NAG Library Function Document

nag_glopt_nlp_pso (e05sbc)

1  Purpose

2  Specification

3  Description

4  References

5  Arguments

1:    $\mathbf{ndim}$ – IntegerInput

On entry: $\mathit{ndim}$, the number of dimensions.

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

2:    $\mathbf{ncon}$ – IntegerInput

On entry: $\mathit{ncon}$, the number of constraints, not including box constraints.

Constraint: $\mathbf{ncon}\ge 0$.

3:    $\mathbf{npar}$ – IntegerInput

On entry: $\mathit{npar}$, the number of particles to be used in the swarm. Assuming all particles remain within constraints, each complete iteration will perform at least npar function evaluations. Otherwise, significantly fewer objective function evaluations may be performed.

Suggested value: $\mathbf{npar}=10\times \mathbf{ndim}$.

Constraint: $\mathbf{npar}\ge 5$.

4:    $\mathbf{xb}\left[\mathbf{ndim}\right]$ – doubleOutput

On exit: the location of the best solution found, $\tilde{\mathbf{x}}$, in $R^{\mathit{ndim}}$.

5:    $\mathbf{fb}$ – double *Output

On exit: the objective value of the best solution, $\tilde{f}=F\left(\tilde{\mathbf{x}}\right)$.

6:    $\mathbf{cb}\left[\mathbf{ncon}\right]$ – doubleOutput

On exit: the constraint violations of the best solution found, $\tilde{\mathbf{e}}=\mathbf{e}\left(\tilde{\mathbf{x}}\right)$. These may have been deemed to be acceptable given the tolerance and scaling of the constraints. See Sections 11 and 12.

7:    $\mathbf{bl}\left[\mathbf{ndim}+\mathbf{ncon}\right]$ – const doubleInput

8:    $\mathbf{bu}\left[\mathbf{ndim}+\mathbf{ncon}\right]$ – const doubleInput

On entry: $\mathbf{bl}$ is $\mathbf{\ell }$, the array of lower bounds, bu is $\mathbf{u}$, the array of upper bounds. The first 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 $\mathbf{Boundary}$ in Section 12).

The next ncon entries must contain the lower and upper bounds for any general constraints respectively.

If $\mathbf{bl}\left[i-1\right]=\mathbf{bu}\left[i-1\right]$ for any $i\in \left\{1,\dots ,\mathbf{ndim}\right\}$, variable $i$ will remain locked to $\mathbf{bl}\left[i-1\right]$ regardless of the $\mathbf{Boundary}$ option selected.

It is strongly advised that you place sensible lower and upper bounds on all variables and constraints, even if your model allows for unbounded variables or constraints.

Constraints:

• $\mathbf{bl}\left[\mathit{i}-1\right]\le \mathbf{bu}\left[\mathit{i}-1\right]$, for $\mathit{i}=1,2,\dots ,\mathbf{ndim}+\mathbf{ncon}$;
• $\mathbf{bl}\left[i-1\right]\ne \mathbf{bu}\left[i-1\right]$ for at least one $i\in \left\{1,\dots ,\mathbf{ndim}\right\}$.

9:    $\mathbf{xbest}\left[\mathbf{ndim}\times \mathbf{npar}\right]$ – doubleInput/Output

Note: the $i$th component of the best position of the $j$th particle, ${\hat{x}}_j\left(i\right)$, is stored in $\mathbf{xbest}\left[\left(j-1\right)\times \mathbf{ndim}+i-1\right]$.

On entry: if using $\mathbf{Start}=\mathrm{WARM}$, the initial particle positions, ${\hat{\mathbf{x}}}_j^0$.

On exit: the best positions found, ${\hat{\mathbf{x}}}_j$, by the npar particles in the swarm.

10:  $\mathbf{fbest}\left[\mathbf{npar}\right]$ – doubleInput/Output

On entry: if using $\mathbf{Start}=\mathrm{WARM}$, objective function values, ${\hat{f}}_j^0=F\left({\hat{\mathbf{x}}}_j^0\right)$, corresponding to the npar particle locations stored in xbest.

On exit: objective function values, ${\hat{f}}_j=F\left({\hat{\mathbf{x}}}_j\right)$, corresponding to the locations returned in xbest.

