nag_opt_bounds_no_deriv (e04jbc) : NAG Library, Mark 24

nag_opt_bounds_no_deriv (e04jbc) is a comprehensive quasi-Newton algorithm for finding:

–	an unconstrained minimum of a function of several variables;
–	a minimum of a function of several variables subject to fixed upper and/or lower bounds on the variables.

No derivatives are required. nag_opt_bounds_no_deriv (e04jbc) is intended for objective functions which have continuous first and second derivatives (although it will usually work even if the derivatives have occasional discontinuities).

nag_opt_bounds_no_deriv (e04jbc) is applicable to problems of the form:

\begin{array}{l} Minimize & F (x_{1}, x_{2}, \dots, x_{n}) \\ subject to & l_{j} \leq x_{j} \leq u_{j}, j = 1, 2, \dots, n . \end{array}

Special provision is made for unconstrained minimization (i.e., 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} = \dots = l_{n}

and

u_{1} = u_{2} = \dots = u_{n}

. It is possible to specify that a particular

x_{j}

should be held constant. You must supply a starting point and a function objfun to calculate the value of

F (x)

at any point

x

.

A typical iteration starts at the current point

x

where

n_{z}

(say) variables are free from both their bounds. The vector

g_{z}

, whose elements are finite difference approximations to the derivatives of

F (x)

with respect to the free variables, is known. A unit lower triangular matrix

L

and a diagonal matrix

D

(both of dimension

n_{z}

), such that

{L D L}^{T}

is a positive definite approximation to the matrix of second derivatives with respect to the free variables, are also stored. The equations

{L D L}^{T} p_{z} = - g_{z}

are solved to give a search direction

p_{z}

, which is expanded to an

n

-vector

p

by the insertion of appropriate zero elements. Then

α

is found such that

F (x + α p)

is approximately a minimum (subject to the fixed bounds) with respect to

α

;

x

is replaced by

x + α p

, and the matrices

L

and

D

are updated so as to be consistent with the change produced in the estimated gradient by the step

α p

. If any variable actually reaches a bound during the search along

p

, it is fixed and

n_{z}

is reduced for the next iteration. Most iterations calculate

g_{z}

using forward differences, but central differences are used when they seem necessary.

There are two sets of convergence criteria – a weaker and a stronger. Whenever the weaker criteria are satisfied, the Lagrange-multipliers are estimated for all the active constraints. If any Lagrange-multiplier estimate is significantly negative, then one of the variables associated with a negative Lagrange-multiplier estimate is released from its bound and the next search direction is computed in the extended subspace (i.e.,

n_{z}

is increased). Otherwise minimization continues in the current subspace provided that this is practicable. When it is not, or when the stronger convergence criteria is already satisfied, then, if one or more Lagrange-multiplier estimates are close to zero, a slight perturbation is made in the values of the corresponding variables in turn until a lower function value is obtained. The normal algorithm is then resumed from the perturbed point.

If a saddle point is suspected, a local search is carried out with a view to moving away from the saddle point. In addition, nag_opt_bounds_no_deriv (e04jbc) gives you the option of specifying that a local search should be performed when a point is found which is thought to be a constrained minimum.

If you specify that the problem is unconstrained, nag_opt_bounds_no_deriv (e04jbc) sets the

l_{j}

to

- 10^{10}

and the

u_{j}

to

10^{10}

. Thus, provided that the problem has been sensibly scaled, no bounds will be encountered during the minimization process and nag_opt_bounds_no_deriv (e04jbc) will act as an unconstrained minimization algorithm. When the problem is unconstrained, the function values used for estimating the first derivatives will always be required in sets of

n

. nag_opt_bounds_no_deriv (e04jbc) enables you to take advantage (via the argument bound) of the fact that such sets can often be evaluated in less computer time than

n

separate function evaluations would take in general.

