NAG Toolbox: nag_opt_bounds_bobyqa_func (e04jc)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{objfun}$ – function handle or string containing name of m-file

objfun must evaluate the objective function $F$ at a specified vector $\mathbf{x}$.

[f, user, inform] = objfun(n, x, user)

Input Parameters

1:     $\mathrm{n}$ – int64int32nag_int scalar

$n$, the number of independent variables.

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

$\mathbf{x}$, the vector at which the objective function is to be evaluated.

3:     $\mathrm{user}$ – Any MATLAB object

objfun is called from nag_opt_bounds_bobyqa_func (e04jc) with the object supplied to nag_opt_bounds_bobyqa_func (e04jc).

Output Parameters

1:     $\mathrm{f}$ – double scalar

Must be set to the value of the objective function at $\mathbf{x}$, unless you have specified termination of the current problem using inform.

2:     $\mathrm{user}$ – Any MATLAB object

3:     $\mathrm{inform}$ – int64int32nag_int scalar

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.

2:     $\mathrm{npt}$ – int64int32nag_int scalar

Suggested value: $\mathbf{npt}=2\times n_r+1$, where $n_r$ denotes the number of non-fixed variables.

$m$, the number of interpolation conditions imposed on the quadratic approximation at each iteration.

Constraint: $n_r+2\le \mathbf{npt}\le \frac{\left(n_r+1\right)\times \left(n_r+2\right)}{2}$, where $n_r$ denotes the number of non-fixed variables.

3:     $\mathrm{x}\left(\mathbf{n}\right)$ – double array

An estimate of the position of the minimum. If any component is out-of-bounds it is replaced internally by the bound it violates.

4:     $\mathrm{bl}\left(\mathbf{n}\right)$ – double array

5:     $\mathrm{bu}\left(\mathbf{n}\right)$ – double array

The fixed vectors of bounds: the lower bounds $\mathbf{\ell }$ and the upper bounds $\mathbf{u}$, respectively. To signify that a variable is unbounded you should choose a large scalar $r$ appropriate to your problem, then set the lower bound on that variable to $-r$ and the upper bound to $r$. For well-scaled problems $r=r_{\max }^{\frac{1}{4}}$ may be suitable, where $r_{\max }$ denotes the largest positive model number (see nag_machine_real_largest (x02al)).

Constraints:

• if $\mathbf{x}\left(\mathit{i}\right)$ is to be fixed at $\mathbf{bl}\left(\mathit{i}\right)$, then $\mathbf{bl}\left(\mathit{i}\right)=\mathbf{bu}\left(\mathit{i}\right)$;
• otherwise $\mathbf{bu}\left(\mathit{i}\right)-\mathbf{bl}\left(\mathit{i}\right)\ge 2.0\times \mathbf{rhobeg}$, for $\mathit{i}=1,2,\dots ,\mathbf{n}$.

6:     $\mathrm{rhobeg}$ – double scalar

Suggested value: 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.

An initial lower bound on the value of the trust-region radius.

Constraints:

• $\mathbf{rhobeg}>0.0$;
• $\mathbf{rhobeg}\ge \mathbf{rhoend}$.

7:     $\mathrm{rhoend}$ – double scalar

Suggested value: rhoend should indicate the absolute accuracy that is required in the final values of the variables.

A final lower bound on the value of the trust-region radius.

Constraint: $\mathbf{rhoend}>0.0$.

8:     $\mathrm{monfun}$ – function handle or string containing name of m-file

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 string nag_opt_bounds_bobyqa_func_dummy_monfun (e04jcp)

[user, inform] = monfun(n, nf, x, f, rho, user)

Input Parameters

1:     $\mathrm{n}$ – int64int32nag_int scalar

$n$, the number of independent variables.

2:     $\mathrm{nf}$ – int64int32nag_int scalar

The cumulative number of calls made to objfun.

3:     $\mathrm{x}\left(\mathbf{n}\right)$ – double array

The current best point.

4:     $\mathrm{f}$ – double scalar

The value of objfun at x.

5:     $\mathrm{rho}$ – double scalar

A lower bound on the current trust-region radius.

6:     $\mathrm{user}$ – Any MATLAB object

monfun is called from nag_opt_bounds_bobyqa_func (e04jc) with the object supplied to nag_opt_bounds_bobyqa_func (e04jc).

Output Parameters

1:     $\mathrm{user}$ – Any MATLAB object

2:     $\mathrm{inform}$ – int64int32nag_int scalar

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.

9:     $\mathrm{maxcal}$ – int64int32nag_int scalar

The maximum permitted number of calls to objfun.

Constraint: $\mathbf{maxcal}\ge 1$.

Optional Input Parameters

1:     $\mathrm{n}$ – int64int32nag_int scalar

Default: the dimension of the arrays x, bl, bu. (An error is raised if these dimensions are not equal.)

$n$, the number of independent variables.

Constraint: $\mathbf{n}\ge 2$ and $n_r\ge 2$, where $n_r$ denotes the number of non-fixed variables.

2:     $\mathrm{user}$ – Any MATLAB object

user is not used by nag_opt_bounds_bobyqa_func (e04jc), but is passed to objfun and monfun. Note that for large objects it may be more efficient to use a global variable which is accessible from the m-files than to use user.