11:  $\mathbf{cbest}\left[\mathbf{ncon}\times \mathbf{npar}\right]$ – doubleInput/Output

Note: the $k$th constraint violation of the $j$th particle is stored in $\mathbf{cbest}\left[\left(j-1\right)\times \mathbf{ncon}+k-1\right]$.

On entry: if using $\mathbf{Start}=\mathrm{WARM}$, the initial constraint violations, ${\hat{\mathbf{e}}}_j^0=\mathbf{e}\left({\hat{\mathbf{x}}}_j^0\right)$, corresponding to the npar particle locations.

On exit: the final constraint violations, ${\hat{\mathbf{e}}}_j$, corresponding to the locations returned in xbest.

12:  $\mathbf{objfun}$ – function, supplied by the userExternal Function

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 $\mathbf{Local\; Minimizer}$ for more information.

The specification of objfun is:

void  objfun (Integer *mode, Integer ndim, const double x[], double *objf, double vecout[], Integer nstate, Nag_Comm *comm)

1:    $\mathbf{mode}$ – Integer *Input/Output

On entry: 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.

$\mathbf{mode}=1$

$\mathbf{Local\; Minimizer}=\mathrm{e04ucc}$ only
First derivatives can be evaluated and returned in vecout. Any unaltered elements of vecout will be approximated using finite differences.

$\mathbf{mode}=2$

$\mathbf{Local\; Minimizer}=\mathrm{e04ucc}$ 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.

$\mathbf{mode}=6$

$\mathbf{Local\; Minimizer}=\mathrm{e04dgc}$ only
All first derivatives must be evaluated and returned in vecout.

$\mathbf{mode}=7$

$\mathbf{Local\; Minimizer}=\mathrm{e04dgc}$ only
$F\left(\mathbf{x}\right)$ must be calculated and returned in objf, and all first derivatives must be evaluated and returned in vecout.

On exit: if the value of mode is set to be negative, then nag_glopt_nlp_pso (e05sbc) will exit as soon as possible with $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_USER_STOP and $\mathbf{inform}=\mathbf{mode}$.

2:    $\mathbf{ndim}$ – IntegerInput

On entry: the number of dimensions.

3:    $\mathbf{x}\left[\mathbf{ndim}\right]$ – const doubleInput

On entry: $\mathbf{x}$, the point at which the objective function and/or its gradient are to be evaluated.

4:    $\mathbf{objf}$ – double *Input/Output

On entry: 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 constraint penalised value of $F\left(\mathbf{x}\right)$ found so far by a given particle if the objective function is strictly positive (see Section 11). 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.

On exit: 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.

5:    $\mathbf{vecout}\left[\mathbf{ndim}\right]$ – doubleInput/Output

On entry: if $\mathbf{Local\; Minimizer}=\mathrm{e04ucc}$, the values of vecout are used internally to indicate whether a finite difference approximation is required. See nag_opt_nlp (e04ucc).

On exit: 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_conj_grad (e04dgc)).

6:    $\mathbf{nstate}$ – IntegerInput

On entry: 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.

$\mathbf{nstate}=2$

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:    $\mathbf{comm}$ – Nag_Comm *

Pointer to structure of type Nag_Comm; the following members are relevant to objfun.

user – double *

iuser – Integer *

p – Pointer 

The type Pointer will be void *. Before calling nag_glopt_nlp_pso (e05sbc) you may allocate memory and initialize these pointers with various quantities for use by objfun when called from nag_glopt_nlp_pso (e05sbc) (see Section 3.2.1.1 in the Essential Introduction).

13:  $\mathbf{confun}$ – function, supplied by the userExternal Function

confun must calculate any constraints other than the box constraints. If no constraints are required, confun may be NULLFN. For information on how a NAG local minimizer will use confun see the documentation for nag_opt_nlp (e04ucc).

The specification of confun is:

void  confun (Integer *mode, Integer ncon, Integer ndim, Integer tdcj, const Integer needc[], const double x[], double c[], double cjac[], Integer nstate, Nag_Comm *comm)

1:    $\mathbf{mode}$ – Integer *Input/Output

On entry: indicates which values must be assigned during each call of confun. Only the following values need be assigned, for each value of $k\in \left\{1,\dots ,\mathbf{ncon}\right\}$ such that $\mathbf{needc}\left[k-1\right]>0$:

$\mathbf{mode}=0$

the constraint values $c_k\left(\mathbf{x}\right)$.

