e05jbf : NAG Library, Mark 26

Note: this routine uses optional parameters to define choices in the problem specification and in the details of the algorithm. If you wish to use default settings for all of the optional parameters, you need only read Sections 1 to 10 of this document. If, however, you wish to reset some or all of the settings please refer to Section 11 for a detailed description of the algorithm, and to Section 12 for a detailed description of the specification of the optional parameters.

e05jbf is designed to find the global minimum or maximum of an arbitrary function, subject to simple bound-constraints using a multi-level coordinate search method. Derivatives are not required, but convergence is only guaranteed if the objective function is continuous in a neighbourhood of a global optimum. It is not intended for large problems.

The initialization routine e05jaf must have been called before calling e05jbf.

Fortran Interface

Subroutine e05jbf (

n, objfun, ibound, iinit, bl, bu, sdlist, list, numpts, initpt, monit, x, obj, comm, lcomm, iuser, ruser, ifail)

Integer, Intent (In)	::	n, ibound, iinit, sdlist, lcomm
Integer, Intent (Inout)	::	numpts(n), initpt(n), iuser(*), ifail
Real (Kind=nag_wp), Intent (Inout)	::	bl(n), bu(n), list(n,sdlist), comm(lcomm), ruser(*)
Real (Kind=nag_wp), Intent (Out)	::	x(n), obj
External	::	objfun, monit

C Header Interface

#include nagmk26.h

void

e05jbf_ ( const Integer *n,
void (NAG_CALL *objfun)( const Integer *n, const double x[], double *f, const Integer *nstate, Integer iuser[], double ruser[], Integer *inform),
const Integer *ibound, const Integer *iinit, double bl[], double bu[], const Integer *sdlist, double list[], Integer numpts[], Integer initpt[],
void (NAG_CALL *monit)( const Integer *n, const Integer *ncall, const double xbest[], const Integer icount[], const Integer *ninit, const double list[], const Integer numpts[], const Integer initpt[], const Integer *nbaskt, const double xbaskt[], const double boxl[], const double boxu[], const Integer *nstate, Integer iuser[], double ruser[], Integer *inform),
double x[], double *obj, double comm[], const Integer *lcomm, Integer iuser[], double ruser[], Integer *ifail)

e05jaf must be called before calling e05jbf, or any of the option-setting or option-getting routines e05jcf, e05jdf, e05jef, e05jff, e05jgf, e05jhf, e05jjf, e05jkf or e05jlf.

You must not alter the number of non-fixed variables in your problem or the contents of the array comm between calls of the routines e05jaf, e05jbf, e05jcf, e05jdf, e05jef, e05jff, e05jgf, e05jhf, e05jjf, e05jkf or e05jlf.

e05jbf is designed to solve modestly sized global optimization problems having simple bound-constraints only; it finds the global optimum of a nonlinear function subject to a set of bound constraints on the variables. Without loss of generality, the problem is assumed to be stated in the following form:

\underset{x \in R^{n}}{minimize} F (x) subject to ℓ \leq x \leq u and ℓ \leq u,

where

F (x)

(the objective function) is a nonlinear scalar function (assumed to be continuous in a neighbourhood of a global minimum), and the bound vectors are elements of

{\bar{R}}^{n}

, where

\bar{R}

denotes the extended reals

R \cup \{- \infty, \infty\}

. Relational operators between vectors are interpreted elementwise.

The optional parameter Maximize should be set if you wish to solve maximization, rather than minimization, problems.

If certain bounds are not present, the associated elements of

ℓ

or

u

can be set to special values that will be treated as

- \infty

or

+ \infty

. See the description of the optional parameter Infinite Bound Size. Phrases in this document containing terms like ‘unbounded values’ should be understood to be taken relative to this optional parameter.

Fixing variables (that is, setting

l_{i} = u_{i}

for some

i

) is allowed in e05jbf.

A typical excerpt from a routine calling e05jbf is:

Call e05jaf (n_r, comm, lcomm, ...)
Call e05jdf (optstr, comm, lcomm, ...)
Call e05jbf (n, objfun, ...)

where e05jdf sets the optional parameter and value specified in optstr.