Baker G A Jr and Graves–Morris P R (1981) Padé approximants, Part 1: Basic theory encyclopaedia of Mathematics and its Applications Addison–Wesley

Gill P E and Murray W (1972) Quasi-Newton methods for unconstrained optimization J. Inst. Math. Appl. 9 91–108

Gill P E and Murray W (1973) Safeguarded steplength algorithms for optimization using descent methods NPL Report NAC 37 National Physical Laboratory

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

Intermediate and final results are printed out by default. The level of printed output can be controlled with the structure member

options . print_level

(see Section 11.2). The default,

options . print_level = Nag_Soln_Iter

, provides a single line of output at each iteration and the final result. This section describes the default printout produced by nag_opt_bounds_no_deriv (e04jbc).

The following line of output is produced at each iteration. In all cases the values of the quantities printed are those in effect on completion of the given iteration.

Itn	the iteration count, $k$ .
Nfun	the cumulative number of calls made to objfun.
Objective	the value of the objective function, $F (x^{(k)})$
Norm g	the Euclidean norm of the projected gradient vector, $‖g_{z} (x^{(k)})‖$ .
Norm x	the Euclidean norm of $x^{(k)}$ .
Norm(x(k-1)-x(k))	the Euclidean norm of $x^{(k - 1)} - x^{(k)}$ .
Step	the step $α^{(k)}$ taken along the computed search direction $p^{(k)}$ .
Cond H	the ratio of the largest to the smallest element of the diagonal factor $D$ of the projected Hessian matrix. This quantity is usually a good estimate of the condition number of the projected Hessian matrix. (If no variables are currently free, this value will be zero.)

The printout of the final result consists of:

x	the final point, $x^{*}$ .
g	the final estimate of the projected gradient vector, $g_{z} (x^{*})$ .
Status	the final state of the variable with respect to its bound.

A successful exit

(fail . code = NE_NOERROR)

is made from nag_opt_bounds_no_deriv (e04jbc) when (B1, B2 and B3) or B4 hold, and the local search (if used) confirms a minimum, where

$B 1 \equiv α^{(k)} \times ‖p^{(k)}‖ < (options . optim_tol + \sqrt{ε}) \times (1.0 + ‖x^{(k)}‖)$
$B 2 \equiv |F^{(k)} - F^{(k - 1)}| < ({options . optim_tol}^{2} + ε) \times (1.0 + |F^{(k)}|)$
$B 3 \equiv ‖g_{z}^{(k)}‖ < (ε^{1 / 3} + options . optim_tol) \times (1.0 + |F^{(k)}|)$
$B 4 \equiv ‖g_{z}^{(k)}‖ < 0.01 \times \sqrt{ε} .$

(Quantities with superscript

k

are the values at the

k

th iteration of the quantities mentioned in Section 3;

ε

is the machine precision,

‖.‖

denotes the Euclidean norm and

options . optim_tol

is described in Section 11.)

If

fail . code = NE_NOERROR

, then the vector in x on exit,

x_{sol}

, is almost certainly an estimate of the position of the minimum,

x_{true}

, to the accuracy specified by

options . optim_tol

.

If

fail . code = NW_COND_MIN

or NW_LOCAL_SEARCH,

x_{sol}

may still be a good estimate of

x_{true}

, but the following checks should be made. Let the largest of the first

n_{z}

elements of

options . hesd

be

options . hesd [b]

, let the smallest be

options . hesd [s]

, and define

k = options . hesd [b] / options . hesd [s]

. The scalar

k

is usually a good estimate of the condition number of the projected Hessian matrix at

x_{sol}

. If

(a)	the sequence $\{F (x^{(k)})\}$ converges to $F (x_{sol})$ at a superlinear or a fast linear rate,
(b)	${‖g_{z} (x_{sol})‖}^{2} < 10.0 \times ε$ , and
(c)	$k < 1.0 / ‖g_{z} (x_{sol})‖$ ,

then it is almost certain that

x_{sol}

is a close approximation to the position of a minimum. When (b) is true, then usually

F (x_{sol})

is a close approximation to

F (x_{true})

. The quantities needed for these checks are all available in the results printout from nag_opt_bounds_no_deriv (e04jbc); in particular the final value of Cond H gives

k

.

Further suggestions about confirmation of a computed solution are given in the e04 Chapter Introduction.

Not applicable.