$\mathbf{mode}=1$

rows of the constraint jacobian, $\frac{\partial c_k}{\partial x_{\mathit{i}}}\left(\mathbf{x}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{ndim}$.

$\mathbf{mode}=2$

the constraint values $c_k\left(\mathbf{x}\right)$ and the corresponding rows of the constraint jacobian, $\frac{\partial c_k}{\partial x_{\mathit{i}}}\left(\mathbf{x}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{ndim}$.

On exit: may be set to a negative value if you wish to terminate the solution to the current problem. In this case nag_glopt_nlp_pso (e05sbc) will terminate with $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_USER_STOP and $\mathbf{inform}=\mathbf{mode}$ as soon as possible.

2:    $\mathbf{ncon}$ – IntegerInput

On entry: the number of constraints, not including box bounds.

3:    $\mathbf{ndim}$ – IntegerInput

On entry: the number of variables.

4:    $\mathbf{tdcj}$ – IntegerInput

On entry: the stride separating matrix column elements in the array cjac.

5:    $\mathbf{needc}\left[\mathbf{ncon}\right]$ – const IntegerInput

On entry: the indices of the elements of c and/or cjac that must be evaluated by confun. If $\mathbf{needc}\left[k-1\right]>0$, the $k-1$th element of c, corresponding to the values of the $k$th constraint, and/or the available elements of the $k-1$th row of cjac, corresponding to the derivatives of the $k$th constraint, must be evaluated at $\mathbf{x}$ (see argument mode).

6:    $\mathbf{x}\left[\mathbf{ndim}\right]$ – const doubleInput

On entry: $\mathbf{x}$, the vector of variables at which the constraint functions and/or the available elements of the constraint Jacobian are to be evaluated.

7:    $\mathbf{c}\left[\mathbf{ncon}\right]$ – doubleOutput

On exit: if $\mathbf{needc}\left[k-1\right]>0$ and $\mathbf{mode}=0$ or $2$, $\mathbf{c}\left[k-1\right]$ must contain the value of $c_k\left(\mathbf{x}\right)$. The remaining elements of c, corresponding to the non-positive elements of needc, need not be set.

8:    $\mathbf{cjac}\left[\mathbf{ncon}\times \mathbf{tdcj}\right]$ – doubleInput/Output

Note: the dimension, dim, of the array cjac is $\mathbf{tdcj}\times \mathbf{ndim}$.

The derivative of the $k$th constraint with respect to the $i$th component, $\frac{\partial c_k}{\partial x_{\mathrm{i}}}$, is stored in $\mathbf{cjac}[\left(k-1\right)\times \mathbf{tdcj}+i-1]$, which we denote as $\mathbf{CJAC}\left(k,i\right)$.

On entry: the elements of cjac are set to special values which enable nag_glopt_nlp_pso (e05sbc) to detect whether they are changed by confun.

On exit: if $\mathbf{needc}\left[k-1\right]>0$ and $\mathbf{mode}=1$ or $2$, the elements of cjac corresponding to the $k$th row of the constraint jacobian should contain the available elements of the vector $\nabla c_k$ given by

$$\nabla c_k=\left(\frac{\partial c_k}{\partial x_1},\frac{\partial c_k}{\partial x_2},\dots ,\frac{\partial c_k}{\partial x_n}\right)\text{,}$$

where $\frac{\partial c_k}{\partial x_i}$ is the partial derivative of the $k$th constraint with respect to the $i$th variable, evaluated at the point $\mathbf{x}$; elements of cjac that remain unaltered will be approximated internally using finite differences. The remaining rows of cjac, corresponding to non-positive elements of needc, need not be set.

It must be emphasized that unassigned elements of cjac are not treated as constant; they are estimated by finite differences, at nontrivial expense. An interval for each element of $\mathbf{x}$ is computed automatically at the start of the optimization. The automatic procedure can usually identify constant elements of cjac, which are then computed once only by finite differences.