The initialization routine e05jaf does not need to be called before each invocation of e05jbf. You should be aware that a call to the initialization routine will reset each optional parameter to its default value, and, if you are using repeatable randomized initialization lists (see the description of the argument iinit), the random state stored in the array comm will be destroyed.

You must supply a subroutine that evaluates

F (x)

; derivatives are not required.

The method used by e05jbf is based on MCS, the Multi-level Coordinate Search method described in Huyer and Neumaier (1999), and the algorithm it uses is described in detail in Section 11.

Huyer W and Neumaier A (1999) Global optimization by multi-level coordinate search Journal of Global Optimization 14 331–355

If on entry

ifail = 0

or

- 1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

If

ifail = 0

on exit, then the vector returned in the array x is an estimate of the solution

x

whose function value satisfies your termination criterion: the function value was static for Static Limit sweeps through levels, or

F (x) - objval \leq \max (objerr \times |objval|, objsfg),

where

objval

is the value of the optional parameter Target Objective Value,

objerr

is the value of the optional parameter Target Objective Error, and

objsfg

is the value of the optional parameter Target Objective Safeguard.

e05jbf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

e05jbf 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.

For each invocation of e05jbf, local workspace arrays of fixed length are allocated internally. The total size of these arrays amounts to

13 n_{r} + smax - 1

integer elements, where

smax

is the value of the optional parameter Splits Limit and

n_{r}

is the number of non-fixed variables, and

(2 + n_{r}) sdlist + 2 n + 22 n_{r} + 3 n_{r}^{2} + 1

real elements. In addition, if you are using randomized initialization lists (see the description of the argument iinit), a further

21

integer elements are allocated internally.

In order to keep track of the regions of the search space that have been visited while looking for a global optimum, e05jbf internally allocates arrays of increasing sizes depending on the difficulty of the problem. Two of the main factors that govern the amount allocated are the number of sub-boxes (call this quantity

nboxes

) and the number of points in the ‘shopping basket’ (the argument nbaskt on entry to monit). Safe, pessimistic upper bounds on these two quantities are so large as to be impractical. In fact, the worst-case number of sub-boxes for even the most simple initialization list (when

ninit = 3

on entry to monit) grows like

{n_{r}}^{n_{r}}

. Thus e05jbf does not attempt to estimate in advance the final values of

nboxes

or nbaskt for a given problem. There are a total of

5

integer arrays and

4 + n_{r} + ninit

real arrays whose lengths depend on

nboxes

, and there are a total of

2

integer arrays and

3 + n + n_{r}

real arrays whose lengths depend on nbaskt. e05jbf makes a fixed initial guess that the maximum number of sub-boxes required will be

10000

and that the maximum number of points in the ‘shopping basket’ will be

1000

. If ever a greater amount of sub-boxes or more room in the ‘shopping basket’ is required, e05jbf performs reallocation, usually doubling the size of the inadequately-sized arrays. Clearly this process requires periods where the original array and its extension exist in memory simultaneously, so that the data within can be copied, which compounds the complexity of e05jbf's memory usage. It is possible (although not likely) that if your problem is particularly difficult to solve, or of a large size (hundreds of variables), you may run out of memory.

One array that could be dynamically resized by e05jbf is the ‘shopping basket’ (xbaskt on entry to monit). If the initial attempt to allocate

1000 n_{r}

reals for this array fails, monit will not be called on exit from e05jbf.

e05jbf performs better if your problem is well-scaled. It is worth trying (by guesswork perhaps) to rescale the problem if necessary, as sensible scaling will reduce the difficulty of the optimization problem, so that e05jbf will take less computer time.

This example finds the global minimum of the ‘peaks’ function in two dimensions

F (x, y) = 3 {(1 - x)}^{2} \exp (- x^{2} - {(y + 1)}^{2}) - 10 (\frac{x}{5} - x^{3} - y^{5}) \exp (- x^{2} - y^{2}) - \frac{1}{3} \exp (- {(x + 1)}^{2} - y^{2})

on the box

[- 3, 3] \times [- 3, 3]

.

The function

F

has several local minima and one global minimum in the given box. The global minimum is approximately located at

(0.23, - 1.63)

, where the function value is approximately

- 6.55

.