The number of iterations required depends on the number of variables, the behaviour of

F (x)

, the accuracy demanded and the distance of the starting point from the solution. The number of multiplications performed in an iteration of nag_opt_bounds_no_deriv (e04jbc) is roughly proportional to

n_{z}^{2}

. In addition, each iteration makes at least

n_{z} + 1

function evaluations. So, unless

F (x)

can be evaluated very quickly, the run time will be dominated by the time spent in objfun.

Ideally, the problem should be scaled so that, at the solution,

F (x)

and the corresponding values of the

x_{j}

are each in the range

(- 1, + 1)

, and so that at points one unit away from the solution,

F (x)

differs from its value at the solution by approximately one unit. This will usually imply that the Hessian matrix at the solution is well conditioned. 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_no_deriv (e04jbc) will take less computer time.

If a problem is genuinely unconstrained and has been scaled sensibly, the following points apply:

(a)	$n_{z}$ will always be $n$ ,
(b)	if $options . init_state = Nag_Init_All$ or $Nag_Init_H_S$ on entry, $options . state [j - 1]$ has simply to be set to $j$ , for $j = 1, 2, \dots, n$ ,
(c)	$options . hesl$ and $options . hesd$ will be factors of the full approximate second derivative matrix with elements stored in the natural order,
(d)	the elements of g should all be close to zero at the final point,
(e)	the Status values given in the printout from nag_opt_bounds_no_deriv (e04jbc) and in $options . state$ on exit are unlikely to be of interest (unless they are negative, which would indicate that the modulus of one of the $x_{j}$ has reached $10^{10}$ for some reason),
(f)	Norm g simply gives the norm of the estimated first derivative vector.

This example minimizes the function

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}

subject to the bounds

\begin{array}{l} 1 \leq x_{1} \leq 3 \\ - 2 \leq x_{2} \leq 0 \\ 1 \leq x_{4} \leq 3 \end{array}

starting from the initial guess

{(3, - 1, 0, 1)}^{T}

.

The example program also shows the use of certain optional arguments. It shows option values being assigned directly within the program text and by reading values from a data file. The options structure is declared and initialized by nag_opt_init (e04xxc). Values are then assigned directly to

options . outfile

and

options . optim_tol

and three further options are read from the data file by use of nag_opt_read (e04xyc). The memory freeing function nag_opt_free (e04xzc) is used to free the memory assigned to the pointers in the option structure. You must not use the standard C function free() for this purpose.

A number of optional input and output arguments to nag_opt_bounds_no_deriv (e04jbc) are available through the structure argument options, type Nag_E04_Opt. An argument may be selected by assigning an appropriate value to the relevant structure member; those arguments not selected will be assigned default values. If no use is to be made of any of the optional arguments you should use the NAG defined null pointer, E04_DEFAULT, in place of options when calling nag_opt_bounds_no_deriv (e04jbc); the default settings will then be used for all arguments.

Before assigning values to options directly the structure must be initialized by a call to the function nag_opt_init (e04xxc). Values may then be assigned to the structure members in the normal C manner.

Option settings may also be read from a text file using the function nag_opt_read (e04xyc) in which case initialization of the options structure will be performed automatically if not already done. Any subsequent direct assignment to the options structure must not be preceded by initialization.

If assignment of functions and memory to pointers in the options structure is required, then this must be done directly in the calling program; they cannot be assigned using nag_opt_read (e04xyc).

For easy reference, the following list shows the members of options which are valid for nag_opt_bounds_no_deriv (e04jbc) together with their default values where relevant. The number

ε

is a generic notation for machine precision (see nag_machine_precision (X02AJC)).

Boolean list	Nag_TRUE
Nag_PrintType print_level	Nag_Soln_Iter
char outfile[80]	stdout
void (*print_fun)()	NULL
Nag_InitType init_state	Nag_Init_None
Integer max_iter	$50 n$
double optim_tol	$10 \sqrt{ε}$
double linesearch_tol	0.5 (0.0 if $n = 1$ )
double step_max	100000.0
double f_est
Boolean local_search	Nag_TRUE
double *delta	size n
Integer *state	size n
double *hesl	size $\max (n [n - 1] / 2, 1)$
double *hesd	size n
Integer iter
Integer nf

On entry: if

options . list = Nag_TRUE

the argument settings in the call to nag_opt_bounds_no_deriv (e04jbc) will be printed.