9:    $\mathbf{nstate}$ – IntegerInput

On entry: nstate indicates various stages of initialization throughout the function. This allows for permanent global arguments to be initialized a minimum number of times. For example, you may initialize a random number generator seed. Note that unless the option $\mathbf{Optimize}=\mathrm{CONSTRAINTS}$ has been set, objfun will be called before confun.

$\mathbf{nstate}=2$

confun is called for the very first time. This argument setting allows you to save computational time if certain data must be read or calculated only once.

$\mathbf{nstate}=1$

confun is called for the first time during a NAG local minimization function. This argument setting allows you to 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.

10:  $\mathbf{comm}$ – Nag_Comm *

Pointer to structure of type Nag_Comm; the following members are relevant to confun.

user – double *

iuser – Integer *

p – Pointer 

The type Pointer will be void *. Before calling nag_glopt_nlp_pso (e05sbc) you may allocate memory and initialize these pointers with various quantities for use by confun when called from nag_glopt_nlp_pso (e05sbc) (see Section 3.2.1.1 in the Essential Introduction).

confun should be tested separately before being used in conjunction with nag_glopt_nlp_pso (e05sbc).

14:  $\mathbf{monmod}$ – function, supplied by the userExternal Function

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 NULLFN.

The specification of monmod is:

void  monmod (Integer ndim, Integer ncon, Integer npar, double x[], const double xb[], double fb, const double cb[], const double xbest[], const double fbest[], const double cbest[], const Integer itt[], Nag_Comm *comm, Integer *inform)

1:    $\mathbf{ndim}$ – IntegerInput

On entry: the number of dimensions.

2:    $\mathbf{ncon}$ – IntegerInput

On entry: the number of constraints.

3:    $\mathbf{npar}$ – IntegerInput

On entry: the number of particles.

4:    $\mathbf{x}\left[\mathbf{ndim}\times \mathbf{npar}\right]$ – doubleInput/Output

Note: the $i$th component of the $j$th particle, $x_j\left(i\right)$, is stored in $\mathbf{x}\left[\left(j-1\right)\times \mathbf{ndim}+i-1\right]$.

On entry: the npar particle locations, ${\mathbf{x}}_j$, which will currently be used during the next iteration unless altered in monmod.

On exit: the particle locations to be used during the next iteration.

5:    $\mathbf{xb}\left[\mathbf{ndim}\right]$ – const doubleInput

On entry: the location, $\tilde{\mathbf{x}}$, of the best solution yet found.

6:    $\mathbf{fb}$ – doubleInput

On entry: the objective value, $\tilde{f}=F\left(\tilde{\mathbf{x}}\right)$, of the best solution yet found.

7:    $\mathbf{cb}\left[\mathbf{ncon}\right]$ – const doubleInput

On entry: the constraint violations, $\tilde{\mathbf{e}}=\mathbf{e}\left(\tilde{\mathbf{x}}\right)$, of the best solution yet found.

8:    $\mathbf{xbest}\left[\mathbf{ndim}\times \mathbf{npar}\right]$ – const doubleInput

Note: the $i$th component of the position of the $j$th particle's cognitive memory, ${\hat{x}}_j\left(i\right)$, is stored in $\mathbf{xbest}\left[\left(j-1\right)\times \mathbf{ndim}+i-1\right]$.

On entry: the locations currently in the cognitive memory, ${\hat{\mathbf{x}}}_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,\mathbf{npar}$ (see Section 11).

9:    $\mathbf{fbest}\left[\mathbf{npar}\right]$ – const doubleInput

On entry: the objective values currently in the cognitive memory, $F\left({\hat{\mathbf{x}}}_{\mathit{j}}\right)$, for $\mathit{j}=1,2,\dots ,\mathbf{npar}$.

10:  $\mathbf{cbest}\left[\mathbf{ncon}\times \mathbf{npar}\right]$ – const doubleInput

Note: the $k$th constraint violation of the $j$th particle's cognitive memory is stored in $\mathbf{cbest}\left[\left(j-1\right)\times \mathbf{ncon}+k-1\right]$.

