}}=10$
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.
 ${\mathbf{ifail}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
 ${\mathbf{ifail}}=399$
Your licence key may have expired or may not have been installed correctly.
 ${\mathbf{ifail}}=999$
Dynamic memory allocation failed.
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.
Accuracy
When a successful exit is made then, for a computer with a mantissa of $t$ decimals, one would expect to get about $t/21$ decimals accuracy in $x$, and about $t1$ decimals accuracy in $F$, provided the problem is reasonably well scaled.
Further Comments
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
nag_opt_bounds_mod_deriv2_easy (e04ly) 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 usersupplied functions
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 nag_opt_bounds_mod_deriv2_easy (e04ly) will take less computer time.
Example
A program to minimize
subject to
starting from the initial guess
$\left(3,1,0,1\right)$. (In practice, it is worth trying to make usersupplied functions
funct2 and
hess2 as efficient as possible. This has not been done in the example program for reasons of clarity.)
Open in the MATLAB editor:
e04ly_example
function e04ly_example
fprintf('e04ly example results\n\n');
bl = [ 1; 2; 1000000; 1];
bu = [ 3; 0; 1000000; 3];
x = [ 3; 1; 0; 1];
ibound = int64(0);
wstat = warning();
warning('OFF');
[bl, bu, x, f, g, user, ifail] = e04ly(ibound, @funct2, @hess2, bl, bu, x);
warning(wstat);
if (ifail == 0  ifail == 5  ifail == 3)
fprintf('Minimum found at x: ');
fprintf(' %9.4f',x);
fprintf('\nGradients at x, g: ');
fprintf(' %9.4f',g);
fprintf('\nMinimum value : %9.4f\n\n',f);
else
fprintf('\n Error: e04ly returns ifail = %d\n',ifail);
end
function [fc, gc, user] = funct2(n, xc, user)
gc = zeros(n, 1);
fc = 0;
x1 = xc(1) + 10*xc(2);
x2 = xc(3)  xc(4);
x3 = xc(2)  2*xc(3);
x4 = xc(1)  xc(4);
fc = x1^2 + 5*x2^2 + x3^4 + 10*x4^4;
gc(1) = 2*x1 + 40*x4^3;
gc(2) = 20*x1 + 4*x3^3;
gc(3) = 10*x2  8*x3^3;
gc(4) = 10*x2  40*x4^3;
function [fhesl, fhesd, user] = hess2(n, xc, lh, user)
fhesl = zeros(lh, 1);
fhesd = zeros(n, 1);
x3 = xc(2)  2*xc(3);
x4 = xc(1)  xc(4);
fhesd(1) = 2 + 120*x4^2;
fhesd(2) = 200 + 12*x3^2;
fhesd(3) = 10 + 48*x3^2;
fhesd(4) = 10 + 120*x4^2;
fhesl(1) = 20;
fhesl(2) = 0;
fhesl(3) = 24*x3^2;
fhesl(4) = 120*x4^2;
fhesl(5) = 0;
fhesl(6) = 10;
e04ly example results
Minimum found at x: 1.0000 0.0852 0.4093 1.0000
Gradients at x, g: 0.2953 0.0000 0.0000 5.9070
Minimum value : 2.4338
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