NAG Library Routine Document

E05SAF

1  Purpose

1.1  Information for users of the NAG Library for SMP & Multicore

2  Specification

3  Description

4  References

5  Parameters

1:     NDIM – INTEGERInput

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

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

2:     NPAR – INTEGERInput

On entry: $\mathit{npar}$, 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.

Suggested value: $\mathbf{NPAR}=10\times \mathbf{NDIM}$.

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 routine.

3:     XB(NDIM) – REAL (KIND=nag_wp) arrayOutput

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

4:     FB – REAL (KIND=nag_wp)Output

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

5:     BL(NDIM) – REAL (KIND=nag_wp) arrayInput

6:     BU(NDIM) – REAL (KIND=nag_wp) arrayInput

On entry: $\mathbf{BL}$ 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 Section 11).

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 $i\in \left\{1,\dots ,\mathbf{NDIM}\right\}$.

7:     OBJFUN – SUBROUTINE, supplied by the user.External Procedure

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.

The specification of OBJFUN is:

SUBROUTINE OBJFUN ( MODE, NDIM, X, OBJF, VECOUT, NSTATE, IUSER, RUSER) INTEGER  MODE, NDIM, NSTATE, IUSER(*) REAL (KIND=nag_wp)  X(NDIM), OBJF, VECOUT(NDIM), RUSER(*)

1:     MODE – INTEGERInput/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.

$\mathbf{MODE}=1$

$\mathbf{Local\; Minimizer}='\mathrm{E04UCF}'$ only
First derivatives can be evaluated and returned in VECOUT. Any unaltered elements of VECOUT will be approximated using finite differences.

$\mathbf{MODE}=2$

$\mathbf{Local\; Minimizer}='\mathrm{E04UCF}'$ 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.

$\mathbf{MODE}=6$

$\mathbf{Local\; Minimizer}='\mathrm{E04DGF}'$ or $'\mathrm{E04KZF}'$ only
All first derivatives must be evaluated and returned in VECOUT.

$\mathbf{MODE}=7$

$\mathbf{Local\; Minimizer}='\mathrm{E04DGF}'$ or $'\mathrm{E04KZF}'$ 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 E05SAF will exit as soon as possible with $\mathbf{IFAIL}=\mathbf{3}$ and $\mathbf{INFORM}=\mathbf{MODE}$.

2:     NDIM – INTEGERInput

On entry: the number of dimensions.

3:     X(NDIM) – REAL (KIND=nag_wp) arrayInput

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

4:     OBJF – REAL (KIND=nag_wp)Input/Output

On entry: the value of OBJF passed to OBJFUN varies with the parameter 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.

On exit: the value of OBJF returned varies with the parameter 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:     VECOUT(NDIM) – REAL (KIND=nag_wp) arrayInput/Output

On entry: if $\mathbf{Local\; Minimizer}=\mathrm{E04UCF}$ or $\mathrm{E04UCA}$, the values of VECOUT are used internally to indicate whether a finite difference approximation is required. See E04UCF/E04UCA.

On exit: the required values of VECOUT returned to the calling routine 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 E04DGF/E04DGA and E04KZF).

6:     NSTATE – INTEGERInput

On entry: NSTATE indicates various stages of initialization throughout the routine. This allows for permanent global parameters 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 IUSER and RUSER.

$\mathbf{NSTATE}=2$

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 routine. 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:     IUSER($*$) – INTEGER arrayUser Workspace

8:     RUSER($*$) – REAL (KIND=nag_wp) arrayUser Workspace

OBJFUN is called with the parameters IUSER and RUSER as supplied to E05SAF. You are free to use the arrays IUSER and RUSER to supply information to OBJFUN as an alternative to using COMMON global variables.

OBJFUN must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which E05SAF is called. Parameters denoted as Input must not be changed by this procedure.

8:     MONMOD – SUBROUTINE, supplied by the NAG Library or the user.External Procedure

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 the dummy monitoring routine E05SXM. (E05SXM is included in the NAG Library) .

The specification of MONMOD is:

SUBROUTINE MONMOD ( NDIM, NPAR, X, XB, FB, XBEST, FBEST, ITT, IUSER, RUSER, INFORM) INTEGER  NDIM, NPAR, ITT(6), IUSER(*), INFORM REAL (KIND=nag_wp)  X(NDIM,NPAR), XB(NDIM), FB, XBEST(NDIM,NPAR), FBEST(NPAR), RUSER(*)

1:     NDIM – INTEGERInput

On entry: the number of dimensions.

2:     NPAR – INTEGERInput

On entry: the number of particles.