We use default values for all the optional parameters, and we instruct e05jbf to use the simple initialization list corresponding to

iinit = 0

. In particular, this will set for us the initial point

(0, 0)

(see Section 10.3).

Note: the remainder of this document is intended for more advanced users. Section 11 contains a detailed description of the algorithm. This information may be needed in order to understand Section 12, which describes the optional parameters that can be set by calls to e05jcf, e05jdf, e05jef, e05jff and/or e05jgf.

Here we summarise the main features of the MCS algorithm used in e05jbf, and we introduce some terminology used in the description of the subroutine and its arguments. We assume throughout that we will only do any work in coordinates

i

in which

x_{i}

is free to vary. The MCS algorithm is fully described in Huyer and Neumaier (1999).

Each sub-box is determined by a basepoint

x

and an opposite point

y

. We denote such a sub-box by

B [x, y]

. The basepoint is allowed to belong to more than one sub-box, is usually a boundary point, and is often a vertex.

An initialization procedure produces an initial set of sub-boxes. Whenever a sub-box is split along a coordinate

i

for the first time (in the initialization procedure or later), the splitting is done at three or more user-defined values

{\{x_{i}^{j}\}}_{j}

at which the objective function is sampled, and at some adaptively chosen intermediate points. At least four children are generated. More precisely, we assume that we are given

ℓ_{i} \leq x_{i}^{1} < x_{i}^{2} < \dots < x_{i}^{L_{i}} \leq u_{i}, L_{i} \geq 3,  for ​ i = 1, 2, \dots, n

and a vector

p

that, for each

i

, locates within

{\{x_{i}^{j}\}}_{j}

the

i

th coordinate of an initial point

x^{0}

; that is, if

x_{i}^{0} = x_{i}^{j}

for some

j = 1, 2, \dots, L_{i}

, then

p_{i} = j

. A good guess for the global optimum can be used as

x^{0}

.

The initialization points and the vectors

ℓ

and

p

are collectively called the initialization list (and sometimes we will refer to just the initialization points as ‘the initialization list’, whenever this causes no confusion). The initialization data may be input by you, or they can be set to sensible default values by e05jbf: if you provide them yourself,

list (i, j)

should contain

x_{i}^{j}

,

numpts (i)

should contain

L_{i}

, and

initpt (i)

should contain

p_{i}

, for

i = 1, 2, \dots, n

and

j = 1, 2, \dots, L_{i}

; if you wish e05jbf to use one of its preset initialization methods, you could choose one of two simple, three-point methods (see Figure 1). If the list generated by one of these methods contains infinite values, attempts are made to generate a safeguarded list using the function

subint (x, y)

(which is also used during the splitting procedure, and is described in Section 11.2). If infinite values persist, e05jbf exits with

ifail = 3

. There is also the option to generate an initialization list with the aid of linesearches (by setting

iinit = 2

). Starting with the absolutely smallest point in the root box, linesearches are made along each coordinate. For each coordinate, the local minimizers found by the linesearches are put into the initialization list. If there were fewer than three minimizers, they are augmented by close-by values. The final preset initialization option (

iinit = 4

) generates a randomized list, so that independent multiple runs may be made if you suspect a global optimum has not been found. Each call to the initialization routine e05jaf resets the initial-state vector for the Wichmann–Hill base-generator that is used. Depending on whether you set the optional parameter Repeatability to

ON

or

OFF

, the random state is initialized to give a repeatable or non-repeatable sequence. Then, a random integer between

3

and sdlist is selected, which is then used to determine the number of points to be generated in each coordinate; that is, numpts becomes a constant vector, set to this value. The components of list are then generated, from a uniform distribution on the root box if the box is finite, or else in a safeguarded fashion if any bound is infinite. The array

initpt

is set to point to the best point in list.

