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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_glopt_bnd_pso (e05sa)

## Purpose

nag_glopt_bnd_pso (e05sa) is designed to search for the global minimum or maximum of an arbitrary function, using Particle Swarm Optimization (PSO). Derivatives are not required, although these may be used by an accompanying local minimization function if desired. nag_glopt_bnd_pso (e05sa) is essentially identical to nag_glopt_nlp_pso (e05sb), but with a simpler interface and with various optional parameters removed; otherwise most arguments are identical. In particular, nag_glopt_bnd_pso (e05sa) does not handle general constraints.

## Syntax

[xb, fb, iopts, opts, user, itt, inform, ifail] = e05sa(bl, bu, objfun, monmod, iopts, opts, 'ndim', ndim, 'npar', npar, 'user', user)
[xb, fb, iopts, opts, user, itt, inform, ifail] = nag_glopt_bnd_pso(bl, bu, objfun, monmod, iopts, opts, 'ndim', ndim, 'npar', npar, 'user', user)
Before calling nag_glopt_bnd_pso (e05sa), nag_glopt_optset (e05zk) must be called with optstr set to ‘’. Optional parameters may also be specified by calling nag_glopt_optset (e05zk) before the call to nag_glopt_bnd_pso (e05sa).
Note: the interface to this routine has changed since earlier releases of the toolbox:
 At Mark 24: npar was made optional

## Description

nag_glopt_bnd_pso (e05sa) uses a stochastic method based on Particle Swarm Optimization (PSO) to search for the global optimum of a nonlinear function $F$, subject to a set of bound constraints on the variables. In the PSO algorithm (see Algorithmic Details), a set of particles is generated in the search space, and advances each iteration to (hopefully) better positions using a heuristic velocity based upon inertia, cognitive memory and global memory. The inertia is provided by a decreasingly weighted contribution from a particles current velocity, the cognitive memory refers to the best candidate found by an individual particle and the global memory refers to the best candidate found by all the particles. This allows for a global search of the domain in question.
Further, this may be coupled with a selection of local minimization functions, which may be called during the iterations of the heuristic algorithm, the interior phase, to hasten the discovery of locally optimal points, and after the heuristic phase has completed to attempt to refine the final solution, the exterior phase. Different options may be set for the local optimizer in each phase.
Without loss of generality, the problem is assumed to be stated in the following form:
 $minimize x∈Rndim Fx subject to ℓ ≤ x ≤ u ,$
where the objective $F\left(\mathbf{x}\right)$ is a scalar function, $\mathbf{x}$ is a vector in ${R}^{\mathit{ndim}}$ and the vectors $\mathbf{\ell }\le \mathbf{u}$ are lower and upper bounds respectively for the $\mathit{ndim}$ variables. The objective function may be nonlinear. Continuity of $F$ is not essential. For functions which are smooth and primarily unimodal, faster solutions will almost certainly be achieved by using Chapter E04 functions directly.
For functions which are smooth and multi-modal, gradient dependent local minimization functions may be coupled with nag_glopt_bnd_pso (e05sa).
For multi-modal functions for which derivatives cannot be provided, particularly functions with a significant level of noise in their evaluation, nag_glopt_bnd_pso (e05sa) should be used either alone, or coupled with nag_opt_uncon_simplex (e04cb).
The $\mathit{ndim}$ lower and upper box bounds on the variable $\mathbf{x}$ are included to initialize the particle swarm into a finite hypervolume, although their subsequent influence on the algorithm is user determinable (see the option Boundary in Optional Parameters). It is strongly recommended that sensible bounds are provided for all variables.
nag_glopt_bnd_pso (e05sa) may also be used to maximize the objective function (see the option Optimize).
Due to the nature of global optimization, unless a predefined target is provided, there is no definitive way of knowing when to end a computation. As such several stopping heuristics have been implemented into the algorithm. If any of these is achieved, nag_glopt_bnd_pso (e05sa) will exit with ${\mathbf{ifail}}={\mathbf{1}}$, and the parameter inform will indicate which criteria was reached. See inform for more information.
In addition, you may provide your own stopping criteria through monmod and objfun.
nag_glopt_nlp_pso (e05sb) provides a comprehensive interface, allowing for the inclusion of general nonlinear constraints.

## References

Gill P E, Murray W and Wright M H (1981) Practical Optimization Academic Press
Kennedy J and Eberhart R C (1995) Particle Swarm Optimization Proceedings of the 1995 IEEE International Conference on Neural Networks 1942–1948
Koh B, George A D, Haftka R T and Fregly B J (2006) Parallel Asynchronous Particle Swarm Optimization International Journal for Numerical Methods in Engineering 67(4) 578–595
Vaz A I and Vicente L N (2007) A Particle Swarm Pattern Search Method for Bound Constrained Global Optimization Journal of Global Optimization 39(2) 197–219 Kluwer Academic Publishers

## Parameters

Note: for descriptions of the symbolic variables, see Algorithmic Details.

### Compulsory Input Parameters

1:     $\mathrm{bl}\left({\mathbf{ndim}}\right)$ – double array
2:     $\mathrm{bu}\left({\mathbf{ndim}}\right)$ – double array
${\mathbf{bl}}$ is $\mathbf{\ell }$, the array of lower bounds, bu is $\mathbf{u}$, the array of upper bounds. The ndim entries in bl and bu must contain the lower and upper simple (box) bounds of the variables respectively. These must be provided to initialize the sample population into a finite hypervolume, although their subsequent influence on the algorithm is user determinable (see the option Boundary in Optional Parameters).
If ${\mathbf{bl}}\left(i\right)={\mathbf{bu}}\left(i\right)$ for any $i\in \left\{1,\dots ,{\mathbf{ndim}}\right\}$, variable $i$ will remain locked to ${\mathbf{bl}}\left(i\right)$ regardless of the Boundary option selected.
It is strongly advised that you place sensible lower and upper bounds on all variables, even if your model allows for variables to be unbounded (using the option ${\mathbf{Boundary}}=\mathrm{ignore}$) since these define the initial search space.
Constraints:
• ${\mathbf{bl}}\left(\mathit{i}\right)\le {\mathbf{bu}}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{ndim}}$;
• ${\mathbf{bl}}\left(i\right)\ne {\mathbf{bu}}\left(i\right)$ for at least one $i\in \left\{1,\dots ,{\mathbf{ndim}}\right\}$.
3:     $\mathrm{objfun}$ – function handle or string containing name of m-file
objfun must, depending on the value of mode, calculate the objective function and/or calculate the gradient of the objective function for a $\mathit{ndim}$-variable vector $\mathbf{x}$. Gradients are only required if a local minimizer has been chosen which requires gradients. See the option Local Minimizer for more information.
[mode, objf, vecout, user] = objfun(mode, ndim, x, objf, vecout, nstate, user)