3:     X(NDIM,NPAR) – REAL (KIND=nag_wp) arrayInput/Output

Note: the $i$th component of the $j$th particle, $x_j\left(i\right)$, is stored in $\mathbf{X}\left(i,j\right)$.

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.

4:     XB(NDIM) – REAL (KIND=nag_wp) arrayInput

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

5:     FB – REAL (KIND=nag_wp)Input

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

6:     XBEST(NDIM,NPAR) – REAL (KIND=nag_wp) arrayInput

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 $\mathbf{XBEST}\left(i,j\right)$.

On entry: the locations currently in the cognitive memory, ${\hat{\mathbf{x}}}_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,\mathbf{NPAR}$ (see Section 10).

7:     FBEST(NPAR) – REAL (KIND=nag_wp) arrayInput

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

8:     ITT($6$) – INTEGER arrayInput

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

9:     IUSER($*$) – INTEGER arrayUser Workspace

10:   RUSER($*$) – REAL (KIND=nag_wp) arrayUser Workspace

MONMOD is called with the parameters IUSER and RUSER as supplied to E05SAF. You are free to use the arrays IUSER and RUSER to supply information to MONMOD as an alternative to using COMMON global variables.

11:   INFORM – INTEGERInput/Output

On entry: $\mathbf{INFORM}=\mathbf{thread\_num}$, 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.

On exit: setting $\mathbf{INFORM}<0$ will cause near immediate exit from E05SAF. This value will be returned as INFORM with $\mathbf{IFAIL}=\mathbf{3}$. You need not set INFORM unless you wish to force an exit.

MONMOD must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which E05SAF is called. Parameters denoted as Input must not be changed by this procedure.

9:     IOPTS($*$) – INTEGER arrayCommunication Array

Note: the contents of IOPTS must not have been altered between calls to E05ZKF, E05ZLF, E05SAF and the selected problem solving routine.

On entry: optional parameter array as generated and possibly modified by calls to E05ZKF. The contents of IOPTS must not be modified directly between calls to E05SAF, E05ZKF or E05ZLF.

10:   OPTS($*$) – REAL (KIND=nag_wp) arrayCommunication Array

Note: the contents of OPTS must not have been altered between calls to E05ZKF, E05ZLF, E05SAF and the selected problem solving routine.

On entry: optional parameter array as generated and possibly modified by calls to E05ZKF. The contents of OPTS must not be modified directly between calls to E05SAF, E05ZKF or E05ZLF.

11:   IUSER($*$) – INTEGER arrayUser Workspace

IUSER is not used by E05SAF, but is passed directly to OBJFUN and MONMOD and may be used to pass information to these routines as an alternative to using COMMON global variables.

With care, you may also write information back into IUSER. This might be useful, for example, should there be a need to preserve the state of a random number generator.

With SMP-enabled versions of E05SAF the array IUSER provided are classified as OpenMP shared memory. Use of IUSER has to take account of this in order to preserve thread safety whenever information is written back to either of these arrays.

12:   RUSER($*$) – REAL (KIND=nag_wp) arrayUser Workspace

RUSER is not used by E05SAF, but is passed directly to OBJFUN and MONMOD and may be used to pass information to these routines as an alternative to using COMMON global variables.

With care, you may also write information back into RUSER. This might be useful, for example, should there be a need to preserve the state of a random number generator.

With SMP-enabled versions of E05SAF the array RUSER provided are classified as OpenMP shared memory. Use of RUSER has to take account of this in order to preserve thread safety whenever information is written back to either of these arrays.

13:   ITT($6$) – INTEGER arrayOutput

On exit: integer iteration counters for E05SAF.

$\mathbf{ITT}\left(1\right)$

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.

14:   INFORM – INTEGEROutput

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

INFORM	Meaning
$<0$	Exit from a user-supplied subroutine.
0	E05SAF 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.

15:   IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

On exit: the most common exit will be $\mathbf{IFAIL}=\mathbf{1}$.

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.

E05SAF 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 E05ZKF if required.

E05SAF 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 routine 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 Sections 6, 7 and 10 for more information.

6  Error Indicators and Warnings

$\mathbf{IFAIL}=1$

The algorithm has reached a finalization criterion that does not guarantee success. You should investigate the returned value of INFORM for more information on why this occurred.

$\mathbf{IFAIL}=2$

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 routine. 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 routine is called, it may imply that the objective is largely unimodal, and that it may be more efficient to use the routine 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.

$\mathbf{IFAIL}=3$

You requested an exit from either OBJFUN or MONMOD. The exit flag you provided will be returned in INFORM.

$\mathbf{IFAIL}=11$

On entry,

$\mathbf{NDIM}<1$.