Given an initialization list (preset or otherwise), e05jbf evaluates

F

at

x^{0}

, and sets the initial estimate of the global minimum,

x^{*}

, to

x^{0}

. Then, for

i = 1, 2, \dots, n

, the objective function

F

is evaluated at

L_{i} - 1

points that agree with

x^{*}

in all but the

i

th coordinate. We obtain pairs

({\hat{x}}^{j}, f_{i}^{j})

, for

j = 1, 2, \dots, L_{i}

, with:

x^{*} = {\hat{x}}^{j_{1}}

, say; with, for

j \neq j_{1}

,

{\hat{x}}_{k}^{j} = \{\begin{cases} x_{k}^{*} & if ​ k \neq i; \\ x_{k}^{j} & otherwise; \end{cases}

and with

f_{i}^{j} = F ({\hat{x}}^{j}) .

The point having the smallest function value is renamed

x^{*}

and the procedure is repeated with the next coordinate.

Once e05jbf has a full set of initialization points and function values, it can generate an initial set of sub-boxes. Recall that the root box is

B [x, y] = [ℓ, u]

, having basepoint

x = x^{0}

. The opposite point

y

is a corner of

[ℓ, u]

farthest away from

x

, in some sense. The point

x

need not be a vertex of

[ℓ, u]

, and

y

is entitled to have infinite coordinates. We loop over each coordinate

i

, splitting the current box along coordinate

i

into

2 L_{i} - 2

,

2 L_{i} - 1

or

2 L_{i}

sub-intervals with exactly one of the

{\hat{x}}_{i}^{j}

as endpoints, depending on whether two, one or none of the

{\hat{x}}_{i}^{j}

are on the boundary. Thus, as well as splitting at

{\hat{x}}_{i}^{j}

, for

j = 1, 2, \dots, L_{i}

, we split at additional points

z_{i}^{j}

, for

j = 2, 3, \dots, L_{i}

. These additional

z_{i}^{j}

are such that

z_{i}^{j} = {\hat{x}}_{i}^{j - 1} + q^{m} ({\hat{x}}_{i}^{j} - {\hat{x}}_{i}^{j - 1}), j = 2, \dots, L_{i},

where

q

is the golden-section ratio

(\sqrt{5} - 1) / 2

, and the exponent

m

takes the value

1

or

2

, chosen so that the sub-box with the smaller function value gets the larger fraction of the interval. Each child sub-box gets as basepoint the point obtained from

x^{*}

by changing

x_{i}^{*}

to the

x_{i}^{j}

that is a boundary point of the corresponding

i

th coordinate interval; this new basepoint therefore has function value

f_{i}^{j}

. The opposite point is derived from

y

by changing

y_{i}

to the other end of that interval.

e05jbf can now rank the coordinates based on an estimated variability of

F

. For each

i

we compute the union of the ranges of the quadratic interpolant through any three consecutive

{\hat{x}}_{i}^{j}

, taking the difference between the upper and lower bounds obtained as a measure of the variability of

F

in coordinate

i

. A vector

π

is populated in such a way that coordinate

i

has the

π_{i}

th highest estimated variability. For tiebreaks, when the

x^{*}

obtained after splitting coordinate

i

belongs to two sub-boxes, the one that contains the minimizer of the quadratic models is designated the current sub-box for coordinate

i + 1

.

Boxes are assigned levels in the following manner. The root box is given level

1

. When a sub-box of level

s

is split, the child with the smaller fraction of the golden-section split receives level

s + 2

; all other children receive level

s + 1

. The box with the better function value is given the larger fraction of the splitting interval and the smaller level because then it is more likely to be split again more quickly. We see that after the initialization procedure the first level is empty and the non-split boxes have levels

2, \dots, n_{r} + 2

, so it is meaningful to choose

s_{\max}

much larger than

n_{r}

. Note that the internal structure of e05jbf demands that

s_{\max}

be at least

n_{r} + 3

.

Examples of initializations in two dimensions are given in Figure 1. In both cases the initial point is

x^{0} = (ℓ + u) / 2

; on the left the initialization points are

x^{1} = ℓ, x^{2} = (ℓ + u) / 2, x^{3} = u,

while on the right the points are

x^{1} = (5 ℓ + u) / 6, x^{2} = (ℓ + u) / 2, x^{3} = (ℓ + 5 u) / 6 .

In Figure 1, basepoints and levels after initialization are displayed. Note that these initialization lists correspond to

iinit = 0

and

iinit = 1

, respectively.

