NAG CL Interfacee04fgc (handle_​solve_​dfls_​rcomm)

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

1Purpose

e04fgc is a reverse communication Derivative-free Optimization (DFO) solver from the NAG optimization modelling suite (DFLS) for small to medium-scale nonlinear least squares problems with bound constraints.

2Specification

 #include
 void e04fgc (void *handle, Integer *irevcm, Integer *neval, Integer maxeval, Integer nvar, double x[], Integer nres, double rx[], double rinfo[], double stats[], NagError *fail)
The function may be called by the names: e04fgc or nag_opt_handle_solve_dfls_rcomm.

3Description

e04fgc uses reverse communication for function evaluations and monitoring steps. Every time the solver requires an evaluation of the objective function, it pauses its progress, exits and waits for the function to be called again with the objective value provided in the argument rx.
e04fgc is aimed at minimizing a sum of squares objective function subject to bound constraints:
 $minimize x∈ℝn ∑ i=1 mr ri x 2 subject to lx ≤ x ≤ ux .$
Here the ${r}_{i}\left(x\right)$ are smooth nonlinear functions called residuals and ${l}_{x}$ and ${u}_{x}$ are $n$-dimensional vectors defining bounds on the variables. Typically, in a calibration or data fitting context, the residuals will be defined as the difference between the data points and a nonlinear model (see Section 2.2.3 in the E04 Chapter Introduction).
e04fgc serves as a solver for compatible problems stored as a handle. The handle points to an internal data structure which defines the problem and serves as a means of communication for functions in the NAG optimization modelling suite. To define a compatible problem handle, you must call e04rac followed by e04rmc to initialize it and optionally call e04rhc to define bounds on the variables. If e04rhc is not called, all the variables will be considered free by the solver. It should be noted that e04fgc always assumes that the Jacobian of the residuals is dense, therefore defining a sparse structure for the residuals in the call to e04rmc will have no effect. See Section 4.1 in the E04 Chapter Introduction for more details about the NAG optimization modelling suite.
The solver allows fixing variables with the definition of the bounds. However, the following constraint must be met in order to be able to call the solver:
• for all non-fixed variable ${x}_{i}$, the value of ${u}_{x}\left(i\right)-{l}_{x}\left(i\right)$ must be at least twice the starting trust region radius (see the consistency constraint of the optional parameter ${\mathbf{DFO Starting Trust Region}}$).
The solver is based on a derivative-free trust region framework. This type of method is well suited for small to medium-scale problems (around 100 variables) for which the derivatives are unavailable or not easy to compute, and/or for which the function evaluations are expensive or noisy. For a detailed description of the algorithm see Section 11.
The algorithm behaviour and solver strategy can be modified by various optional parameters (see Section 12) which can be set by e04zmc and e04zpc at any time between the initialization of the handle by e04rac and a call to the solver. The optional parameters' names specific for this solver start either with the prefix DFO (Derivative-free Optimization) or DFLS (Derivative-free Least Squares). The default values for these optional parameters are chosen to work well in the general case, but it is recommended you tune them to your particular problem. In particular, if the objective function is known to be noisy, it is highly recommended to set the optional parameter ${\mathbf{DFO Noisy Problem}}$ to $\mathrm{YES}$. Once the solver has finished, options may be modified for the next solve. The solver may be called repeatedly with various starting points and/or optional parameters.
The underlying algorithm implemented for e04fgc is the same as the one used by e04ffc. e04fgc serves as a reverse communication interface to the derivative-free solver for nonlinear least squares problems.

4References

Cartis C, Fiala J, Marteau B and Roberts L (2018) Improving the Flexibility and Robustness of Model-Based Derivative-Free Optimization Solvers Technical Report University of Oxford
Cartis C and Roberts L (2017) A Derivative-Free Gauss-Newton Method
Conn A R, Scheinberg K and Vicente L N (2009) Introduction to Derivative-Free Optimization, vol. 8 of MPS-SIAM Series on Optimization MPS/SIAM, Philadelphia
Powell M J D (2009) The BOBYQA algorithm for bound constrained optimization without derivatives Report DAMTP 2009/NA06 University of Cambridge https://www.damtp.cam.ac.uk/user/na/NA_papers/NA2009_06.pdf
Zhang H, Conn A R and Scheinberg K (2010) A Derivative-Free Algorithm for Least-Squares Minimization SIAM J. Optim. 20(6) 3555–3576