On entry: the constraint violations currently in the cognitive memory, $\hat{\mathbf{e}}=\mathbf{e}\left({\hat{\mathbf{x}}}_{\mathit{j}}\right)$, for $\mathit{j}=1,2,\dots ,\mathbf{npar}$, evaluated at ${\hat{\mathbf{x}}}_j$.

11:  $\mathbf{itt}\left[7\right]$ – const IntegerInput

On entry: iteration and function evaluation counters (see description of itt below).

12:  $\mathbf{comm}$ – Nag_Comm *

Pointer to structure of type Nag_Comm; the following members are relevant to monmod.

user – double *

iuser – Integer *

p – Pointer 

The type Pointer will be void *. Before calling nag_glopt_nlp_pso (e05sbc) you may allocate memory and initialize these pointers with various quantities for use by monmod when called from nag_glopt_nlp_pso (e05sbc) (see Section 3.2.1.1 in the Essential Introduction).

13:  $\mathbf{inform}$ – Integer *Input/Output

On entry: $\mathbf{inform}=0$

On exit: setting $\mathbf{inform}<0$ will cause near immediate exit from nag_glopt_nlp_pso (e05sbc). This value will be returned as inform with $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_USER_STOP. You need not set inform unless you wish to force an exit.

15:  $\mathbf{iopts}\left[\mathit{dim}\right]$ – IntegerCommunication Array

Note: the dimension, $\mathit{dim}$, 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_opt_set (e05zkc).

On entry: optional argument array as generated and possibly modified by calls to nag_glopt_opt_set (e05zkc). The contents of iopts MUST NOT be modified directly between calls to nag_glopt_nlp_pso (e05sbc), nag_glopt_opt_set (e05zkc) or nag_glopt_opt_get (e05zlc).

16:  $\mathbf{opts}\left[\mathit{dim}\right]$ – doubleCommunication Array

Note: the dimension, $\mathit{dim}$, 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_opt_set (e05zkc).

On entry: optional argument array as generated and possibly modified by calls to nag_glopt_opt_set (e05zkc). The contents of opts MUST NOT be modified directly between calls to nag_glopt_nlp_pso (e05sbc), nag_glopt_opt_set (e05zkc) or nag_glopt_opt_get (e05zlc).

17:  $\mathbf{comm}$ – Nag_Comm *

The NAG communication argument (see Section 3.2.1.1 in the Essential Introduction).

18:  $\mathbf{itt}\left[7\right]$ – IntegerOutput

On exit: integer iteration counters for nag_glopt_nlp_pso (e05sbc).

$\mathbf{itt}\left[0\right]$

Number of complete iterations.

$\mathbf{itt}\left[1\right]$

Number of complete iterations without improvement to the current optimum.

$\mathbf{itt}\left[2\right]$

Number of particles converged to the current optimum.

$\mathbf{itt}\left[3\right]$

Number of improvements to the optimum.

$\mathbf{itt}\left[4\right]$

Number of function evaluations performed.

$\mathbf{itt}\left[5\right]$

Number of particles reset.

$\mathbf{itt}\left[6\right]$

Number of violated constraints at completion. Note this is always calculated using the $L^1$ norm and a nonzero result does not necessarily mean that the algorithm did not find a suitably constrained point with respect to the single norm used.

19:  $\mathbf{inform}$ – Integer *Output

On exit: indicates which finalization criterion was reached. The possible values of inform are:

inform	Meaning
$<0$	Exit from a user-supplied subroutine.
0	nag_glopt_nlp_pso (e05sbc) has detected an error and terminated.
1	The provided objective target has been achieved. ($\mathbf{Target\; Objective\; Value}$).
2	The standard deviation of the location of all the particles is below the set threshold ($\mathbf{Swarm\; Standard\; Deviation}$). If the solution returned is not satisfactory, you may try setting a smaller value of $\mathbf{Swarm\; Standard\; Deviation}$, or try adjusting the options governing the repulsive phase ($\mathbf{Repulsion\; Initialize}$, $\mathbf{Repulsion\; Finalize}$).
3	The total number of particles converged ($\mathbf{Maximum\; Particles\; Converged}$) to the current global optimum has reached the set limit. This is the number of particles which have moved to a distance less than $\mathbf{Distance\; Tolerance}$ from the optimum with regard to the $L^2$ norm. If the solution is not satisfactory, you may consider lowering the $\mathbf{Distance\; Tolerance}$. However, this may hinder the global search capability of the algorithm.
4	The maximum number of iterations without improvement ($\mathbf{Maximum\; Iterations\; Static}$) has been reached, and the required number of particles ($\mathbf{Maximum\; Iterations\; Static\; Particles}$) 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 ($\mathbf{Maximum\; Iterations\; Completed}$) 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 $\mathbf{Local\; Minimizer}$ 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 ($\mathbf{Repulsion\; Initialize}$, $\mathbf{Repulsion\; Finalize}$).
6	The maximum allowed number of function evaluations ($\mathbf{Maximum\; Function\; Evaluations}$) 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.
7	A feasible point has been found. The objective has not been minimized, although it has been evaluated at the final solutions given in xb and xbest ($\mathbf{Optimize}=\mathrm{CONSTRAINTS}$).

If you wish to continue from the final position gained from a previous simulation with adjusted options, you may set the option $\mathbf{Start}=\mathrm{WARM}$, and pass back in the returned arrays xbest, fbest, and cbest. You should either record the returned values of xb, fb and cb for comparison, as these will not be re-used by the algorithm, or include them in xbest, fbest and cbest respectively by overwriting the entries corresponding to one particle with the relevant information.

20:  $\mathbf{fail}$ – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

nag_glopt_nlp_pso (e05sbc) returns $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_NOERROR if and only if a finalization criterion has been reached which can guarantee success. This may only happen if:

These finalization criteria are not active using default option settings, and must be explicitly set using nag_glopt_opt_set (e05zkc) if required.

nag_glopt_nlp_pso (e05sbc) will return $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NW_SOLUTION_NOT_GUARANTEED 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.

6  Error Indicators and Warnings

NE_ALLOC_FAIL

Dynamic memory allocation failed.
See Section 3.2.1.2 in the Essential Introduction for further information.

NE_BAD_PARAM

On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.

NE_BOUND

On entry, $\mathbf{bl}\left[i\right]=\mathbf{bu}\left[i\right]$ for all box bounds $i$.
Constraint: $\mathbf{bu}\left[i\right]>\mathbf{bl}\left[i\right]$ for at least one box bound $i$.

On entry, $\mathbf{bl}\left[〈\mathit{\text{value}}〉\right]=〈\mathit{\text{value}}〉$ and $\mathbf{bu}\left[〈\mathit{\text{value}}〉\right]=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{bu}\left[i\right]\ge \mathbf{bl}\left[i\right]$ for all $i$.

NE_DERIV_ERRORS

Derivative checks indicate possible errors in the supplied derivatives. Gradient checks may be disabled by setting $\mathbf{Verify\; Gradients}=\mathrm{OFF}$.

NE_ILLEGAL_CALLBACK

nag_glopt_nlp_pso (e05sbc) has been called with $\mathbf{ncon}>0$ and $\mathbf{confun}\text{is}\mathbf{NULL}$. Only use NULL with $\mathbf{ncon}=0$.

NE_INT

On entry, $\mathbf{ncon}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{ncon}\ge 0$.

On entry, $\mathbf{ndim}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{ndim}\ge 1$.

On entry, $\mathbf{npar}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{npar}\ge 5\times \mathbf{num\_threads}$, 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.

NE_INTERNAL_ERROR

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 3.6.6 in the Essential Introduction for further information.

NE_INVALID_OPTION

Either the option arrays have not been initialized for nag_glopt_nlp_pso (e05sbc), or they have become corrupted.

The option $\mathbf{Optimize}=\mathrm{CONSTRAINTS}$ is active, however $\mathbf{ncon}=0$.

NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.

NE_USER_STOP

User requested exit $〈\mathit{\text{value}}〉$ during call to confun.

User requested exit $〈\mathit{\text{value}}〉$ during call to monmod.

User requested exit $〈\mathit{\text{value}}〉$ during call to objfun.

NW_FAST_SOLUTION

If the option $\mathbf{Target\; Warning}$ has been activated, this indicates that the $\mathbf{Target\; Objective\; Value}$ 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 $\mathbf{Local\; Minimizer}$ 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 $\mathbf{Local\; Minimizer}$ directly.