After initialization, a series of sweeps through levels is begun. A sweep is defined by three steps:

(i)	scan the list of non-split sub-boxes. Fill a record list $b$ according to $b_{s} = 0$ if there is no box at level $s$ , and with $b_{s}$ pointing to a sub-box with the lowest function value among all sub-boxes with level $s$ otherwise, for $0 < s < s_{\max}$ ;
(ii)	the sub-box with label $b_{s}$ is a candidate for splitting. If the sub-box is not to be split, according to the rules described in Section 11.2, increase its level by $1$ and update $b_{s + 1}$ if necessary. If the sub-box is split, mark it so, insert its children into the list of sub-boxes, and update $b$ if any child with level $s^{'}$ yields a strict improvement of $F$ over those sub-boxes at level $s^{'}$ ;
(iii)	increment $s$ by $1$ . If $s = s_{\max}$ then displaying monitoring information and start a new sweep; else if $b_{s} = 0$ then repeat this step; else display monitoring information and go to the previous step.

Clearly, each sweep ends after at most

s_{\max} - 1

visits of the third step.

Each sub-box is stored by e05jbf as a set of information about the history of the sub-box: the label of its parent, a label identifying which child of the parent it is, etc. Whenever a sub-box

B [x, y]

of level

s < s_{\max}

is a candidate for splitting, as described in Section 11.1, we recover

x

,

y

, and the number,

n_{j}

, of times coordinate

j

has been split in the history of

B

. Sub-box

B

could be split in one of two ways.

Eventually, a sub-box that is not eligible for splitting by expected gain will reach level

2 n_{r} (\min n_{j} + 1) + 1

and then be split by rank, as long as

s_{\max}

is large enough. As

s_{\max} \to \infty

, the rule for splitting by rank ensures that each coordinate is split arbitrarily often.

Before describing the details of each splitting method, we introduce the procedure for correctly handling splitting at adaptive points and for dealing with unbounded intervals. Suppose we want to split the

i

th coordinate interval

▯ \{x_{i}, y_{i}\}

, where we define

▯ \{x_{i}, y_{i}\} = [\min (x_{i}, y_{i}), \max (x_{i}, y_{i})]

, for

x_{i} \in R

and

y_{i} \in \bar{R}

, and where

x

is the basepoint of the sub-box being considered. The descendants of the sub-box should shrink sufficiently fast, so we should not split too close to

x_{i}

. Moreover, if

y_{i}

is large we want the new splitting value to not be too large, so we force it to belong to some smaller interval

▯ \{ξ^{'}, ξ^{''}\}

, determined by

ξ^{''} = subint (x_{i}, y_{i}), ξ^{'} = x_{i} + (ξ^{''} - x_{i}) / 10,

where the function

subint

is defined by

subint (x, y) = \{\begin{cases} sign (y) & if ​ 1000 |x| < 1 ​ and ​ |y| > 1000; \\ 10 sign (y) |x| & if ​ 1000 |x| \geq 1 ​ and ​ |y| > 1000 |x|; \\ y & otherwise. \end{cases}

Consider a sub-box

B

with level

s > 2 n_{r} (\min n_{j} + 1)

. Although the sub-box has reached a high level, there is at least one coordinate along which it has not been split very often. Among the

i

such that

n_{i} = \min n_{j}

for

B

, select the splitting index to be the coordinate with the lowest

π_{i}

(and hence highest variability rank). ‘Splitting by rank’ refers to the ranking of the coordinates by

n_{i}

and

π_{i}

.

If

n_{i} = 0

, so that

B

has never been split along coordinate

i

, the splitting is done according to the initialization list and the adaptively chosen golden-section split points, as described in Section 11.1. Also as covered there, new basepoints and opposite points are generated. The children having the smaller fraction of the golden-section split (that is, those with larger function values) are given level

\min \{s + 2, s_{\max}\}

. All other children are given level

s + 1

.