5Arguments

Note: this function uses reverse communication. Its use involves an initial entry, intermediate exits and re-entries, and a final exit, as indicated by the argument irevcm. Between intermediate exits and re-entries, all arguments other than those specified by the value of irevcm must remain unchanged.
1: $\mathbf{handle}$void * Input
On entry: the handle to the problem. It needs to be initialized by e04rac and must not be changed before the call to e04fgc.
2: $\mathbf{irevcm}$Integer * Input/Output
On entry: does not need to be set on the first call of e04fgc. On subsequent calls, irevcm must be set to a positive integer if all the required function evaluations have been correctly provided in rx. Otherwise, if a problem occurred during a monitoring step or while providing objective values, it is possible to set it to a negative value:
${\mathbf{irevcm}}=-1$
If function evaluations were required, the solver will attempt a rescue procedure and request an alternative point. If no function were required (monitoring step), the solver will stop with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_USER_STOP.
${\mathbf{irevcm}}\le -2$
The solver will cleanly exit and return the best available point as a well as the solve statistics.
On exit: indicates what action is to be performed before the next call to e04fgc.
${\mathbf{irevcm}}=0$
Final exit of the solver.
${\mathbf{irevcm}}=1$
neval objective evaluations are required.
${\mathbf{irevcm}}=2$
Monitoring step, no evaluation is required, x and rx contain the best evaluation of the objective yet.
3: $\mathbf{neval}$Integer * Output
On exit: indicates the number of objective evaluations required for the next call of e04fgc in rx. The coordinates of the points to evaluate are provided in the first neval columns of x.
4: $\mathbf{maxeval}$Integer Input
On entry: the second dimension of the arrays x and rx. The maximum number of function evaluations that can be requested at the same time. The value of maxeval must remain constant between all the calls to e04fgc. See Section 8 for a short discussion on when evaluations are requested simultaneously by the solver.
Constraint: ${\mathbf{maxeval}}\ge 1$.
5: $\mathbf{nvar}$Integer Input
On entry: $n$, the number of variables in the problem. It must be unchanged from the value set during the initialization of the handle by e04rac.
Constraint: ${\mathbf{nvar}}\ge 1$.
6: $\mathbf{x}\left[{\mathbf{nvar}}×{\mathbf{maxeval}}\right]$double Input/Output
Note: the $\left(i,j\right)$th element of the matrix $X$ is stored in ${\mathbf{x}}\left[\left(j-1\right)×{\mathbf{nvar}}+i-1\right]$.
On entry: the first column contains ${x}_{0}$, the initial estimates of the variables $x$.
On exit: if ${\mathbf{irevcm}}=0$ or $2$, the first column contains the best computed estimate of the solution.
If ${\mathbf{irevcm}}=1$, the first neval columns contain the coordinates of the points to evaluate.
7: $\mathbf{nres}$Integer Input
On entry: ${m}_{r}$, the number of residuals in the problem. It must be unchanged from the value set during the definition of the objective structure by e04rmc.
Constraint: ${\mathbf{nres}}\ge 0$.
8: $\mathbf{rx}\left[{\mathbf{nres}}×{\mathbf{maxeval}}\right]$double Input/Output
Note: the $\left(i,j\right)$th element of the matrix is stored in ${\mathbf{rx}}\left[\left(j-1\right)×{\mathbf{nres}}+i-1\right]$.
On entry: does not need to be set on the first call to e04fgc.
If ${\mathbf{irevcm}}=1$ after the last call of e04fgc, the first neval columns must contain the residuals of the requested points.
On exit: if ${\mathbf{irevcm}}=0$ or $2$, the first column contains the residuals of the best computed point.
9: $\mathbf{rinfo}\left[100\right]$double Output
On exit: optimal objective value and various indicators at monitoring steps or at the end of the final iteration. The measures are given in the table below:
 $0$ Objective function value $f\left(x\right)$ (sum of the squared residuals). $1$ $\rho$, the current lower bound of the trust region. $2$ $\Delta$, the current size of the trust region. $3$ The number of interpolation points used by the solver. $4-99$ Reserved for future use.
10: $\mathbf{stats}\left[100\right]$double Output
On exit: solver statistics at monitoring steps or at the end of the final iteration as given in the table below:
 $0$ Number of calls to the objective function. $1$ Total time spent in the solver (including time spent evaluating the objective). $2$ Total time spent evaluating the objective function. $3$ Number of steps. $4-99$ Reserved for future use.
11: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_HANDLE
The supplied handle does not define a valid handle to the data structure for the NAG optimization modelling suite. It has not been initialized by e04rac or it has been corrupted.
NE_INT
Initial number of interpolation points $\mathit{ninit}=〈\mathit{\text{value}}〉$, total number of interpolation points $\mathit{npts}=〈\mathit{\text{value}}〉$, number of variables $\mathit{nvar}=〈\mathit{\text{value}}〉$.