Input Parameters

1:     $\mathrm{mode}$int64int32nag_int scalar
Indicates which functionality is required.
${\mathbf{mode}}=0$
$F\left(\mathbf{x}\right)$ should be returned in objf. The value of objf on entry may be used as an upper bound for the calculation. Any expected value of $F\left(\mathbf{x}\right)$ that is greater than objf may be approximated by this upper bound; that is objf can remain unaltered.
${\mathbf{mode}}=1$
${\mathbf{Local Minimizer}}=\mathrm{e04uc}$ only
First derivatives can be evaluated and returned in vecout. Any unaltered elements of vecout will be approximated using finite differences.
${\mathbf{mode}}=2$
${\mathbf{Local Minimizer}}=\mathrm{e04uc}$ only
$F\left(\mathbf{x}\right)$ must be calculated and returned in objf, and available first derivatives can be evaluated and returned in vecout. Any unaltered elements of vecout will be approximated using finite differences.
${\mathbf{mode}}=5$
$F\left(\mathbf{x}\right)$ must be calculated and returned in objf. The value of objf on entry may not be used as an upper bound.
${\mathbf{mode}}=6$
${\mathbf{Local Minimizer}}=\mathrm{e04dg}$ or $\mathrm{e04kz}$ only
All first derivatives must be evaluated and returned in vecout.
${\mathbf{mode}}=7$
${\mathbf{Local Minimizer}}=\mathrm{e04dg}$ or $\mathrm{e04kz}$ only
$F\left(\mathbf{x}\right)$ must be calculated and returned in objf, and all first derivatives must be evaluated and returned in vecout.
2:     $\mathrm{ndim}$int64int32nag_int scalar
The number of dimensions.
3:     $\mathrm{x}\left({\mathbf{ndim}}\right)$ – double array
$\mathbf{x}$, the point at which the objective function and/or its gradient are to be evaluated.
4:     $\mathrm{objf}$ – double scalar
The value of objf passed to objfun varies with the argument mode.
${\mathbf{mode}}=0$
objf is an upper bound for the value of $F\left(\mathbf{x}\right)$, often equal to the best value of $F\left(\mathbf{x}\right)$ found so far by a given particle. Only objective function values less than the value of objf on entry will be used further. As such this upper bound may be used to stop further evaluation when this will only increase the objective function value above the upper bound.
${\mathbf{mode}}=1$, $2$, $5$, $6$ or $7$
objf is meaningless on entry.
5:     $\mathrm{vecout}\left({\mathbf{ndim}}\right)$ – double array
If ${\mathbf{Local Minimizer}}=\mathrm{e04uc}$ or $\mathrm{e04uc}$, the values of vecout are used internally to indicate whether a finite difference approximation is required. See nag_opt_nlp1_solve (e04uc).
6:     $\mathrm{nstate}$int64int32nag_int scalar
nstate indicates various stages of initialization throughout the function. This allows for permanent global arguments to be initialized the least number of times. For example, you may initialize a random number generator seed.
${\mathbf{nstate}}=3$
SMP users only. objfun is called for the first time in a parallel region on a new thread other than the master thread. You may use this opportunity to set up any thread-dependent information in user and user.
${\mathbf{nstate}}=2$
objfun is called for the very first time. You may save computational time if certain data must be read or calculated only once.
${\mathbf{nstate}}=1$
objfun is called for the first time by a NAG local minimization function. You may save computational time if certain data required for the local minimizer need only be calculated at the initial point of the local minimization.
${\mathbf{nstate}}=0$
Used in all other cases.
7:     $\mathrm{user}$ – Any MATLAB object
objfun is called from nag_glopt_bnd_pso (e05sa) with the object supplied to nag_glopt_bnd_pso (e05sa).

Output Parameters

1:     $\mathrm{mode}$int64int32nag_int scalar
If the value of mode is set to be negative, then nag_glopt_bnd_pso (e05sa) will exit as soon as possible with ${\mathbf{ifail}}={\mathbf{3}}$ and ${\mathbf{inform}}={\mathbf{mode}}$.
2:     $\mathrm{objf}$ – double scalar
The value of objf returned varies with the argument mode.
${\mathbf{mode}}=0$
objf must be the value of $F\left(\mathbf{x}\right)$. Only values of $F\left(\mathbf{x}\right)$ strictly less than objf on entry need be accurate.
${\mathbf{mode}}=1$ or $6$
Need not be set.
${\mathbf{mode}}=2$, $5$ or $7$
$F\left(\mathbf{x}\right)$ must be calculated and returned in objf. The entry value of objf may not be used as an upper bound.
3:     $\mathrm{vecout}\left({\mathbf{ndim}}\right)$ – double array
The required values of vecout returned to the calling function depend on the value of mode.
${\mathbf{mode}}=0$ or $5$
The value of vecout need not be set.
${\mathbf{mode}}=1$ or $2$
vecout can contain components of the gradient of the objective function $\frac{\partial F}{\partial {x}_{i}}$ for some $i=1,2,\dots {\mathbf{ndim}}$, or acceptable approximations. Any unaltered elements of vecout will be approximated using finite differences.
${\mathbf{mode}}=6$ or $7$
vecout must contain the gradient of the objective function $\frac{\partial F}{\partial {x}_{i}}$ for all $i=1,2,\dots {\mathbf{ndim}}$. Approximation of the gradient is strongly discouraged, and no finite difference approximations will be performed internally (see nag_opt_uncon_conjgrd_comp (e04dg) and nag_opt_bounds_mod_deriv_easy (e04kz)).
4:     $\mathrm{user}$ – Any MATLAB object
4:     $\mathrm{monmod}$ – function handle or string containing name of m-file
A user-specified monitoring and modification function. monmod is called once every complete iteration after a finalization check. It may be used to modify the particle locations that will be evaluated at the next iteration. This permits the incorporation of algorithmic modifications such as including additional advection heuristics and genetic mutations. monmod is only called during the main loop of the algorithm, and as such will be unaware of any further improvement from the final local minimization. If no monitoring and/or modification is required, monmod may be string nag_glopt_bnd_pso_dummy_monmod (e05sxm)nag_glopt_bnd_pso_dummy_monmod (e05sxm) .
[x, user, inform] = monmod(ndim, npar, x, xb, fb, xbest, fbest, itt, user, inform)

Input Parameters

1:     $\mathrm{ndim}$int64int32nag_int scalar
The number of dimensions.
2:     $\mathrm{npar}$int64int32nag_int scalar
The number of particles.
3:     $\mathrm{x}\left({\mathbf{ndim}},{\mathbf{npar}}\right)$ – double array
Note: the $i$th component of the $j$th particle, ${x}_{j}\left(i\right)$, is stored in ${\mathbf{x}}\left(i,j\right)$.
The npar particle locations, ${\mathbf{x}}_{j}$, which will currently be used during the next iteration unless altered in monmod.
4:     $\mathrm{xb}\left({\mathbf{ndim}}\right)$ – double array
The location, $\stackrel{~}{\mathbf{x}}$, of the best solution yet found.
5:     $\mathrm{fb}$ – double scalar
The objective value, $\stackrel{~}{f}=F\left(\stackrel{~}{\mathbf{x}}\right)$, of the best solution yet found.
6:     $\mathrm{xbest}\left({\mathbf{ndim}},{\mathbf{npar}}\right)$ – double array
Note: the $i$th component of the position of the $j$th particle's cognitive memory, ${\stackrel{^}{x}}_{j}\left(i\right)$, is stored in ${\mathbf{xbest}}\left(i,j\right)$.
The locations currently in the cognitive memory, ${\stackrel{^}{\mathbf{x}}}_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,{\mathbf{npar}}$ (see Algorithmic Details).
7:     $\mathrm{fbest}\left({\mathbf{npar}}\right)$ – double array
The objective values currently in the cognitive memory, $F\left({\stackrel{^}{\mathbf{x}}}_{\mathit{j}}\right)$, for $\mathit{j}=1,2,\dots ,{\mathbf{npar}}$.
8:     $\mathrm{itt}\left(6\right)$int64int32nag_int array
Iteration and function evaluation counters (see description of itt below).
9:     $\mathrm{user}$ – Any MATLAB object
monmod is called from nag_glopt_bnd_pso (e05sa) with the object supplied to nag_glopt_bnd_pso (e05sa).
10:   $\mathrm{inform}$int64int32nag_int scalar
${\mathbf{inform}}=\mathbf{thread_num}$, where thread_num is the value returned by a call of the OpenMP function OMP_GET_THREAD_NUM(). If running in serial this will always be zero.

Output Parameters

1:     $\mathrm{x}\left({\mathbf{ndim}},{\mathbf{npar}}\right)$ – double array
Note: the $i$th component of the $j$th particle, ${x}_{j}\left(i\right)$, is stored in ${\mathbf{x}}\left(i,j\right)$.
The particle locations to be used during the next iteration.
2:     $\mathrm{user}$ – Any MATLAB object
3:     $\mathrm{inform}$int64int32nag_int scalar
Setting ${\mathbf{inform}}<0$ will cause near immediate exit from nag_glopt_bnd_pso (e05sa). This value will be returned as inform with ${\mathbf{ifail}}={\mathbf{3}}$. You need not set inform unless you wish to force an exit.
5:     $\mathrm{iopts}\left(:\right)$int64int32nag_int array
Note: the dimension of this array is dictated by the requirements of associated functions that must have been previously called. This array must be the same array passed as argument iopts in the previous call to nag_glopt_optset (e05zk).
Optional parameter array as generated and possibly modified by calls to nag_glopt_optset (e05zk). The contents of iopts must not be modified directly between calls to nag_glopt_bnd_pso (e05sa), nag_glopt_optset (e05zk) or nag_glopt_optget (e05zl).
6:     $\mathrm{opts}\left(:\right)$ – double array
Note: the dimension of this array is dictated by the requirements of associated functions that must have been previously called. This array must be the same array passed as argument opts in the previous call to nag_glopt_optset (e05zk).
Optional parameter array as generated and possibly modified by calls to nag_glopt_optset (e05zk). The contents of opts must not be modified directly between calls to nag_glopt_bnd_pso (e05sa), nag_glopt_optset (e05zk) or nag_glopt_optget (e05zl).