Otherwise,

B

ranges between

x_{i}

and

y_{i}

in the

i

th coordinate direction. The splitting value is selected to be

z_{i} = x_{i} + 2 (subint (x_{i}, y_{i}) - x_{i}) / 3

; we are not attempting to split based on a large reduction in function value, merely in order to reduce the size of a large interval, so

z_{i}

may not be optimal. Sub-box

B

is split at

z_{i}

and the golden-section split point, producing three parts and requiring only one additional function evaluation, at the point

x^{'}

obtained from

x

by changing the

i

th coordinate to

z_{i}

. The child with the smaller fraction of the golden-section split is given level

\min \{s + 2, s_{\max}\}

, while the other two parts are given level

s + 1

. Basepoints are assigned as follows: the basepoint of the first child is taken to be

x

, and the basepoint of the second and third children is the point

x^{'}

. Opposite points are obtained by changing

y_{i}

to the other end of the

i

th coordinate-interval of the corresponding child.

When a sub-box

B

has level

s \leq 2 n_{r} (\min n_{j} + 1)

, we compute the optimal splitting index and splitting value from a local separable quadratic used as a simple local approximation of the objective function. To fit this curve, for each coordinate we need two additional points and their function values. Such data may be recoverable from the history of

B

: whenever the

i

th coordinate was split in the history of

B

, we obtained values that can be used for the current quadratic interpolation in coordinate

i

.

We loop over

i

; for each coordinate we pursue the history of

B

back to the root box, and we take the first two points and function values we find, since these are expected to be closest to the current basepoint

x

. If the current coordinate has not yet been split we use the initialization list. Then we generate a local separable model

e (ξ)

for

F (ξ)

by interpolation at

x

and the

2 n_{r}

additional points just collected:

e (ξ) = F (x) + \sum_{i = 1}^{n} e_{i} (ξ_{i}) .

We define the expected gain

{\hat{e}}_{i}

in function value when we evaluate at a new point obtained by changing coordinate

i

in the basepoint, for each

i

, based on two cases:

(i)

n_{i} = 0

. We compute the expected gain as

{\hat{e}}_{i} = \min_{1 \leq j \leq L_{i}} \{f_{i}^{j}\} - f_{i}^{p_{i}} .

Again, we split according to the initialization list, with the new basepoints and opposite points being as before.

(ii)

n_{i} > 0

. Now, the

i

th component of our sub-box ranges from

x_{i}

to

y_{i}

. Using the quadratic partial correction function

e_{i} (ξ_{i}) = α_{i} (ξ_{i} - x_{i}) + β_{i} {(ξ_{i} - x_{i})}^{2}

we can approximate the maximal gain expected when changing

x_{i}

only. We will choose the splitting value from

▯ \{ξ^{'}, ξ^{''}\}

. We compute

{\hat{e}}_{i} = \min_{ξ_{i} \in ▯ \{ξ^{'}, ξ^{''}\}} e_{i} (ξ_{i})

and call

z_{i}

the minimizer in

▯ \{ξ^{'}, ξ^{''}\}

.

If the expected best function value

f_{\exp}

satisfies

f_{\exp} = F (x) + \min_{1 \leq i \leq n} {\hat{e}}_{i} < f_{best},

(1)

where

f_{best}

is the current best function value (including those function values obtained by local optimization), we expect the sub-box to contain a better point and so we split it, using as splitting index the component with minimal

{\hat{e}}_{i}

. Equation (1) prevents wasting function calls by avoiding splitting sub-boxes whose basepoints have bad function values. These sub-boxes will eventually be split by rank anyway.

We now have a splitting index and a splitting value

z_{i}

. The sub-box is split at

z_{i}

as long as

z_{i} \neq y_{i}

, and at the golden-section split point; two or three children are produced. The larger fraction of the golden-section split receives level

s + 1

, while the smaller fraction receives level

\min \{s + 2, s_{\max}\}

. If it is the case that

z_{i} \neq y_{i}

and the third child is larger than the smaller of the two children from the golden-section split, the third child receives level

s + 1

. Otherwise it is given the level

\min \{s + 2, s_{\max}\}

. The basepoint of the first child is set to

x

, and the basepoint of the second (and third if it exists) is obtained by changing the

i

th coordinate of

x

to

z_{i}

. The opposite points are again derived by changing

y_{i}

to the other end of the

i

th coordinate interval of

B

.

If equation (1) does not hold, we expect no improvement. We do not split, and we increase the level of

B

by

1

.