Constraint: growing interpolation set is only supported for linear models ($\mathit{npts}=\mathit{nvar}+1$).
Use ${\mathbf{DFO Number Interp Points}}$ and ${\mathbf{DFO Number Initial Points}}$ to control the number of interpolation points.
The number of initial interpolation points is greater than the maximum.
Use ${\mathbf{DFO Number Interp Points}}$ and ${\mathbf{DFO Number Initial Points}}$ to control the number of interpolation points.
There were ${n}_{r}=〈\mathit{\text{value}}〉$ unequal bounds and the optional parameter ${\mathbf{DFO Number Interp Points}}$ $\mathit{npt}=〈\mathit{\text{value}}〉$.
Constraint: ${n}_{r}+1\le \mathit{npt}\le \frac{\left({n}_{r}+1\right)×\left({n}_{r}+2\right)}{2}$.
Use e04zmc to set compatible option values.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_IMPROVEMENT
No progress, the solver was stopped after $〈\mathit{\text{value}}〉$ consecutive slow steps.
Use the optional parameter ${\mathbf{DFO Maximum Slow Steps}}$ to modify the maximum number of slow steps accepted.
The solver stopped after $5×{\mathbf{DFO Maximum Slow Steps}}$ consecutive slow steps and a trust region above the tolerance set by ${\mathbf{DFO Trust Region Slow Tol}}$.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_PHASE
The problem is already being solved.
NE_REAL_2
Inconsistent optional parameters ${\mathbf{DFO Trust Region Tolerance}}$ ${\rho }_{\mathrm{end}}$ and ${\mathbf{DFO Trust Region Slow Tol}}$ ${\rho }_{\mathrm{tol}}$.
Constraint: ${\rho }_{\mathrm{end}}<{\rho }_{\mathrm{tol}}$.
Use e04zmc to set compatible option values.
Inconsistent optional parameters ${\mathbf{DFO Trust Region Tolerance}}$ ${\rho }_{\mathrm{end}}$ and ${\mathbf{DFO Starting Trust Region}}$ ${\rho }_{\mathrm{beg}}$.
Constraint: ${\rho }_{\mathrm{end}}<{\rho }_{\mathrm{beg}}$.
Use e04zmc to set compatible option values.
Optional parameter ${\mathbf{DFO Starting Trust Region}}$ ${\rho }_{\mathrm{beg}}=〈\mathit{\text{value}}〉$, ${l}_{x}\left(i\right)=〈\mathit{\text{value}}〉$, ${u}_{x}\left(i\right)=〈\mathit{\text{value}}〉$ and $i=〈\mathit{\text{value}}〉$.
Constraint: if ${l}_{x}\left(i\right)\ne {u}_{x}\left(i\right)$ in coordinate $i$, then ${u}_{x}\left(i\right)-{l}_{x}\left(i\right)\ge 2×{\rho }_{\mathrm{beg}}$.
Use e04zmc to set compatible option values.
NE_REF_MATCH
On entry, ${\mathbf{maxeval}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{maxeval}}>0$.
The information supplied does not match with that previously stored.
On entry, ${\mathbf{maxeval}}=〈\mathit{\text{value}}〉$ must match that given during the first call of the function, i.e., $〈\mathit{\text{value}}〉$.
The information supplied does not match with that previously stored.
On entry, ${\mathbf{nres}}=〈\mathit{\text{value}}〉$ must match that given during the definition of the objective in the handle, i.e., $〈\mathit{\text{value}}〉$.
The information supplied does not match with that previously stored.
On entry, ${\mathbf{nvar}}=〈\mathit{\text{value}}〉$ must match that given during initialization of the handle, i.e., $〈\mathit{\text{value}}〉$.
NE_RESCUE_FAILED
Rescue failed: the trust region could not be reduced further after some function evaluation could not be provided. Check the specification of your objective and whether it needs rescaling. Try a different initial x.
Some initial interpolation points were not provided. Rescue cannot be attempted at this stage.
Check the specification of your objective and whether it needs rescaling. Try a different initial x.
NE_SETUP_ERROR
This solver does not support the model defined in the handle.
Maximization is not possible for a nonlinear least squares problem.
${\mathbf{nres}}=0$. There are no residuals, the objective function is empty.
NE_TIME_LIMIT
The solver terminated after the maximum time allowed was exceeded.
Maximum number of seconds exceeded. Use optional parameter ${\mathbf{Time Limit}}$ to reset the limit.
NE_TOO_MANY_ITER
Maximum number of function evaluations exceeded.
NE_TR_STEP_FAILED
The predicted reduction in a trust region step was non-positive. Check the specification of your objective and whether it needs rescaling. Try a different initial x.
NE_USER_STOP
User requested termination during a monitoring step. irevcm was set to a value lower than $-1$ after a monitoring step.
User requested termination during an objective evaluation step.
irevcm was set to a value lower than $\text{}-1$ after the solver requested objective function values.
NW_NOT_CONVERGED
The problem was solved to an acceptable level after $〈\mathit{\text{value}}〉$ consecutive slow iterations.
Use the optional parameter ${\mathbf{DFO Maximum Slow Steps}}$ to modify the maximum number of slow steps accepted.
The solver stopped after ${\mathbf{DFO Maximum Slow Steps}}$ consecutive slow steps and a trust region below the tolerance set by ${\mathbf{DFO Trust Region Slow Tol}}$.