On entry: the level of results printout produced by nag_opt_bounds_no_deriv (e04jbc). The following values are available:

$Nag_NoPrint$	No output.
$Nag_Soln$	The final solution.
$Nag_Iter$	One line of output for each iteration.
$Nag_Soln_Iter$	The final solution and one line of output for each iteration.
$Nag_Soln_Iter_Full$	The final solution and detailed printout at each iteration.

Details of each level of results printout are described in Section 11.3.

Constraint:

options . print_level = Nag_NoPrint

,

Nag_Soln

,

Nag_Iter

,

Nag_Soln_Iter

or

Nag_Soln_Iter_Full

.

On entry: the name of the file to which results should be printed. If

options . outfile [0] ='
  \0
'

then the stdout stream is used.

On entry: printing function defined by you; the prototype of

options . print_fun

is

void (*print_fun)(const Nag_Search_State *st, Nag_Comm *comm);

See Section 11.3.1 below for further details.

On entry:

options . init_state

specifies which of the arguments objf, g,

options . hesl

,

options . hesd

and

options . state

are actually being initialized. Such information will generally reduce the time taken by nag_opt_bounds_no_deriv (e04jbc).

$options . init_state = Nag_Init_None$: No values are assumed to have been set in any of objf, g, $options . hesl$ , $options . hesd$ or $options . state$ . (nag_opt_bounds_no_deriv (e04jbc) will use the unit matrix as the initial estimate of the Hessian matrix.)
$options . init_state = Nag_Init_All$: The arguments objf and g must contain the value of $F (x)$ and estimates of its first derivatives at the starting point. All $n$ elements of $options . state$ must have been set to indicate which variables are on their bounds and which are free. The pointer $options . delta$ must give the $n$ finite difference intervals. $options . hesl$ and $options . hesd$ must contain the Cholesky factors of a positive definite approximation to the $n_{z}$ by $n_{z}$ Hessian matrix for the subspace of free variables. (This option is useful for restarting the minimization process if $options . max_iter$ is reached.)
$options . init_state = Nag_Init_H_S$: No values are assumed to have been set in objf or g, but $options . hesl$ , $options . hesd$ , $options . state$ and $options . delta$ must have been set as for $options . init_state = Nag_Init_All$ . (This option is useful for starting off a minimization run using second derivative information from a previous, similar, run.)

Constraint:

options . init_state = Nag_Init_None

,

Nag_Init_All

or

Nag_Init_H_S

.

On entry: the limit on the number of iterations allowed before termination.

Constraint:

options . max_iter \geq 0

.

On entry: the accuracy in

x

to which the solution is required. If

x_{true}

is the true value of

x

at the minimum, then

x_{sol}

, the estimated position prior to a normal exit, is such that

‖x_{sol} - x_{true}‖ < options . optim_tol \times (1.0 + ‖x_{true}‖),

where

‖y‖ = {(\sum_{j = 1}^{n} y_{j}^{2})}^{1 / 2}

. For example, if the elements of

x_{sol}

are not much larger than 1.0 in modulus and if

options . optim_tol

is set to

10^{- 5}

, then

x_{sol}

is usually accurate to about 5 decimal places. (For further details see Section 9.) If the problem is scaled roughly as described in Section 9 and

ε

is the machine precision, then

\sqrt{ε}

is probably the smallest reasonable choice for

options . optim_tol

. (This is because, normally, to machine accuracy,

F (x + \sqrt{ε} e_{j}) = F (x)

where

e_{j}

is any column of the identity matrix.)

Constraint:

ε \leq options . optim_tol < 1.0

.

On entry: every iteration of nag_opt_bounds_no_deriv (e04jbc) involves a linear minimization (i.e., minimization of

F (x + α p)

with respect to

α

).

options . linesearch_tol

specifies how accurately these linear minimizations are to be performed. The minimum with respect to

α

will be located more accurately for small values of

options . linesearch_tol

(say 0.01) than for large values (say 0.9).

Although accurate linear minimizations will generally reduce the number of iterations performed by nag_opt_bounds_no_deriv (e04jbc), they will increase the number of function evaluations required for each iteration. On balance, it is usually more efficient to perform a low accuracy linear minimization.