The local optimization algorithm used by e05jbf uses linesearches along directions that are determined by minimizing quadratic models, all subject to bound constraints. Triples of vectors are computed using coordinate searches based on linesearches. These triples are used in triple search procedures to build local quadratic models for

F

. A trust region-type approach to minimize these models is then carried out, and more information about the coordinate search and the triple search can be found in Huyer and Neumaier (1999).

The local search starts by looking for better points without being too local, by making a triple search using points found by a coordinate search. This yields a new point and function value, an approximation of the gradient of the objective, and an approximation of the Hessian of the objective. Then the quadratic model for

F

is minimized over a small box, with the solution to that minimization problem then being used as a linesearch direction to minimize the objective. A measure

r

is computed to quantify the predictive quality of the quadratic model.

The third stage is the checking of termination criteria. The local search will stop if more than

loclim

visits to this part of the local search have occurred, where

loclim

is the value of the optional parameter Local Searches Limit. If that is not the case, it will stop if the limit on function calls has been exceeded (see the description of the optional parameter Function Evaluations Limit). The final criterion checks if no improvement can be made to the function value, or whether the approximated gradient

g

is small, in the sense that

{|g|}^{T} \max (|x|, |x_{old}|) < loctol (f_{0} - f) .

The vector

x_{old}

is the best point at the start of the current loop in this iterative local-search procedure, the constant

loctol

is the value of the optional parameter Local Searches Tolerance,

f

is the objective value at

x

, and

f_{0}

is the smallest function value found by the initialization procedure.

Next, e05jbf attempts to move away from the boundary, if any components of the current point lie there, using linesearches along the offending coordinates. Local searches are terminated if no improvement could be made.

The fifth stage carries out another triple search, but this time it does not use points from a coordinate search, rather points lying within the trust region box are taken.

The final stage modifies the trust region box to be bigger or smaller, depending on the quality of the quadratic model, minimizes the new quadratic model on that box, and does a linesearch in the direction of the minimizer. The value of

r

is updated using the new data, and then we go back to the third stage (checking of termination criteria).

The Hessians of the quadratic models generated by the local search may not be positive definite, so e05jbf uses the general nonlinear optimizer e04vhf to minimize the models.

Several optional parameters in e05jbf define choices in the problem specification or the algorithm logic. In order to reduce the number of formal arguments of e05jbf these optional parameters have associated default values that are appropriate for most problems. Therefore, you need only specify those optional parameters whose values are to be different from their default values.

The remainder of this section can be skipped if you wish to use the default values for all optional parameters.

The following is a list of the optional parameters available. A full description of each optional parameter is provided in Section 12.1.

Optional parameters may be specified by calling one, or more, of the routines e05jcf, e05jdf, e05jef, e05jff and e05jgf before a call to e05jbf.

e05jcf reads options from an external options file, with Begin and End as the first and last lines respectively, and with each intermediate line defining a single optional parameter. For example,

 Begin
   Static Limit = 50
 End

The call

Call e05jcf (iopts, comm, lcomm, ifail)

can then be used to read the file on unit iopts. ifail will be zero on successful exit. e05jcf should be consulted for a full description of this method of supplying optional parameters.

e05jdf, e05jef, e05jff or e05jgf can be called to supply options directly, one call being necessary for each optional parameter. e05jdf, e05jef, e05jff or e05jgf should be consulted for a full description of this method of supplying optional parameters.

All optional parameters not specified by you are set to their default values. Valid values of optional parameters specified by you are unaltered by e05jbf and so remain in effect for subsequent calls to e05jbf, unless you explicitly change them.

For each option, we give a summary line, a description of the optional parameter and details of constraints.

The summary line contains:

Option names are case-insensitive and must be provided in full; abbreviations are not recognized.

This special keyword is used to reset all optional parameters to their default values, and any random state stored in the array comm will be destroyed.

Any option value given with this keyword will be ignored. This optional parameter cannot be queried or got.

This puts an approximate limit on the number of function calls allowed. The total number of calls made is checked at the top of an internal iteration loop, so it is possible that a few calls more than

nf

may be made.

Constraint:

nf > 0

.