Assuming that objfun is correct, you may wish to set a better $\mathbf{Target\; Objective\; Value}$, or a stricter $\mathbf{Target\; Objective\; Tolerance}$.

NW_NOT_FEASIBLE

Unable to locate strictly feasible point. $〈\mathit{\text{value}}〉$ constraints remain violated. This exit may be suppressed using the option $\mathbf{Constraint\; Warning}$.

NW_SOLUTION_NOT_GUARANTEED

A finalization criterion was reached that cannot guarantee success.
On exit, $\mathbf{inform}=〈\mathit{\text{value}}〉$.

7  Accuracy

8  Parallelism and Performance

9  Further Comments

10  Example

10.1  Program Text

Program Text (e05sbce.c)

10.2  Program Data

10.3  Program Results

Program Results (e05sbce.r)

11  Algorithmic Details

12  Optional Arguments

• Advance Cognitive
• Advance Global
• Boundary
• Constraint Norm
• Constraint Scale Maximum
• Constraint Scaling
• Constraint Superiority
• Constraint Tolerance
• Constraint Warning
• Distance Scaling
• Distance Tolerance
• Function Precision
• Local Boundary Restriction
• Local Exterior Iterations
• Local Exterior Major Iterations
• Local Exterior Minor Iterations
• Local Exterior Tolerance
• Local Interior Iterations
• Local Interior Major Iterations
• Local Interior Minor Iterations
• Local Interior Tolerance
• Local Minimizer
• Maximum Function Evaluations
• Maximum Iterations Completed
• Maximum Iterations Static
• Maximum Iterations Static Particles
• Maximum Particles Converged
• Maximum Particles Reset
• Maximum Variable Velocity
• Objective Scale
• Objective Scaling
• Optimize
• Repeatability
• Repulsion Finalize
• Repulsion Initialize
• Repulsion Particles
• Start
• Swarm Standard Deviation
• Target Objective
• Target Objective Safeguard
• Target Objective Tolerance
• Target Objective Value
• Target Warning
• Verify Gradients
• Weight Decrease
• Weight Initial
• Weight Initialize
• Weight Maximum
• Weight Minimum
• Weight Reset
• Weight Value

12.1  Description of the Optional Arguments

• 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 (X02AJC)), and $\mathit{Imax}$ represents the largest representable integer value (see nag_max_integer (X02BBC)).

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. ${\hat{\mathbf{x}}}_j$, ${\hat{f}}_j$ and ${\hat{\mathbf{e}}}_j$ 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 $0.0$.

L1

The $L^1$ norm will be used, ${‖\mathbf{e}‖}_1=\frac{1}{\mathbf{ncon}}{\displaystyle \sum _1^{\mathbf{ncon}}}\left|e_k\right|$

L2

The $L^2$ norm will be used, ${‖\mathbf{e}‖}_2=\frac{1}{\mathbf{ncon}}\sqrt{{\displaystyle \sum _1^{\mathbf{ncon}}}{e}_k^2}$

L2SQ

The square of the $L^2$ norm will be used, ${‖\mathbf{e}‖}_{2^2}=\frac{1}{\mathbf{ncon}}{\displaystyle \sum _1^{\mathbf{ncon}}}{e}_k^2$

LMAX

The $L^{\infty }$ norm will be used, ${‖\mathbf{e}‖}_{\infty }={\displaystyle \underset{0<k\le \mathbf{ncon}}{\max }}\left(\left|e_k\right|\right)$

OFF

Neither the constraint violations nor the objective will be scaled automatically. This should only be used if the constraints and objective are similarly scaled everywhere throughout the domain.

INITIAL

The maximum of the initial cognitive memories, ${\hat{f}}_j$ and ${\hat{\mathbf{e}}}_j$, will be used to scale the objective function and constraint violations respectively.

ADAPTIVE

Initially, the maximum of the initial cognitive memories, ${\hat{f}}_j$ and ${\hat{\mathbf{e}}}_j$, will be used to scale the objective function and constraint violations respectively. If a significant change is detected in the behaviour of the constraints or the objective, these will be rescaled with respect to the current state of the cognitive memory.