### Optional Input Parameters

1:     $\mathrm{ndim}$int64int32nag_int scalar
Default: the dimension of the arrays bl, bu. (An error is raised if these dimensions are not equal.)
$\mathit{ndim}$, the number of dimensions.
Constraint: ${\mathbf{ndim}}\ge 1$.
2:     $\mathrm{npar}$int64int32nag_int scalar
Default: ${\mathbf{npar}}=10×{\mathbf{ndim}}$
$\mathit{npar}$, the number of particles to be used in the swarm. Assuming all particles remain within bounds, each complete iteration will perform at least npar function evaluations. Otherwise, significantly fewer objective function evaluations may be performed.
Constraint: ${\mathbf{npar}}\ge 5×\mathbf{num_threads}$, where num_threads is the value returned by the OpenMP environment variable OMP_NUM_THREADS, or num_threads is $1$ for a serial version of this function.
3:     $\mathrm{user}$ – Any MATLAB object
user is not used by nag_glopt_bnd_pso (e05sa), but is passed to objfun and monmod. 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:     $\mathrm{xb}\left({\mathbf{ndim}}\right)$ – double array
The location of the best solution found, $\stackrel{~}{\mathbf{x}}$, in ${R}^{\mathit{ndim}}$.
2:     $\mathrm{fb}$ – double scalar
The objective value of the best solution, $\stackrel{~}{f}=F\left(\stackrel{~}{\mathbf{x}}\right)$.
3:     $\mathrm{iopts}\left(:\right)$int64int32nag_int array
Note: the dimension of this array is dictated by the requirements of associated functions that must have been previously called. This array must be the same array passed as argument iopts in the previous call to nag_glopt_optset (e05zk).
Communication array, used to store information between calls to nag_glopt_bnd_pso (e05sa).
4:     $\mathrm{opts}\left(:\right)$ – double array
Note: the dimension of this array is dictated by the requirements of associated functions that must have been previously called. This array must be the same array passed as argument opts in the previous call to nag_glopt_optset (e05zk).
Communication array, used to store information between calls to nag_glopt_bnd_pso (e05sa).
5:     $\mathrm{user}$ – Any MATLAB object
6:     $\mathrm{itt}\left(6\right)$int64int32nag_int array
Integer iteration counters for nag_glopt_bnd_pso (e05sa).
${\mathbf{itt}}\left(1\right)$
Number of complete iterations.
${\mathbf{itt}}\left(2\right)$
Number of complete iterations without improvement to the current optimum.
${\mathbf{itt}}\left(3\right)$
Number of particles converged to the current optimum.
${\mathbf{itt}}\left(4\right)$
Number of improvements to the optimum.
${\mathbf{itt}}\left(5\right)$
Number of function evaluations performed.
${\mathbf{itt}}\left(6\right)$
Number of particles reset.
7:     $\mathrm{inform}$int64int32nag_int scalar
Indicates which finalization criterion was reached. The possible values of inform are:
 inform Meaning $<0$ Exit from a user-supplied subroutine. 0 nag_glopt_bnd_pso (e05sa) has detected an error and terminated. 1 The provided objective target has been achieved. (Target Objective Value). 2 The standard deviation of the location of all the particles is below the set threshold (Swarm Standard Deviation). If the solution returned is not satisfactory, you may try setting a smaller value of Swarm Standard Deviation, or try adjusting the options governing the repulsive phase (Repulsion Initialize, Repulsion Finalize). 3 The total number of particles converged (Maximum Particles Converged) to the current global optimum has reached the set limit. This is the number of particles which have moved to a distance less than Distance Tolerance from the optimum with regard to the ${L}^{2}$ norm. If the solution is not satisfactory, you may consider lowering the Distance Tolerance. However, this may hinder the global search capability of the algorithm. 4 The maximum number of iterations without improvement (Maximum Iterations Static) has been reached, and the required number of particles (Maximum Iterations Static Particles) have converged to the current optimum. Increasing either of these options will allow the algorithm to continue searching for longer. Alternatively if the solution is not satisfactory, re-starting the application several times with ${\mathbf{Repeatability}}=\mathrm{OFF}$ may lead to an improved solution. 5 The maximum number of iterations (Maximum Iterations Completed) has been reached. If the number of iterations since improvement is small, then a better solution may be found by increasing this limit, or by using the option Local Minimizer with corresponding exterior options. Otherwise if the solution is not satisfactory, you may try re-running the application several times with ${\mathbf{Repeatability}}=\mathrm{OFF}$ and a lower iteration limit, or adjusting the options governing the repulsive phase (Repulsion Initialize, Repulsion Finalize). 6 The maximum allowed number of function evaluations (Maximum Function Evaluations) has been reached. As with ${\mathbf{inform}}=5$, increasing this limit if the number of iterations without improvement is small, or decreasing this limit and running the algorithm multiple times with ${\mathbf{Repeatability}}=\mathrm{OFF}$, may provide a superior result.
8:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).
For this reason, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended; otherwise, the recommended value is $-1$. When the value $-\mathbf{1}\text{​ or ​}1$ is used it is essential to test the value of ifail on exit.
nag_glopt_bnd_pso (e05sa) will return ${\mathbf{ifail}}={\mathbf{0}}$ if and only if a finalization criterion has been reached which can guarantee success. This may only happen if the option Target Objective Value has been set and reached at a point within the search domain. The finalization criterion Target Objective Value is not activated using default option settings, and must be explicitly set using nag_glopt_optset (e05zk) if required.
nag_glopt_bnd_pso (e05sa) will return ${\mathbf{ifail}}={\mathbf{1}}$ if no error has been detected, and a finalization criterion has been achieved which cannot guarantee success. This does not indicate that the function has failed, merely that the returned solution cannot be guaranteed to be the true global optimum.
The value of inform should be examined to determine which finalization criterion was reached.
Other positive values of ifail indicate that either an error or a warning has been triggered. See Error Indicators and Warnings, Accuracy and Algorithmic Details for more information.

## Error Indicators and 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.

W  ${\mathbf{ifail}}=1$
A finalization criterion was reached that cannot guarantee success.
W  ${\mathbf{ifail}}=2$
If the option Target Warning has been activated, this indicates that the Target Objective Value has been achieved to specified tolerances at a sufficiently constrained point, either during the initialization phase, or during the first two iterations of the algorithm. While this is not necessarily an error, it may occur if:
 (i) The target was achieved at the first point sampled by the function. This will be the mean of the lower and upper bounds. (ii) The target may have been achieved at a randomly generated sample point. This will always be a possibility provided that the domain under investigation contains a point with a target objective value. (iii) If the Local Minimizer has been set, then a sample point may have been inside the basin of attraction of a satisfactory point. If this occurs repeatedly when the function is called, it may imply that the objective is largely unimodal, and that it may be more efficient to use the function selected as the Local Minimizer directly.
Assuming that objfun is correct, you may wish to set a better Target Objective Value, or a stricter Target Objective Tolerance.
W  ${\mathbf{ifail}}=3$
User requested exit during call to monmod.
User requested exit during call to objfun.
${\mathbf{ifail}}=11$
Constraint: ${\mathbf{ndim}}\ge 1$.
${\mathbf{ifail}}=12$
Constraint: ${\mathbf{npar}}\ge 5×\mathbf{num_threads}$, where num_threads is the value returned by the OpenMP environment variable OMP_NUM_THREADS, or num_threads is $1$ for a serial version of this function.
${\mathbf{ifail}}=14$
Constraint: ${\mathbf{bu}}\left(i\right)\ge {\mathbf{bl}}\left(i\right)$ for all $i$.
On entry, ${\mathbf{bl}}\left(i\right)={\mathbf{bu}}\left(i\right)$ for all $i$.
Constraint: ${\mathbf{bu}}\left(i\right)>{\mathbf{bl}}\left(i\right)$ for at least one $i$.
${\mathbf{ifail}}=19$
Error $_$ occurred whilst adjusting to exterior local minimizer options.
Error $_$ occurred whilst adjusting to interior local minimizer options.
${\mathbf{ifail}}=21$
Either the option arrays have not been initialized for nag_glopt_bnd_pso (e05sa), or they have become corrupted.
W  ${\mathbf{ifail}}=32$
Derivative checks indicate possible errors in the supplied derivatives. Gradient checks may be disabled by setting ${\mathbf{Verify Gradients}}=\mathrm{OFF}$.
${\mathbf{ifail}}=-99$
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