This defines the ‘infinite’ bound

infbnd

in the definition of the problem constraints. Any upper bound greater than or equal to

infbnd

will be regarded as

\infty

(and similarly any lower bound less than or equal to

- infbnd

will be regarded as

- \infty

).

Constraint:

r_{\max}^{\frac{1}{4}} \leq infbnd \leq r_{\max}^{\frac{1}{2}}

.

If you want to try to accelerate convergence of e05jbf by starting local searches from candidate minima, you will require

lcsrch

to be

ON

.

Constraint:

lcsrch = ON ​ or ​ OFF

.

This defines the maximal number of iterations to be used in the trust region loop of the local-search procedure.

Constraint:

loclim > 0

.

The value of

loctol

is the multiplier used during local searches as a stopping criterion for when the approximated gradient is small, in the sense described in Section 11.3.

Constraint:

loctol \geq 2 ε

.

These keywords specify the required direction of optimization. Any option value given with these keywords will be ignored.

These options control the echoing of each optional parameter specification as it is supplied. List turns printing on, Nolist turns printing off. The output is sent to the current advisory message unit (as defined by x04abf).

Any option value given with these keywords will be ignored. This optional parameter cannot be queried or got.

For use with random initialization lists (

iinit = 4

). When set to

ON

, an internally-initialized random state is stored in the array comm for use in subsequent calls to e05jbf.

Constraint:

repeat = ON ​ or ​ OFF

.

Along with the initialization list list, this defines a limit on the number of times the root box will be split along any single coordinate direction. If Local Searches is

OFF

you may find the default value to be too small.

Constraint:

smax > n_{r} + 2

.

As the default termination criterion, computation stops when the best function value is static for

stclim

sweeps through levels. This parameter is ignored if you have specified a target value to reach in Target Objective Value.

Constraint:

stclim > 0

.

If you have given a target objective value to reach in

objval

(the value of the optional parameter Target Objective Value),

objerr

sets your desired relative error (from above if Minimize is set, from below if Maximize is set) between obj and

objval

, as described in Section 7. See also the description of the optional parameter Target Objective Safeguard.

Constraint:

objerr \geq 2 ε

.

If you have given a target objective value to reach in

objval

(the value of the optional parameter Target Objective Value),

objsfg

sets your desired safeguarded termination tolerance, for when

objval

is close to zero.

Constraint:

objsfg \geq 2 ε

.

This parameter may be set if you wish e05jbf to use a specific value as the target function value to reach during the optimization. Setting

objval

overrides the default termination criterion determined by the optional parameter Static Limit.

NAG Library Routine Document

e05jbf (bnd_mcs_solve)

▸▿ Contents

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

6

Error Indicators and Warnings

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

10.2

Program Data

10.3

Program Results

11

Algorithmic Details

11.1

Initialization and Sweeps

11.2

Splitting

11.2.1

Splitting by rank

11.2.2

Splitting by expected gain

11.3

Local Search

12

Optional Parameters

12.1

Description of the Optional Parameters

(i)	Splitting by rank If $s > 2 n_{r} (\min n_{j} + 1)$ , the box is always split. The splitting index is set to a coordinate $i$ such that $n_{i} = \min n_{j}$ .
(ii)	Splitting by expected gain If $s \leq 2 n_{r} (\min n_{j} + 1)$ , the sub-box could be split along a coordinate where a maximal gain in function value is expected. This gain is estimated according to a local separable quadratic model obtained by fitting to $2 n_{r} + 1$ function values. If the expected gain is too small the sub-box is not split at all, and its level is increased by $1$ .

NAG Library Routine Document

e05jbf (bnd_mcs_solve)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

11 Algorithmic Details

11.1 Initialization and Sweeps

11.2 Splitting

11.2.1 Splitting by rank

11.2.2 Splitting by expected gain

11.3 Local Search

12 Optional Parameters

12.1 Description of the Optional Parameters

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

6

Error Indicators and Warnings

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

10.2

Program Data

10.3

Program Results

11

Algorithmic Details

11.1

Initialization and Sweeps

11.2

Splitting

11.2.1

Splitting by rank

11.2.2

Splitting by expected gain

11.3

Local Search

12

Optional Parameters

12.1

Description of the Optional Parameters