$\mathbf{IFAIL}=12$

On entry,

$\mathbf{NPAR}<5\times \mathbf{num\_threads}$, 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 routine

$\mathbf{IFAIL}=14$

On entry, at least one lower bound $\mathbf{BL}\left(\mathit{i}\right)>\mathbf{BU}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{NDIM}$, or all of the bounds $\mathbf{BL}\left(\mathit{i}\right)=\mathbf{BU}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{NDIM}$.

$\mathbf{IFAIL}=21$

On entry, either the option arrays IOPTS and OPTS have not been initialized, or they have been corrupted. Re-initialize the arrays using E05ZKF.

Alternatively, both Advance Cognitive and Advance Global may have been set to $0.0$.

$\mathbf{IFAIL}=32$

The gradient check has indicated an error may be present in the objective gradient. These checks are not infallible. If you are sure your gradients are correct then the gradient checks may be disabled by setting $\mathbf{Verify\; Gradients}=\mathrm{OFF}$.

$\mathbf{IFAIL}=51$

(SMP parallel version only). Multiple threads have been detected, however you have not set the option SMP Callback Thread Safe to declare that the provided callback routines are thread-safe, or that you want the algorithm to run in serial. See Section 11.2 for more information.

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (e05safe.f90)

Program Text (e05safe_smp.f90)

9.2  Program Data

9.3  Program Results

Program Results (e05safe.r)

Program Results (e05safe_smp.r)

10  Algorithmic Details

10.1  Details of SMP parallelization

11  Optional Parameters

• Advance Cognitive
• Advance Global
• Boundary
• 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
• Optimize
• Repeatability
• Repulsion Finalize
• Repulsion Initialize
• Repulsion Particles
• SMP Callback Thread Safe
• SMP Local Minimizer External
• SMP Monitor
• SMP Monmod
• SMP Subswarm
• SMP Thread Overrun
• 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

11.1  Description of the Optional Parameters

• the keywords;
• a parameter value, where the letters $a$, $i\text{ and }r$ 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 X02AJF), and $\mathit{Imax}$ represents the largest representable integer value (see X02BBF).

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

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.

E04CBF

Use E04CBF as the local minimizer. This does not require the calculation of derivatives.

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

E04KZF

Use E04KZF as the local minimizer. This requires the calculation of derivatives in OBJFUN, as indicated by MODE.

The box bounds forwarded to this routine from E05SAF will have been acted upon by Local Boundary Restriction. As such, the domain exposed may be greatly smaller than that provided to E05SAF.

Accurate derivatives must be provided to this routine, 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, $\mathbf{MODE}=7$.

E04JYF

Use E04JYF 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 routine from E05SAF will have been acted upon by Local Boundary Restriction. As such, the domain exposed may be greatly smaller than that provided to E05SAF.

E04DGF
E04DGA

Use E04DGA 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 $7$.

E04UCF
E04UCA

Use E04UCA 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 routine from E05SAF will have been acted upon by Local Boundary Restriction. As such, the domain exposed may be greatly smaller than that provided to E05SAF.

MINIMIZE

The objective function will be minimized.

MAXIMIZE

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

$r$

Once a point is found with an objective value within the Target Objective Tolerance of $r$, E05SAF 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.

OFF

No error will be returned, and the routine will exit normally.

ON

An error will be returned if the target objective is reached prematurely, and the routine will exit with $\mathbf{IFAIL}=\mathbf{2}$.

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.

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

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.

11.2  Description of the SMP optional parameters

YES

The callback routines have been programmed in a thread safe way. The algorithm will use OMP_NUM_THREADS threads.

NO

The callback routines are not thread safe. Setting this option will force the algorithm to run on a single thread only, and is advisable only for debugging purposes, or if you wish to parallelize your callback functions.

WARNING

This will cause an immediate exit from E05SAF with $\mathbf{IFAIL}=\mathbf{51}$ if multiple threads are detected. This is to inform you that you have not declared the callback functions either to be thread safe, or that they are thread unsafe and you wish the algorithm to run in serial.

MASTER

Only the master thread will attempt to find any improvement. The local minimization will be launched from the best known solution. All other threads will remain effectively idle.

ALL

The master thread will perform a local minimization from the best known solution, while all other threads will perform a local minimization from randomly generated perturbations of the best known solution, increasing the chance of an improvement. Assuming all local minimizations will take approximately the same amount of computation, this will be effectively free in terms of real time. It will however increase the number of function evaluations performed.

SINGLE

Only one thread will invoke MONMOD, after all threads have performed at least one sub-iteration.

ALL

Each thread will invoke MONMOD each time it completes a sub-iteration. If you wish to alter X using MONMOD you should use this option, as MONMOD will only receive the arrays X, XBEST and FBEST private to the calling thread.