A smaller value such as 0.01 may be worthwhile:

(a)	if $F (x)$ can be evaluated unusually quickly (since it may be worth using extra function evaluations to reduce the number of iterations and associated matrix calculations);
(b)	if $F (x)$ is a penalty or barrier function arising from a constrained minimization problem (since such problems are very difficult to solve).

If

n = 1

, the default for

options . linesearch_tol = 0.0

(if the problem is effectively one-dimensional then

options . linesearch_tol

should be set to 0.0 even though

n > 1

; i.e., if for all except one of the variables the lower and upper bounds are equal).

Constraint:

0.0 \leq options . linesearch_tol < 1.0

.

On entry: an estimate of the Euclidean distance between the solution and the starting point supplied. (For maximum efficiency a slight overestimate is preferable.) nag_opt_bounds_no_deriv (e04jbc) will ensure that, for each iteration,

{(\sum_{j = 1}^{n} {[x_{j}^{(k)} - x_{j}^{(k - 1)}]}^{2})}^{1 / 2} \leq options . step_max,

where

k

is the iteration number. Thus, if the problem has more than one solution, nag_opt_bounds_no_deriv (e04jbc) is most likely to find the one nearest the starting point. On difficult problems, a realistic choice can prevent the sequence of

x^{(k)}

entering a region where the problem is ill-behaved and can also help to avoid possible overflow in the evaluation of

F (x)

. However an underestimate of

options . step_max

can lead to inefficiency.

Constraint:

options . step_max \geq options . optim_tol

.

On entry: an estimate of the function value at the minimum. This estimate is just used for calculating suitable step lengths for starting linear minimizations off, so the choice is not too critical. However, it is better for

options . f_est

to be set to an underestimate rather than to an overestimate. If no value is supplied then an initial step length of

1.0

is used, though this may be reduced to ensure that the bounds are not overstepped.

On entry:

options . local_search

must specify whether or not you wish a ‘local search’ to be performed when a point is found which is thought to be a constrained minimum.

If

options . local_search = Nag_TRUE

and either the quasi-Newton direction of search fails to produce a lower function value or the convergence criteria are satisfied, then a local search will be performed. This may move the search away from a saddle point or confirm that the final point is a minimum.

If

options . local_search = Nag_FALSE

there will be no local search when a point is found which is thought to be a minimum.

The amount of work involved in a local search is comparable to twice that required in a normal iteration to minimize

F (x + α p)

with respect to

α

. For most problems this will be small (relative to the total time required for the minimization).

options . local_search

could be set Nag_FALSE if:

–	it is known from the physical properties of a problem that a stationary point will be the required minimum;
–	a point which is not a minimum could be easily recognized, for example if the value of $F (x)$ at the minimum is known.

On entry: suitable step lengths for making difference approximations to the partial derivatives of

F (x)

.

If

options . delta

is not allocated memory and

options . init_state = Nag_Init_None

then nag_opt_bounds_no_deriv (e04jbc) will allocate memory to

options . delta

and assign a suitable set of difference intervals.

If

options . delta

is allocated memory, i.e.,

options . delta

is not NULL, and

options . init_state = Nag_Init_None

then difference intervals are assumed to be supplied by

options . delta

.

When

options . init_state \neq Nag_Init_None

then

options . delta

must hold the finite difference intervals; these may be the values output from a previous call to nag_opt_bounds_no_deriv (e04jbc).

If you wish to supply difference intervals then the following advice can be given. When the problem is scaled roughly as described in Section 9 and

ε

is the machine precision, values in the range

\sqrt{ε}

to

ε^{2 / 3}

may be suitable. Otherwise, you must choose suitable settings, bearing in mind that, when forward differences are used, the approximation is

\frac{\partial F}{\partial x_{j}} = \frac{F (x + options . delta [j] \times e_{j}) - F (x)}{options . delta [j]}

where

e_{j}

is the

j

th coordinate direction, for

j = 1, 2, \dots, n

.

On exit: the n finite difference intervals used by nag_opt_bounds_no_deriv (e04jbc). If

options . delta

is NULL on entry and

options . init_state = Nag_Init_None

then memory will have been automatically allocated to

options . delta

and suitable values assigned.