7Accuracy

In a non-noisy case, the solver can declare convergence on two conditions.
1. (i)The trust region radius is below the tolerance ${\rho }_{\mathrm{end}}$ set by the optional parameter ${\mathbf{DFO Trust Region Tolerance}}$. When this condition is met, the corresponding solution will generally be at a distance smaller than $10×{\rho }_{\mathrm{end}}$ of a local minimum.
2. (ii)The sum of the square of the residuals is below the tolerance set by the optional parameter ${\mathbf{DFLS Small Residuals Tol}}$. In a data fitting context, this condition means that the error between the observed data and the model is smaller than the requested tolerance.
If the objective is declared as noisy by the optional parameter ${\mathbf{DFO Noisy Problem}}$, the solver declares convergence more conservatively. Instead of stopping with the first condition, the solver will trigger soft restarts (see Section 11 for more details) to ensure it did not get stuck in a flat region because of the noise. The solver then declares convergence when it is reasonably sure that it has reached a local minimum.
1. (i)The total number of restarts is greater than the limit set by optional parameter ${\mathbf{DFO Max Soft Restarts}}$ and the trust region radius is below the tolerance.
2. (ii)The number of consecutive restarts that did not manage to decrease the objective function is greater than the limit set by the optional parameter ${\mathbf{DFO Max Unsucc Soft Restarts}}$.
In addition, this solver can stop if the convergence is deemed too slow on two conditions.
1. (i)The trust region lower bound is lower than the value set by the optional parameter ${\mathbf{DFO Trust Region Slow Tol}}$ and the number of consecutive slow steps is greater than the value set by ${\mathbf{DFO Maximum Slow Steps}}$.
2. (ii)The trust region lower bound is greater than the value set by the optional parameter ${\mathbf{DFO Trust Region Slow Tol}}$ and the number of consecutive slow steps is greater than five times the value set by ${\mathbf{DFO Maximum Slow Steps}}$.
The slow convergence detection can be deactivated by setting ${\mathbf{DFO Maximum Slow Steps}}$ to $0$.