OFF

$\mathbf{fail}\mathbf{.}\mathbf{code}=$ NW_NOT_FEASIBLE will not be returned.

ON

$\mathbf{fail}\mathbf{.}\mathbf{code}=$ NW_NOT_FEASIBLE will be returned if any constraints are sufficiently violated at the end of the simulation.

ON

When a distance is calculated between $\mathbf{x}$ and $\mathbf{y}$, a scaled $L^2$ norm is used.

$$L^2\left(\mathbf{x},\mathbf{y}\right)={\left(\sum _{\left\{i|{\mathbf{u}}_i\ne {\mathbf{\ell }}_i,i\le \mathit{ndim}\right\}}{\left(\frac{x_i-y_i}{{\mathbf{u}}_i-{\mathbf{\ell }}_i}\right)}^2\right)}^{\frac{1}{2}}\text{.}$$

OFF

Distances are calculated as the standard $L^2$ norm without any rescaling.

$$L^2\left(\mathbf{x},\mathbf{y}\right)={\left(\sum _{i=1}^{\mathit{ndim}}{\left(x_i-y_i\right)}^2\right)}^{\frac{1}{2}}\text{.}$$

OFF

No local minimization will be performed in either the INTERIOR or EXTERIOR sections of the algorithm.

e04cbc

Use nag_opt_simplex_easy (e04cbc) as the local minimizer. This does not require the calculation of derivatives.

On a call to objfun during a local minimization, $\mathbf{mode}=5$.

e04dgc

Use nag_opt_conj_grad (e04dgc) 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 $7$.

e04ucc

Use nag_opt_nlp (e04ucc) as the local minimizer. This operates such that any derivatives of either the objective function or the constraint Jacobian, which 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_nlp_pso (e05sbc) will have been acted upon by $\mathbf{Local\; Boundary\; Restriction}$. As such, the domain exposed may be greatly smaller than that provided to nag_glopt_nlp_pso (e05sbc).

MAXIMUM

The objective is rescaled with respect to the maximum absolute value of the objective in the cognitive and global memory.

MEAN

The objective is rescaled with respect to the mean absolute value of the objective in the cognitive and global memory.

USER

The scale remains fixed at the value set using $\mathbf{Objective\; Scale}$.

MINIMIZE

The objective function will be minimized.

MAXIMIZE

The objective function will be maximized. This is accomplished by minimizing the negative of the objective.

CONSTRAINTS

The objective function will be ignored, and the algorithm will attempt to find a feasible point given the provided constraints. The objective function will be evaluated at the best point found with regards to constraint violations, and the final positions returned in xbest. The objective will be calculated at the best point found in terms of constraints only. Should a constrained point be found, nag_glopt_nlp_pso (e05sbc) will exit with $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_NOERROR and $\mathbf{inform}=6$.

$r$

Once a point is found with an objective value within the $\mathbf{Target\; Objective\; Tolerance}$ of $r$, nag_glopt_nlp_pso (e05sbc) will exit successfully with $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_NOERROR 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.

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 $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NW_FAST_SOLUTION.

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 and those provided via the constraint Jacobian through confun.

OBJECTIVE

A more expensive gradient check will be performed on the gradients corresponding to the objective objfun. The gradients of the constraints will not be checked.

CONSTRAINTS

A more expensive check will be performed on the elements of cjac provided via confun. The objective gradient will not be checked.

FULL

A more expensive check will be performed on both the gradient of the objective and the constraint Jacobian.

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 $\mathbf{Weight\; Value}$ and $\mathit{IT}$ is the current number of iterations.

LINEAR

Weights decrease linearly following $w_{\mathit{IT}+1}=w_{\mathit{IT}}-\mathit{IT}\times \left(W_{\mathit{max}}-W_{\mathit{min}}\right)/{\mathit{IT}}_{\mathit{max}}$, where $\mathit{IT}$ is the iteration number and ${\mathit{IT}}_{\mathit{max}}$ is the maximum number of iterations as set by $\mathbf{Maximum\; Iterations\; Completed}$.

INITIAL

Weights are re-initialized at the initial weight if set. If $\mathbf{Weight\; Initial}$ 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 $\mathbf{Weight\; Initial}$ has been set.