Output Parameters

1:     $\mathrm{x}\left(\mathbf{n}\right)$ – double array

The lowest point found during the calculations. Thus, if $\mathbf{ifail}=\mathbf{0}$ on exit, x is the position of the minimum.

2:     $\mathrm{f}$ – double scalar

The function value at the lowest point found (x).

3:     $\mathrm{nf}$ – int64int32nag_int scalar

Unless $\mathbf{ifail}=\mathbf{1}$ or $-\mathbf{999}$ on exit, the total number of calls made to objfun.

4:     $\mathrm{user}$ – Any MATLAB object

5:     $\mathrm{ifail}$ – int64int32nag_int scalar

$\mathbf{ifail}=\mathbf{0}$ unless the function detects an error (see Error Indicators and Warnings).

nag_opt_bounds_bobyqa_func (e04jc) returns with $\mathbf{ifail}=\mathbf{0}$ if the final trust-region radius has reached its lower bound rhoend.

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

   $\mathbf{ifail}=1$

Constraint: if $\mathbf{bl}\left(i\right)\ne \mathbf{bu}\left(i\right)$ in coordinate $i$, then $\mathbf{bu}\left(i\right)-\mathbf{bl}\left(i\right)\ge 2\times \mathbf{rhobeg}$.

Constraint: $\mathbf{maxcal}\ge 1$.

Constraint: $\mathbf{rhobeg}>0.0$.

Constraint: $\mathbf{rhoend}>0.0$.

Constraint: $\mathbf{rhoend}\le \mathbf{rhobeg}$.

There were $n_r=\_$ unequal bounds and $\mathbf{npt}=\_$ on entry.
Constraint: $n_r+2\le \mathbf{npt}\le \frac{\left(n_r+1\right)\times \left(n_r+2\right)}{2}$.

There were unequal bounds.
Constraint: $n_r\ge 2$.

   $\mathbf{ifail}=2$

The function evaluations limit was reached: objfun has been called maxcal times.

   $\mathbf{ifail}=3$

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.

   $\mathbf{ifail}=4$

A rescue procedure has been called in order to correct damage from rounding errors when computing an update to a quadratic approximation of $F$, but no further progess could be made. Check your specification of objfun and whether the function needs rescaling. Try a different initial x.

W  $\mathbf{ifail}=5$

User-supplied monitoring function requested termination.

User-supplied objective function requested termination.

   $\mathbf{ifail}=-99$

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.

Accuracy

Further Comments

Example

Open in the MATLAB editor: e04jc_example

function e04jc_example


fprintf('e04jc example results\n\n');

maxcal = int64(500);
rhobeg = 0.1;
rhoend = 1.0e-6;
n = 4;
npt = int64(2*n + 1);

% x(3) is unconstrained, so we're going to set bl(3) to a large
% negative number and bu(3) to a large positive number.
infbnd = x02al^0.25;
bl = [1, -2, -infbnd, 1];
bu = [3, 0, infbnd, 3];
x  = [3, -1, 0, 1];

[x, f, nf, user, ifail] = ...
  e04jc( ...
         @objfun, npt, x, bl, bu, rhobeg, rhoend, @monfun, maxcal);

fprintf('\nFunction value at lowest point found =  %12.5e\n',f);
fprintf(' The corresponding X is:\n');
fprintf('%14.5e',x);
fprintf('\n');



function [f, user, inform] = objfun(n, x, user)
  inform = int64(0);

  f = (x(1)+10*x(2))^2 + 5*(x(3)-x(4))^2 + (x(2)-2*x(3))^4 + 10*(x(1)-x(4))^4;

function [user, inform] = monfun(n, nf, x, f, rho, user)
  inform = int64(0);

  fprintf('\nNew rho = %13.5e, number of function evaluations = %d\n', rho, nf);
  fprintf('Current function value = %13.5e\n', f);
  fprintf('The corresponding X is:');
  fprintf(' %13.5e', x);
  fprintf('\n');

e04jc example results


New rho =   1.00000e-02, number of function evaluations = 25
Current function value =   4.09399e+00
The corresponding X is:   1.60106e+00  -1.03604e-01   4.51135e-01   1.02335e+00

New rho =   1.00000e-03, number of function evaluations = 67
Current function value =   2.43397e+00
The corresponding X is:   1.00000e+00  -8.59741e-02   4.06744e-01   1.00000e+00

New rho =   1.00000e-04, number of function evaluations = 77
Current function value =   2.43379e+00
The corresponding X is:   1.00000e+00  -8.52328e-02   4.09342e-01   1.00000e+00

New rho =   1.00000e-05, number of function evaluations = 81
Current function value =   2.43379e+00
The corresponding X is:   1.00000e+00  -8.52328e-02   4.09342e-01   1.00000e+00

New rho =   1.00000e-06, number of function evaluations = 93
Current function value =   2.43379e+00
The corresponding X is:   1.00000e+00  -8.52329e-02   4.09303e-01   1.00000e+00

Function value at lowest point found =   2.43379e+00
 The corresponding X is:
   1.00000e+00  -8.52326e-02   4.09303e-01   1.00000e+00