If ${\mathbf{ifail}}={\mathbf{0}}$ (or ${\mathbf{ifail}}={\mathbf{2}}$) or ${\mathbf{ifail}}={\mathbf{1}}$ on exit, either a Target Objective Value or finalization criterion has been reached, depending on user selected options. As with all global optimization software, the solution achieved may not be the true global optimum. Various options allow for either greater search diversity or faster convergence to a (local) optimum (See Algorithmic Details and Optional Parameters).
Provided the objective function and constraints are sufficiently well behaved, if a local minimizer is used in conjunction with nag_glopt_bnd_pso (e05sa), then it is more likely that the final result will at least be in the near vicinity of a local optimum, and due to the global search characteristics of the particle swarm, this solution should be superior to many other local optima.
Caution should be used in accelerating the rate of convergence, as with faster convergence, less of the domain will remain searchable by the swarm, making it increasingly difficult for the algorithm to detect the basin of attraction of superior local optima. Using the options Repulsion Initialize and Repulsion Finalize described in Optional Parameters will help to overcome this, by causing the swarm to diverge away from the current optimum once no more local improvement is likely.
On successful exit with guaranteed success, ${\mathbf{ifail}}={\mathbf{0}}$. This may only happen if a Target Objective Value is assigned and is reached by the algorithm.
On successful exit without guaranteed success, ${\mathbf{ifail}}={\mathbf{1}}$ is returned. This will happen if another finalization criterion is achieved without the detection of an error.
In both cases, the value of inform provides further information as to the cause of the exit.

The memory used by nag_glopt_bnd_pso (e05sa) is relatively static throughout. As such, nag_glopt_bnd_pso (e05sa) may be used in problems with high dimension number (${\mathbf{ndim}}>100$) without the concern of computational resource exhaustion, although the probability of successfully locating the global optimum will decrease dramatically with the increase in dimensionality.
Due to the stochastic nature of the algorithm, the result will vary over multiple runs. This is particularly true if arguments and options are chosen to accelerate convergence at the expense of the global search. However, the option ${\mathbf{Repeatability}}=\mathrm{ON}$ may be set to initialize the internal random number generator using a preset seed, which will result in identical solutions being obtained.

## Example

This example uses a particle swarm to find the global minimum of the Schwefel function:
 $minimize x∈Rndim f = ∑ i=1 ndim xi sinxi$
 $xi ∈ -500,500 , for ​ i=1,2,…,ndim .$
In two dimensions the optimum is ${f}_{\mathrm{min}}=-837.966$, located at $\mathbf{x}=\left(-420.97,-420.97\right)$.
The example demonstrates how to initialize and set the options arrays using nag_glopt_optset (e05zk), how to query options using nag_glopt_optget (e05zl), and finally how to search for the global optimum using nag_glopt_bnd_pso (e05sa). The function is minimized several times to demonstrate using nag_glopt_bnd_pso (e05sa) alone, and coupled with local minimizers. This program uses the non-default option ${\mathbf{Repeatability}}=\mathrm{ON}$ to produce repeatable solutions.
```function e05sa_example

fprintf('e05sa example results\n\n');

bu = [ 500,  500];
bl = [-500, -500];

npar = int64(5);

x_target = [-420.9687463599820;-420.9687463599820];
f_target = -837.9657745448674;

iopts = zeros(100, 1, 'int64');
opts  = zeros(100, 1);

% Initialize the option arrays for e05sa
[iopts, opts, ifail] = e05zk('Initialize = e05sa', iopts, opts);

% Query some default option values.
[ivalue, rvalue, boundary, optype, ifail] = ...
e05zl(...
'Boundary', iopts, opts);
[maxits, rvalue, itsstr, optype, ifail] = ...
e05zl(...
'Maximum Iterations Completed', iopts, opts);
[ivalue, distol, cvalue, optype, ifail] = ...
e05zl(...
'Distance Tolerance', iopts, opts);
fprintf('\nDefault Option Queries:\n\n');
fprintf('Boundary                     : %s\n', boundary);
fprintf('Maximum Iterations Completed : %d (%s)\n', maxits, strtrim(itsstr));
fprintf('Distance Tolerance           : %10.4e\n', distol);

fprintf('\n1. Solution without using coupled local minimizer.\n');

% Set various options to non-default values if required.
[iopts, opts, ifail] = e05zk(...
'Repeatability = On', iopts, opts);
[iopts, opts, ifail] = e05zk(...
'Verify Gradients = Off', iopts, opts);
[iopts, opts, ifail] = e05zk(...
'Boundary = Hyperspherical', iopts, opts);
[iopts, opts, ifail] = e05zk(...
'Maximum iterations static = 150', iopts, opts);
[iopts, opts, ifail] = e05zk(...
'Repulsion Initialize = 30', iopts, opts);
[iopts, opts, ifail] = e05zk(...
'Repulsion Finalize = 30', iopts, opts);

% Catch warnings and assume ifail=1,3 give a reasonable estimate
wstat = warning();
warning('OFF');
% Call e05sa to search for the global optimum.
[xb, fb, iopts, opts, user, itt, inform, ifail] = ...
e05sa(bl, bu, @objfun, 'e05sxm', iopts, opts, 'npar', npar);

switch ifail
case {0,1}
% e05sa encountered no errors during operation,
% and will have returned the best optimum found.
display_result(x_target, f_target, xb,fb,itt,inform);
case 3
% An instruction to exit was received by e05sa from objfun or monmod.
% The exit flag will have been returned in inform.
display_result(x_target, f_target, xb,fb,itt,inform);
otherwise
% An error was detected, and a warning has been displayed
end

fprintf('\n2. Solution using coupled local minimizer e04cb.\n');

% Set an objective target
optstr = sprintf('Target Objective Value = %32.16e', f_target);
[iopts, opts, ifail] = e05zk(optstr, iopts, opts);
[iopts, opts, ifail] = e05zk(...
'Target Objective Tolerance = 1.0e-5', iopts, opts);
[iopts, opts, ifail] = e05zk(...
'Target Objective Safeguard = 1.0e-8', iopts, opts);

% Set the local minimizer to be e04cb and set corresponding options
[iopts, opts, ifail] = e05zk(...
'Local Minimizer = e04cb', iopts, opts);
[iopts, opts, ifail] = e05zk(...
'Local Interior Iterations = 10', iopts, opts);
[iopts, opts, ifail] = e05zk(...
'Local Exterior Iterations = 20', iopts, opts);
[iopts, opts, ifail] = e05zk(...
'Local Interior Tolerance = 1.0e-4', iopts, opts);
[iopts, opts, ifail] = e05zk(...
'Local Exterior Tolerance = 1.0e-4', iopts, opts);

% Call e05sa to search for the global optimum.
[xb, fb, iopts, opts, user, itt, inform, ifail] = ...
e05sa(...
bl, bu, @objfun, 'e05sxm', iopts, opts, 'npar', npar);

switch ifail
case {0,1}
% e05sa encountered no errors during operation,
% and will have returned the best optimum found.
display_result(x_target, f_target, xb,fb,itt,inform);
case 3
% An instruction to exit was received by e05sa from objfun or monmod.
% The exit flag will have been returned in inform.
display_result(x_target, f_target, xb,fb,itt,inform);
otherwise
% An error was detected, and a warning has been displayed
end

fprintf('\n3. Solution using coupled local minimizer e04dg.\n');

% Set the local minimizer to be e04dg and set corresponding options
[iopts, opts, ifail] = e05zk(...
'Local Minimizer = e04dg', iopts, opts);
[iopts, opts, ifail] = e05zk(...
'Local Interior Iterations = 5', iopts, opts);
[iopts, opts, ifail] = e05zk(...
'Local Exterior Iterations = 20', iopts, opts);

%[iopts, opts, ifail] = e05zk('Verify Gradients = off', iopts, opts);
% Call e05sa to search for the global optimum.
[xb, fb, iopts, opts, user, itt, inform, ifail] = ...
e05sa(...
bl, bu, @objfun, 'e05sxm', iopts, opts, 'npar', npar);

switch ifail
case {0,1}
% e05sa encountered no errors during operation,
% and will have returned the best optimum found.
display_result(x_target, f_target, xb,fb,itt,inform);
case 3
% An instruction to exit was received by e05sa from objfun or monmod.
% The exit flag will have been returned in inform.
display_result(x_target, f_target, xb,fb,itt,inform);
otherwise
% An error was detected, and a warning has been displayed
end

warning(wstat);

function [mode, objf, vecout, user] = ...
objfun(mode, ndim, x, objf, vecout, nstate, user)
% Test nstate to indicate what stage of computation has been reached.
switch nstate
case (2)
% objfun is called for the very first time.
case (-1)
% objfun is called for the first time on a new slave thread.
% This will never happen if running in a serial context.
case (1)
% objfun is called on entry to a NAG local minimiser.
case (0)
% This will be the normal value of NSTATE.
otherwise
% This is extremely unlikely, and indicates that an error has
% occurred on the system
mode = int64(-1);
error('*** Error detected in objfun');
end

% Test mode to determine whether to calculate objf and/or objgrd.
evalobjf = false;
evalobjg = false;
switch mode
case {0,5}
% Only the value of the objective function is needed.
evalobjf = true;
case {1,6}
% Only the values of the NDIM gradients are required.
evalobjg = true;
case {2,7}
% Both the objective function and the NDIM gradients are required.
evalobjf = true;
evalobjg = true;
otherwise
mode = int64(-1);
error('*** Illegal value of mode (%d) in objfun', mode);
end

if evalobjf
% Evaluate the objective function.
objf = sum(x(1:double(ndim)).*sin(sqrt(abs(x(1:double(ndim))))));
end

if evalobjg
% Calculate the gradient of the objective function,
% and return the result in vecout.
vecout = sqrt(abs(x));
for i=1:double(ndim)
if abs(x(i)) < x02aj
vecout(i) = 0;
else
v = vecout(i);
vecout(i) = sin(v) + signtransfer(x(i)*cos(v)/(2*v), x(i));
end
end
end

function [r] = signtransfer(x, y)
if y >= 0
r = abs(x);
else
r = -abs(x);
end

function [] = display_result(x_target, f_target, xb,fb,itt,inform)

% Display final counters.
fprintf('\nAlgorithm Statistics\n--------------------\n');
fprintf('Total complete iterations             : %4d\n', itt(1));
fprintf('Complete iterations since improvement : %4d\n', itt(2));
fprintf('Total particles converged to xb       : %4d\n', itt(3));
fprintf('Total improvements to global optimum  : %4d\n', itt(4));
fprintf('Total function evaluations            : %4d\n', itt(5));
fprintf('Total particles re-initialized        : %4d\n\n', itt(6));

% Display why finalization occurred.
switch inform
case 0
fprintf('Solution Status : An error was detected by e05sa\n');
case 1
fprintf('Solution Status : Target value achieved\n');
case 2
fprintf('Solution Status : Minimum swarm standard deviation obtained\n');
case 3
fprintf('Solution Status : Sufficient particles converged\n');
case 4
fprintf('Solution Status : No improvement in preset iteration limit\n');
case 5
fprintf('Solution Status : Maximum complete iterations attained\n');
case 6
fprintf('Solution Status : Maximum function evaluations exceeded\n');
otherwise
fprintf('User termination case:  %d\n', inform);
end

% Display final objective value and location.
fprintf('\n  Known objective optimum  : %13.5f\n', f_target);
fprintf('  Achieved objective value : %13.5f\n\n', fb);

title = 'Comparison between known and achieved optima.';
clabs = {'x_target'; 'xb      '};
ncols =  int64(80);
indent = int64(0);
[ifail] = x04cb(...
'G', 'N', [x_target, xb], 'f9.2', title, 'I', clabs, ...
'C', clabs, ncols, indent);
fprintf('\n');
```
```e05sa example results

Default Option Queries:

Boundary                     : FLOATING
Maximum Iterations Completed : 1000 (DEFAULT)
Distance Tolerance           : 1.0000e-04

1. Solution without using coupled local minimizer.

Algorithm Statistics
--------------------
Total complete iterations             :  395
Complete iterations since improvement :  152
Total particles converged to xb       :    2
Total improvements to global optimum  :   59
Total function evaluations            : 2773
Total particles re-initialized        :    2

Solution Status : No improvement in preset iteration limit

Known objective optimum  :    -837.96577
Achieved objective value :    -837.96567

Comparison between known and achieved optima.
x_target       xb
1   -420.97  -420.95
2   -420.97  -420.94

2. Solution using coupled local minimizer e04cb.

Algorithm Statistics
--------------------
Total complete iterations             :   51
Complete iterations since improvement :    1
Total particles converged to xb       :    0
Total improvements to global optimum  :   12
Total function evaluations            :  537
Total particles re-initialized        :    0

Solution Status : Target value achieved

Known objective optimum  :    -837.96577
Achieved objective value :    -837.96577

Comparison between known and achieved optima.
x_target       xb
1   -420.97  -420.97
2   -420.97  -420.97

3. Solution using coupled local minimizer e04dg.

Algorithm Statistics
--------------------
Total complete iterations             :    8
Complete iterations since improvement :    1
Total particles converged to xb       :    0
Total improvements to global optimum  :   10
Total function evaluations            :  120
Total particles re-initialized        :    0

Solution Status : Target value achieved

Known objective optimum  :    -837.96577
Achieved objective value :    -837.96561

Comparison between known and achieved optima.
x_target       xb
1   -420.97  -420.94
2   -420.97  -420.98