Constraint:

options . delta [j] \geq 0.0, ​ x [j] + options . delta [j] \neq x [j]

.

On entry:

options . state

need not be set if the default option of

options . init_state = Nag_Init_None

is used as n values of memory will be automatically allocated by nag_opt_bounds_no_deriv (e04jbc).

If the option

options . init_state = Nag_Init_All

or

Nag_Init_H_S

has been chosen,

options . state

must point to a minimum of n elements of memory. This memory will already be available if the calling program has used the options structure in a previous call to nag_opt_bounds_no_deriv (e04jbc) with

options . init_state = Nag_Init_None

and the same value of n. If a previous call has not been made, you must allocate sufficient memory.

When

options . init_state = Nag_Init_All

or

Nag_Init_H_S

then

options . state

must specify information about which variables are currently on their bounds and which are free. If

x_{j}

is:

(a)	fixed on its upper bound, $options . state [j - 1]$ is $- 1$
(b)	fixed on its lower bound, $options . state [j - 1]$ is $- 2$
(c)	effectively a constant (i.e., $l_{j} = u_{j}$ ), $options . state [j - 1]$ is $- 3$
(d)	free, $options . state [j - 1]$ gives its position in the sequence of free variables.

If

options . init_state = Nag_Init_None

,

options . state

will be initialized by nag_opt_bounds_no_deriv (e04jbc).

If

options . init_state = Nag_Init_All

or

Nag_Init_H_S

,

options . state

must be initialized before nag_opt_bounds_no_deriv (e04jbc) is called.

On exit:

options . state

gives information as above about the final point given in n.

On entry:

options . hesl

and

options . hesd

need not be set if the default of

options . init_state = Nag_Init_None

is used as sufficient memory will be automatically allocated by nag_opt_bounds_no_deriv (e04jbc).

If

options . init_state = Nag_Init_All

or

options . init_state = Nag_Init_H_S

has been set then

options . hesl

must point to a minimum of

\max (n [n - 1] / 2, 1)

elements of memory.

options . hesd

must point to at least n elements of memory if

options . init_state = Nag_Init_All

or

Nag_Init_H_S

has been chosen.

The appropriate amount of memory will already be available for

options . hesl

and

options . hesd

if the calling program has used the options structure in a previous call to nag_opt_bounds_no_deriv (e04jbc) with

options . init_state = Nag_Init_None

and the same value of n. If a previous call has not been made, you must allocate sufficient memory.

options . hesl

and

options . hesd

are used to store the factors

L

and

D

of the current approximation to the matrix of second derivatives with respect to the free variables (see Section 3). (The elements of the matrix are assumed to be ordered according to the permutation specified by the positive elements of

options . state

, see above.)

options . hesl

holds the lower triangle of

L

, omitting the unit diagonal, stored by rows.

options . hesd

stores the diagonal elements of

D

. Thus if

n_{z}

elements of

options . state

are positive, the strict lower triangle of

L

will be held in the first

n_{z} (n_{z} - 1) / 2

elements of

options . hesl

and the diagonal elements of

D

in the first

n_{z}

elements of

options . hesd

.

If

options . init_state = Nag_Init_None

(the default),

options . hesl

and

options . hesd

will be initialized within nag_opt_bounds_no_deriv (e04jbc) to the factors of the unit matrix.

If you set

options . init_state = Nag_Init_All

or

Nag_Init_H_S

,

options . hesl

and

options . hesd

must contain on entry the Cholesky factors of a positive definite approximation to the

n_{z}

by

n_{z}

matrix of second derivatives for the subspace of free variables as specified by your setting of

options . state

.

On exit:

options . hesl

and

options . hesd

hold the factors

L

and

D

corresponding to the final point given in x. The elements of

options . hesd

are useful for deciding whether to accept the result produced by nag_opt_bounds_no_deriv (e04jbc) (see Section 9).

On exit: the number of iterations which have been performed in nag_opt_bounds_no_deriv (e04jbc).

On exit: the number of times the residuals have been evaluated.