```

## Algorithmic Details

The following pseudo-code describes the algorithm used with the repulsion mechanism.
 $INITIALIZE for ​ j=1, ​ np xj = R ∈ Uℓbox,ubox x^ j = R∈ Uℓbox,ubox vj = R ∈ U -V max ,Vmax f^j = F x^ j initialize ​wj wj = Wmax Weight Initialize=MAXIMUM Wini Weight Initialize=INITIAL R ∈ U W min , W max Weight Initialize=RANDOMIZED end for x~ = 12 ℓbox + ubox f~ = F x~ Ic = Is = 0 SWARM while (not finalized), Ic = Ic + 1 for ​ j = 1 , np xj = BOUNDARYxj,ℓbox,ubox fj = F xj if ​ fj < f^j f^j = fj , ​ x^j = xj if ​ fj < f~ f~ = fj , ​ x~ = xj end for if ​ new f~ LOCMINx~,f~,Oi , ​ Is=0 [see note on repulsion below for code insertion] else Is = Is + 1 for ​ j = 1 , np vj = wj vj + Cs D1 x^j - xj + Cg D2 x~ - xj xj = xj + vj if ​ xj - x~ < dtol reset ​ xj, ​ vj, ​ wj; ​ x^j = xj else update ​wj end for if (target achieved or termination criterion satisfied) finalized=true monmod xj end LOCMINx~,f~,Oe$
The definition of terms used in the above pseudo-code are as follows.
 ${n}_{p}$ the number of particles, npar ${\mathbf{\ell }}_{\mathrm{box}}$ array of ndim lower box bounds ${\mathbf{u}}_{\mathrm{box}}$ array of ndim upper box bounds ${\mathbf{x}}_{j}$ position of particle $j$ ${\stackrel{^}{\mathbf{x}}}_{j}$ best position found by particle $j$ $\stackrel{~}{\mathbf{x}}$ best position found by any particle ${f}_{j}$ $F\left({\mathbf{x}}_{j}\right)$ ${\stackrel{^}{f}}_{j}$ $F\left({\stackrel{^}{\mathbf{x}}}_{j}\right)$, best value found by particle $j$ $\stackrel{~}{f}$ $F\left(\stackrel{~}{\mathbf{x}}\right)$, best value found by any particle ${\mathbf{v}}_{j}$ velocity of particle $j$ ${w}_{j}$ weight on ${\mathbf{v}}_{j}$ for velocity update, decreasing according to Weight Decrease ${V}_{\mathrm{max}}$ maximum absolute velocity, dependent upon Maximum Variable Velocity ${I}_{c}$ swarm iteration counter ${I}_{s}$ iterations since $\stackrel{~}{\mathbf{x}}$ was updated ${\mathbf{D}}_{1}$,${\mathbf{D}}_{2}$ diagonal matrices with random elements in range $\left(0,1\right)$ ${C}_{s}$ the cognitive advance coefficient which weights velocity towards ${\stackrel{^}{\mathbf{x}}}_{j}$, adjusted using Advance Cognitive ${C}_{g}$ the global advance coefficient which weights velocity towards $\stackrel{~}{\mathbf{x}}$, adjusted using Advance Global $\mathit{dtol}$ the Distance Tolerance for resetting a converged particle $\mathbf{R}\in U\left({\mathbf{\ell }}_{\mathrm{box}},{\mathbf{u}}_{\mathrm{box}}\right)$ an array of random numbers whose $i$-th element is drawn from a uniform distribution in the range $\left({{\mathbf{\ell }}_{\mathrm{box}}}_{\mathit{i}},{{\mathbf{u}}_{\mathrm{box}}}_{\mathit{i}}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{ndim}}$ ${O}_{i}$ local optimizer interior options ${O}_{e}$ local optimizer exterior options $\mathrm{LOCMIN}\left(\mathbf{x},f,O\right)$ apply local optimizer using the set of options $O$ using the solution $\left(\mathbf{x},f\right)$ as the starting point, if used (not default) monmod monitor progress and possibly modify ${\mathbf{x}}_{j}$ BOUNDARY apply required behaviour for ${\mathbf{x}}_{j}$ outside bounding box, (see Boundary) new ($\stackrel{~}{f}$) true if $\stackrel{~}{\mathbf{x}}$, $\stackrel{~}{\mathbf{c}}$, $\stackrel{~}{f}$ were updated at this iteration
Additionally a repulsion phase can be introduced by changing from the default values of options Repulsion Finalize (${r}_{f}$), Repulsion Initialize (${r}_{i}$) and Repulsion Particles (${r}_{p}$). If the number of static particles is denoted ${n}_{s}$ then the following can be inserted after the new($\stackrel{~}{f}$) check in the pseudo-code above.
 $else ​ if ​ ( ns ≥ rp ​ and ​ ri ≤ Is ≤ ri + rf ) LOCMINx~,f~,Oi use ​ -Cg ​ instead of ​ Cg ​ in velocity updates if ​ Is = ri + rf Is = 0$

## Optional Parameters

This section can be skipped if you wish to use the default values for all optional parameters, otherwise, the following is a list of the optional parameters available and a full description of each optional parameter is provided in Description of the s.

### Description of the Optional Parameters

For each option, we give a summary line, a description of the optional parameter and details of constraints.
The summary line contains:
• the keywords;
• a parameter value, where the letters $a$, $i$ and $r$ denote options that take character, integer and real values respectively;
• the default value, where the symbol $\epsilon$ is a generic notation for machine precision (see nag_machine_precision (x02aj)), and $\mathit{Imax}$ represents the largest representable integer value (see nag_machine_integer_max (x02bb)).
All options accept the value ‘DEFAULT’ in order to return single options to their default states.
Keywords and character values are case insensitive, however they must be separated by at least one space.
For nag_glopt_bnd_pso (e05sa) the maximum length of the argument cvalue used by nag_glopt_optget (e05zl) is $15$.
Advance Cognitive  $r$
Default$\text{}=2.0$
The cognitive advance coefficient, ${C}_{s}$. When larger than the global advance coefficient, this will cause particles to be attracted toward their previous best positions. Setting $r=0.0$ will cause nag_glopt_bnd_pso (e05sa) to act predominantly as a local optimizer. Setting $r>2.0$ may cause the swarm to diverge, and is generally inadvisable. At least one of the global and cognitive coefficients must be nonzero.
Advance Global  $r$
Default$\text{}=2.0$
The global advance coefficient, ${C}_{g}$. When larger than the cognitive coefficient this will encourage convergence toward the best solution yet found. Values $r\in \left(0,1\right)$ will inhibit particles overshooting the optimum. Values $r\in \left[1,2\right)$ cause particles to fly over the optimum some of the time. Larger values can prohibit convergence. Setting $r=0.0$ will remove any attraction to the current optimum, effectively generating a Monte–Carlo multi-start optimization algorithm. At least one of the global and cognitive coefficients must be nonzero.
Boundary  $a$
Default$\text{}=\mathrm{FLOATING}$
Determines the behaviour if particles leave the domain described by the box bounds. This only affects the general PSO algorithm, and will not pass down to any NAG local minimizers chosen.
This option is only effective in those dimensions for which ${\mathbf{bl}}\left(i\right)\ne {\mathbf{bu}}\left(i\right)$, $i=1,2,\dots ,{\mathbf{ndim}}$.
IGNORE
The box bounds are ignored. The objective function is still evaluated at the new particle position.
RESET
The particle is re-initialized inside the domain. ${\stackrel{^}{\mathbf{x}}}_{j}$ and ${\stackrel{^}{f}}_{j}$ are not affected.
FLOATING
The particle position remains the same, however the objective function will not be evaluated at the next iteration. The particle will probably be advected back into the domain at the next advance due to attraction by the cognitive and global memory.
HYPERSPHERICAL
The box bounds are wrapped around an $\mathit{ndim}$-dimensional hypersphere. As such a particle leaving through a lower bound will immediately re-enter through the corresponding upper bound and vice versa. The standard distance between particles is also modified accordingly.
FIXED
The particle rests on the boundary, with the corresponding dimensional velocity set to $0.0$.
Distance Scaling  $a$
Default$\text{}=\mathrm{ON}$
Determines whether distances should be scaled by box widths.
ON
When a distance is calculated between $\mathbf{x}$ and $\mathbf{y}$, a scaled ${L}^{2}$ norm is used.
 $L2 x,y = ∑ i | ui ≠ ℓi , i≤ndim xi - yi ui - ℓi 2 1 2 .$
OFF
Distances are calculated as the standard ${L}^{2}$ norm without any rescaling.
 $L2 x,y = ∑ i=1 ndim xi - yi 2 1 2 .$
Distance Tolerance  $r$
Default$\text{}={10}^{-4}$
This is the distance, $\mathit{dtol}$ between particles and the global optimum which must be reached for the particle to be considered converged, i.e., that any subsequent movement of such a particle cannot significantly alter the global optimum. Once achieved the particle is reset into the box bounds to continue searching.
Constraint: $r>0.0$.
Function Precision  $r$
Default$\text{}={\epsilon }^{0.9}$
The parameter defines ${\epsilon }_{r}$, which is intended to be a measure of the accuracy with which the problem function $F\left(\mathbf{x}\right)$ can be computed. If $r<\epsilon$ or $r\ge 1$, the default value is used.
The value of ${\epsilon }_{r}$ should reflect the relative precision of $1+\left|F\left(\mathbf{x}\right)\right|$; i.e., ${\epsilon }_{r}$ acts as a relative precision when $\left|F\right|$ is large, and as an absolute precision when $\left|F\right|$ is small. For example, if $F\left(\mathbf{x}\right)$ is typically of order $1000$ and the first six significant digits are known to be correct, an appropriate value for ${\epsilon }_{r}$ would be ${10}^{-6}$. In contrast, if $F\left(\mathbf{x}\right)$ is typically of order ${10}^{-4}$ and the first six significant digits are known to be correct, an appropriate value for ${\epsilon }_{r}$ would be ${10}^{-10}$. The choice of ${\epsilon }_{r}$ can be quite complicated for badly scaled problems; see Chapter 8 of Gill et al. (1981) for a discussion of scaling techniques. The default value is appropriate for most simple functions that are computed with full accuracy. However when the accuracy of the computed function values is known to be significantly worse than full precision, the value of ${\epsilon }_{r}$ should be large enough so that no attempt will be made to distinguish between function values that differ by less than the error inherent in the calculation.
Local Boundary Restriction  $r$
Default$\text{}=0.5$
Contracts the box boundaries used by a box constrained local minimizer to, $\left[{\beta }_{l},{\beta }_{u}\right]$, containing the start point $x$, where
 $∂i = r × ui - ℓi βli = maxℓi, xi - ∂i2 βui = minui, xi + ∂i2 , i=1,…,ndim .$
Smaller values of $r$ thereby restrict the size of the domain exposed to the local minimizer, possibly reducing the amount of work done by the local minimizer.
Constraint: $0.0\le r\le 1.0$.
Local Interior Iterations  ${i}_{1}$
Local Interior Major Iterations  ${i}_{1}$
Local Exterior Iterations  ${i}_{2}$
Local Exterior Major Iterations  ${i}_{2}$
The maximum number of iterations or function evaluations the chosen local minimizer will perform inside (outside) the main loop if applicable. For the NAG minimizers these correspond to:
 Minimizer Parameter/option Default Interior Default Exterior nag_opt_uncon_simplex (e04cb) maxcal ${\mathbf{ndim}}+10$ $2×{\mathbf{ndim}}+15$ nag_opt_uncon_conjgrd_comp (e04dg) Iteration Limit $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(30,3×{\mathbf{ndim}}\right)$ $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(50,5×{\mathbf{ndim}}\right)$ nag_opt_nlp1_solve (e04uc) Major Iteration Limit $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(10,2×{\mathbf{ndim}}\right)$ $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(30,3×{\mathbf{ndim}}\right)$
Unless set, these are functions of the parameters passed to nag_glopt_bnd_pso (e05sa).
Setting $i=0$ will disable the local minimizer in the corresponding algorithmic region. For example, setting ${\mathbf{Local Interior Iterations}}=0$ and ${\mathbf{Local Exterior Iterations}}=30$ will cause the algorithm to perform no local minimizations inside the main loop of the algorithm, and a local minimization with upto $30$ iterations after the main loop has been exited.