The level of printed output can be controlled with the structure members

options . list

and

options . print_level

(see Section 11.2). If

options . list = Nag_TRUE

then the argument values to nag_opt_bounds_no_deriv (e04jbc) are listed, whereas the printout of results is governed by the value of

options . print_level

. The default of

options . print_level = Nag_Soln_Iter

provides a single line of output at each iteration and the final result. This section describes all of the possible levels of printout available from nag_opt_bounds_no_deriv (e04jbc).

When

options . print_level = Nag_Iter

or

Nag_Soln_Iter

a single line of output is produced on completion of each iteration, this gives the following values:

Itn	the iteration count, $k$ .
Nfun	the cumulative number of objective function evaluations.
Objective	the value of the objective function, $F (x^{(k)})$
Norm g	the Euclidean norm of the projected gradient vector, $‖g_{z} (x^{(k)})‖$ .
Norm x	the Euclidean norm of $x^{(k)}$ .
Norm(x(k-1)-x(k))	the Euclidean norm of $x^{(k - 1)} - x^{(k)}$ .
Step	the step $α^{(k)}$ taken along the computed search direction $p^{(k)}$ .
Cond H	the ratio of the largest to the smallest element of the diagonal factor $D$ of the projected Hessian matrix. This quantity is usually a good estimate of the condition number of the projected Hessian matrix. (If no variables are currently free, this value will be zero.)

When

options . print_level = Nag_Soln_Iter_Full

more detailed results are given at each iteration. Additional values output are:

x	the current point $x^{(k)}$ .
g	the current estimate of the projected gradient vector, $g_{z} (x^{(k)})$ .
Status	the current state of the variable with respect to its bound(s).

If

options . print_level = Nag_Soln

,

Nag_Soln_Iter

or

Nag_Soln_Iter_Full

the final result is printed out. This consists of:

x	the final point, $x^{*}$ .
g	the final estimate of the projected gradient vector, $g_{z} (x^{*})$ .
Status	the final state of the variable with respect to its bound(s).

If

options . print_level = Nag_NoPrint

then printout will be suppressed; you can print the final solution when nag_opt_bounds_no_deriv (e04jbc) returns to the calling program.

You may also specify your own print function for output of iteration results and the final solution by use of the

options . print_fun

function pointer, which has prototype

void (*print_fun)(const Nag_Search_State *st, Nag_Comm *comm);

The rest of this section can be skipped if the default printing facilities provide the required functionality.

When a user-defined function is assigned to

options . print_fun

this will be called in preference to the internal print function of nag_opt_bounds_no_deriv (e04jbc). Calls to the user-defined function are again controlled by means of the

options . print_level

member. Information is provided through st and comm, the two structure arguments to

options . print_fun

.

The results contained in the members of st are those on completion of the last iteration or those after a local search. (An iteration may be followed by a local search (see

options . local_search

, Section 11.2) in which case

options . print_fun

is called with the results of the last iteration (

st \to local_search = Nag_FALSE

) and then again when the local search has been completed (

st \to local_search = Nag_TRUE

).)

If

comm \to it_prt = Nag_TRUE

then the results on completion of an iteration of nag_opt_bounds_no_deriv (e04jbc) are contained in the members of st. If

comm \to sol_prt = Nag_TRUE

then the final results from nag_opt_bounds_no_deriv (e04jbc), including details of the final iteration, are contained in the members of st. In both cases, the same members of st are set, as follows:

The relevant members of the structure comm are:

NAG Library Function Document

nag_opt_bounds_no_deriv (e04jbc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

5.1 Description of Printed Output

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

9.1 Timing

9.2 Scaling

9.3 Unconstrained Minimization

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

11 Optional Arguments

11.1 Optional Argument Checklist and Default Values

11.2 Description of the Optional Arguments

11.3 Description of Printed Output

11.3.1 Output of results via a user-defined printing function

NAG Library Function Documentnag_opt_bounds_no_deriv (e04jbc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

5.1 Description of Printed Output

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

9.1 Timing

9.2 Scaling

9.3 Unconstrained Minimization

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

11 Optional Arguments

11.1 Optional Argument Checklist and Default Values

11.2 Description of the Optional Arguments

11.3 Description of Printed Output

11.3.1 Output of results via a user-defined printing function

NAG Library Function Document

nag_opt_bounds_no_deriv (e04jbc)