Note: currently nag_opt_bounds_quasi_func_easy (e04jy) or nag_opt_bounds_mod_deriv_easy (e04kz) are restricted to using $400×{\mathbf{ndim}}$ and $50×{\mathbf{ndim}}$ as function evaluation limits respectively. This applies to both local minimizations inside and outside the main loop. They may still be deactivated in either phase by setting $i=0$, and may subsequently be reactivated in either phase by setting $i>0$.
Constraint: ${i}_{1}\ge 0$, ${i}_{2}\ge 0$.
Local Interior Tolerance  ${r}_{1}$
Default$\text{}={10}^{-4}$
Local Exterior Tolerance  ${r}_{2}$
Default$\text{}={10}^{-4}$
This is the tolerance provided to a local minimizer in the interior (exterior) of the main loop of the algorithm.
Constraint: ${r}_{1}>0.0$, ${r}_{2}>0.0$.
Local Interior Minor Iterations  ${i}_{1}$
Local Exterior Minor Iterations  ${i}_{2}$
Where applicable, the secondary number of iterations the chosen local minimizer will use inside (outside) the main loop. Currently the relevant default values are:
 Minimizer Parameter/option Default Interior Default Exterior nag_opt_nlp1_solve (e04uc) Minor Iteration Limit $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(10,2×{\mathbf{ndim}}\right)$ $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(30,3×{\mathbf{ndim}}\right)$
Constraint: ${i}_{1}\ge 0$, ${i}_{2}\ge 0$.
Local Minimizer  $a$
Default$\text{}=\mathrm{OFF}$
Allows for a choice of Chapter E04 functions to be used as a coupled, dedicated local minimizer.
OFF
No local minimization will be performed in either the INTERIOR or EXTERIOR sections of the algorithm.
nag_opt_uncon_simplex (e04cb)
Use nag_opt_uncon_simplex (e04cb) as the local minimizer. This does not require the calculation of derivatives.
On a call to objfun during a local minimization, ${\mathbf{mode}}=5$.
nag_opt_bounds_mod_deriv_easy (e04kz)
Use nag_opt_bounds_mod_deriv_easy (e04kz) as the local minimizer. This requires the calculation of derivatives in objfun, as indicated by mode.
The box bounds forwarded to this function from nag_glopt_bnd_pso (e05sa) will have been acted upon by Local Boundary Restriction. As such, the domain exposed may be greatly smaller than that provided to nag_glopt_bnd_pso (e05sa).
Accurate derivatives must be provided to this function, and will not be approximated internally. Each iteration of this local minimizer also requires the calculation of both the objective function and its derivative. Hence on a call to objfun during a local minimization, ${\mathbf{mode}}=7$.
nag_opt_bounds_quasi_func_easy (e04jy)
Use nag_opt_bounds_quasi_func_easy (e04jy) as the local minimizer. This does not require the calculation of derivatives.
On a call to objfun during a local minimization, ${\mathbf{mode}}=5$.
The box bounds forwarded to this function from nag_glopt_bnd_pso (e05sa) will have been acted upon by Local Boundary Restriction. As such, the domain exposed may be greatly smaller than that provided to nag_glopt_bnd_pso (e05sa).
nag_opt_uncon_conjgrd_comp (e04dg)
nag_opt_uncon_conjgrd_comp (e04dg)
Use nag_opt_uncon_conjgrd_comp (e04dg) as the local minimizer.
Accurate derivatives must be provided, and will not be approximated internally. Additionally, each call to objfun during a local minimization will require either the objective to be evaluated alone, or both the objective and its gradient to be evaluated. Hence on a call to objfun, ${\mathbf{mode}}=5$ or $7$.
nag_opt_nlp1_solve (e04uc)
nag_opt_nlp1_solve (e04uc)
Use nag_opt_nlp1_solve (e04uc) as the local minimizer. This operates such that any derivatives of the objective function that you cannot supply, will be approximated internally using finite differences.
Either, the objective, objective gradient, or both may be requested during a local minimization, and as such on a call to objfun, ${\mathbf{mode}}=1$, $2$ or $5$.
The box bounds forwarded to this function from nag_glopt_bnd_pso (e05sa) will have been acted upon by Local Boundary Restriction. As such, the domain exposed may be greatly smaller than that provided to nag_glopt_bnd_pso (e05sa).
Maximum Function Evaluations  $i$
Default $=\mathit{Imax}$
The maximum number of evaluations of the objective function. When reached this will return ${\mathbf{ifail}}={\mathbf{1}}$ and ${\mathbf{inform}}=6$.
Constraint: $i>0$.
Maximum Iterations Completed  $i$
Default$\text{}=1000×{\mathbf{ndim}}$
The maximum number of complete iterations that may be performed. Once exceeded nag_glopt_bnd_pso (e05sa) will exit with ${\mathbf{ifail}}={\mathbf{1}}$ and ${\mathbf{inform}}=5$.
Unless set, this adapts to the parameters passed to nag_glopt_bnd_pso (e05sa).
Constraint: $i\ge 1$.
Maximum Iterations Static  $i$
Default$\text{}=100$
The maximum number of iterations without any improvement to the current global optimum. If exceeded nag_glopt_bnd_pso (e05sa) will exit with ${\mathbf{ifail}}={\mathbf{1}}$ and ${\mathbf{inform}}=4$. This exit will be hindered by setting Maximum Iterations Static Particles to larger values.
Constraint: $i\ge 1$.
Maximum Iterations Static Particles  $i$
Default$\text{}=0$
The minimum number of particles that must have converged to the current optimum before the function may exit due to Maximum Iterations Static with ${\mathbf{ifail}}={\mathbf{1}}$ and ${\mathbf{inform}}=4$.
Constraint: $i\ge 0$.
Maximum Particles Converged  $i$
Default $=\mathit{Imax}$
The maximum number of particles that may converge to the current optimum. When achieved, nag_glopt_bnd_pso (e05sa) will exit with ${\mathbf{ifail}}={\mathbf{1}}$ and ${\mathbf{inform}}=3$. This exit will be hindered by setting ‘Repulsion’ options, as these cause the swarm to re-expand.
Constraint: $i>0$.
Maximum Particles Reset  $i$
Default $=\mathit{Imax}$
The maximum number of particles that may be reset after converging to the current optimum. Once achieved no further particles will be reset, and any particles within Distance Tolerance of the global optimum will continue to evolve as normal.
Constraint: $i>0$.
Maximum Variable Velocity  $r$
Default$\text{}=0.25$
Along any dimension $j$, the absolute velocity is bounded above by $\left|{\mathbf{v}}_{j}\right|\le r×\left({\mathbf{u}}_{j}-{\mathbf{\ell }}_{j}\right)={\mathbf{V}}_{\mathrm{max}}$. Very low values will greatly increase convergence time. There is no upper limit, although larger values will allow more particles to be advected out of the box bounds, and values greater than $4.0$ may cause significant and potentially unrecoverable swarm divergence.
Constraint: $r>0.0$.
Optimize  $a$
Default$\text{}=\mathrm{MINIMIZE}$
Determines whether to maximize or minimize the objective function.
MINIMIZE
The objective function will be minimized.
MAXIMIZE
The objective function will be maximized. This is accomplished by minimizing the negative of the objective.
Repeatability  $a$
Default$\text{}=\mathrm{OFF}$
Allows for the same random number generator seed to be used for every call to nag_glopt_bnd_pso (e05sa). ${\mathbf{Repeatability}}=\mathrm{OFF}$ is recommended in general.
OFF
The internal generation of random numbers will be nonrepeatable.
ON
The same seed will be used.
Repulsion Finalize  $i$
Default $=\mathit{Imax}$
The number of iterations performed in a repulsive phase before re-contraction. This allows a re-diversified swarm to contract back toward the current optimum, allowing for a finer search of the near optimum space.
Constraint: $i\ge 2$.
Repulsion Initialize  $i$
Default $=\mathit{Imax}$
The number of iterations without any improvement to the global optimum before the algorithm begins a repulsive phase. This phase allows the particle swarm to re-expand away from the current optimum, allowing more of the domain to be investigated. The repulsive phase is automatically ended if a superior optimum is found.
Constraint: $i\ge 2$.
Repulsion Particles  $i$
Default$\text{}=0$
The number of particles required to have converged to the current optimum before any repulsive phase may be initialized. This will prevent repulsion before a satisfactory search of the near optimum area has been performed, which may happen for large dimensional problems.
Constraint: $i\ge 0$.
Swarm Standard Deviation  $r$
Default$\text{}=0.1$
The target standard deviation of the particle distances from the current optimum. Once the standard deviation is below this level, nag_glopt_bnd_pso (e05sa) will exit with ${\mathbf{ifail}}={\mathbf{1}}$ and ${\mathbf{inform}}=2$. This criterion will be penalized by the use of ‘Repulsion’ options, as these cause the swarm to re-expand, increasing the standard deviation of the particle distances from the best point.
In SMP parallel implementations of nag_glopt_bnd_pso (e05sa), the standard deviation will be calculated based only on the particles local to the particular thread that checks for finalization. Considerably fewer particles may be used in this calculation than when the algorithm is run in serial. It is therefore recommended that you provide a smaller value of Swarm Standard Deviation when running in parallel than when running in serial.
Constraint: $r\ge 0.0$.
Target Objective  $a$
Default$\text{}=\mathrm{OFF}$
Target Objective Value  $r$
Default$\text{}=0.0$
Activate or deactivate the use of a target value as a finalization criterion. If active, then once the supplied target value for the objective function is found (beyond the first iteration if Target Warning is active) nag_glopt_bnd_pso (e05sa) will exit with ${\mathbf{ifail}}={\mathbf{0}}$ and ${\mathbf{inform}}=1$. Other than checking for feasibility only (${\mathbf{Optimize}}=\mathrm{CONSTRAINTS}$), this is the only finalization criterion that guarantees that the algorithm has been successful. If the target value was achieved at the initialization phase or first iteration and Target Warning is active, nag_glopt_bnd_pso (e05sa) will exit with ${\mathbf{ifail}}={\mathbf{2}}$. This option may take any real value $r$, or the character ON/OFF as well as DEFAULT. If this option is queried using nag_glopt_optget (e05zl), the current value of $r$ will be returned in rvalue, and cvalue will indicate whether this option is ON or OFF. The behaviour of the option is as follows:
$r$
Once a point is found with an objective value within the Target Objective Tolerance of $r$, nag_glopt_bnd_pso (e05sa) will exit successfully with ${\mathbf{ifail}}={\mathbf{0}}$ and ${\mathbf{inform}}=1$.
OFF
The current value of $r$ will remain stored, however it will not be used as a finalization criterion.
ON
The current value of $r$ stored will be used as a finalization criterion.
DEFAULT
The stored value of $r$ will be reset to its default value ($0.0$), and this finalization criterion will be deactivated.
Target Objective Safeguard  $r$
Default$\text{}=100.0\epsilon$
If you have given a target objective value to reach in $\mathit{objval}$ (the value of the optional parameter Target Objective Value), $\mathit{objsfg}$ sets your desired safeguarded termination tolerance, for when $\mathit{objval}$ is close to zero.
Constraint: $\mathit{objsfg}\ge 2\epsilon$.
Target Objective Tolerance  $r$
Default$\text{}=0.0$
The optional tolerance to a user-specified target value.
Constraint: $r\ge 0.0$.
Target Warning  $a$
Default$\text{}=\mathrm{OFF}$
Activates or deactivates the error exit associated with the target value being achieved before entry into the main loop of the algorithm, ${\mathbf{ifail}}={\mathbf{2}}$.
OFF
No error will be returned, and the function will exit normally.
ON
An error will be returned if the target objective is reached prematurely, and the function will exit with ${\mathbf{ifail}}={\mathbf{2}}$.
Verify Gradients  $a$
Default$\text{}=\mathrm{ON}$
Adjusts the level of gradient checking performed when gradients are required. Gradient checks are only performed on the first call to the chosen local minimizer if it requires gradients. There is no guarantee that the gradient check will be correct, as the finite differences used in the gradient check are themselves subject to inaccuracies.
OFF
No gradient checking will be performed.
ON
A cheap gradient check will be performed on both the gradients corresponding to the objective through objfun.
OBJECTIVE
FULL
A more expensive gradient check will be performed on the gradients corresponding to the objective objfun.
Weight Decrease  $a$
Default$\text{}=\mathrm{INTEREST}$
Determines how particle weights decrease.
OFF
Weights do not decrease.
INTEREST
Weights decrease through compound interest as ${w}_{\mathit{IT}+1}={w}_{\mathit{IT}}\left(1-{W}_{\mathit{val}}\right)$, where ${W}_{\mathit{val}}$ is the Weight Value and $\mathit{IT}$ is the current number of iterations.
LINEAR
Weights decrease linearly following ${w}_{\mathit{IT}+1}={w}_{\mathit{IT}}-\mathit{IT}×\left({W}_{\mathit{max}}-{W}_{\mathit{min}}\right)/{\mathit{IT}}_{\mathit{max}}$, where $\mathit{IT}$ is the iteration number and ${\mathit{IT}}_{\mathit{max}}$ is the maximum number of iterations as set by Maximum Iterations Completed.
Weight Initial  $r$
Default$\text{}={W}_{\mathit{max}}$
The initial value of any particle's inertial weight, ${W}_{\mathit{ini}}$, or the minimum possible initial value if initial weights are randomized. When set, this will override ${\mathbf{Weight Initialize}}=\mathrm{RANDOMIZED}$ or $\mathrm{MAXIMUM}$, and as such these must be set afterwards if so desired.
Constraint: ${W}_{\mathit{min}}\le r\le {W}_{\mathit{max}}$.
Weight Initialize  $a$
Default$\text{}=\mathrm{MAXIMUM}$
Determines how the initial weights are distributed.
INITIAL
All weights are initialized at the initial weight, ${W}_{\mathit{ini}}$, if set. If Weight Initial has not been set, this will be the maximum weight, ${W}_{\mathit{max}}$.
MAXIMUM
All weights are initialized at the maximum weight, ${W}_{\mathit{max}}$.
RANDOMIZED
Weights are uniformly distributed in $\left({W}_{\mathit{min}},{W}_{\mathit{max}}\right)$ or $\left({W}_{\mathit{ini}},{W}_{\mathit{max}}\right)$ if Weight Initial has been set.
Weight Maximum  $r$
Default$\text{}=1.0$
The maximum particle weight, ${W}_{\mathit{max}}$.
Constraint: $1.0\ge r\ge {W}_{\mathit{min}}$ (If ${W}_{\mathit{ini}}$ has been set then $1.0\ge r\ge {W}_{\mathit{ini}}$.)
Weight Minimum  $r$
Default$\text{}=0.1$
The minimum achievable weight of any particle, ${W}_{\mathit{min}}$. Once achieved, no further weight reduction is possible.
Constraint: $0.0\le r\le {W}_{\mathit{max}}$ (If ${W}_{\mathit{ini}}$ has been set then $0.0\le r\le {W}_{\mathit{ini}}$.)
Weight Reset  $a$
Default$\text{}=\mathrm{MAXIMUM}$
Determines how particle weights are re-initialized.
INITIAL
Weights are re-initialized at the initial weight if set. If Weight Initial has not been set, this will be the maximum weight.
MAXIMUM
Weights are re-initialized at the maximum weight.
RANDOMIZED
Weights are uniformly distributed in $\left({W}_{\mathit{min}},{W}_{\mathit{max}}\right)$ or $\left({W}_{\mathit{ini}},{W}_{\mathit{max}}\right)$ if Weight Initial has been set.
Weight Value  $r$
Default$\text{}=0.01$
The constant ${W}_{\mathit{val}}$ used with ${\mathbf{Weight Decrease}}=\mathrm{INTEREST}$.
Constraint: $0.0\le r\le \frac{1}{3}$.

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Chapter Contents
Chapter Introduction
NAG Toolbox

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