8Parallelism and Performance

The solver can request up to maxeval evaluations of the objective function at the same time, which you can parallelize. In this release, only the initial interpolation points used to build the first model can be requested simultaneously, all subsequent objective requests will be done one by one. The maximum number of interpolation points used to build the models of the objective residuals is set by the optional parameter ${\mathbf{DFO Number Interp Points}}$, maxeval should therefore be chosen to be lower than this value.
e04fgc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
e04fgc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9.1Description of the Printed Output

The solver can print information to give an overview of the problem and the progress of the computation. The output may be sent to two independent file ID which are set by optional parameters ${\mathbf{Print File}}$ and ${\mathbf{Monitoring File}}$. Optional parameters ${\mathbf{Print Level}}$, ${\mathbf{Print Options}}$, ${\mathbf{Monitoring Level}}$ and ${\mathbf{Print Solution}}$ determine the exposed level of detail. This allows, for example, a detailed log file to be generated while the condensed information is displayed on the screen.
By default (${\mathbf{Print File}}=6$, ${\mathbf{Print Level}}=2$), four sections are printed to the standard output: a header, a list of options, an iteration log and a summary.
```---------------------------------------------------
E04F(G|F)), Derivative-free solver for data fitting
(nonlinear least squares problems)
---------------------------------------------------

Problem statistics
Number of variables                 10
Number of unconstrained variables   10
Number of fixed variables            0
Starting interpolation points       11
Total interpolation points          11
Number of residuals                 10```
Optional parameters list
If ${\mathbf{Print Options}}=\mathrm{YES}$, a list of the optional parameters and their values is printed. The list shows all options of the solver, each displayed on one line. The line contains the option name, its current value and an indicator for how it was set. The options left at their defaults are noted by ‘d’ and the ones you set are noted by ‘U’. Note that the output format is compatible with the file format expected by e04zpc. The output looks as follows:
```Stats Time                    =                 Yes     * U
Dfo Trust Region Tolerance    =         1.00000E-07     * U
Dfo Max Objective Calls       =                 500     * d
Dfo Starting Trust Region     =         1.10000E-01     * U
Dfo Number Interp Points      =                   0     * d```
Iteration log
If ${\mathbf{Print Level}}\ge 2$, the solver will print a summary line for each step. An iteration is considered successful when it yields a decrease of the objective sufficiently close to the decrease predicted by the quadratic model. Each line shows the step number (step), the value of the objective function (obj), the lower bound on the radius of the trust region (rho), and the cumulative number of objective function evaluations (nf). The output looks as follows:
```----------------------------------------
step |    obj        rho    |    nf   |
----------------------------------------
1 |  3.87E+01  1.00E-01  |    12   |
2 |  3.39E+01  1.00E-01  |    13   |
3 |  1.78E+01  1.00E-01  |    14   |
4 |  3.95E-29  1.00E-01  |    15   |```
Occasionally, the letter ‘s’ is printed at the end of the line indicating that the progress is considered slow by the slow convergence detection heuristic. After a certain number of consecutive slow steps, the solver is stopped. The limit on the number of slow iterations can be controlled by the optional parameter ${\mathbf{DFO Maximum Slow Steps}}$ and the tolerance on the trust region radius before the solver can be stopped is driven by ${\mathbf{DFO Trust Region Slow Tol}}$.
If ${\mathbf{Print Level}}\ge 3$, each line additionally shows the current value of the trust region radius (delta) as well as the step length (||d||) taken. It might look as follows:
```------------------------------------------------------------
step |    obj        rho      delta     ||d||   |    nf   |
------------------------------------------------------------
1 |  4.02E+00  1.00E-01  4.00E-01  1.00E-01  |     4   |
2 |  3.66E+00  1.00E-01  4.00E-01  4.00E-01  |     5   |
3 |  3.48E+00  1.00E-01  4.00E-01  4.00E-01  |     6   |
4 |  2.32E+00  1.00E-01  4.00E-01  1.00E-01  |     9   |```
Summary
Once the solver finishes, a summary is produced:
```Status: Converged, small residuals

Value of the objective                    3.95417E-29
Number of objective function evaluations           15
Number of steps                                     4```
Note that only the iterations that decrease the objective function are printed in the iteration log, meaning that objective evaluations are likely to happen between the last printed iteration and the convergence. This leads to a small difference between the last line of the iteration log and the final summary in terms of the number of function evaluations.
Optionally, if ${\mathbf{Stats Time}}=\mathrm{YES}$, the timings are printed:
```Timings
Total time spent in the solver            0.056
Time spent in the objective evaluation    0.012```
Additionally, if ${\mathbf{Print Solution}}=\mathrm{YES}$, the solution is printed along with the bounds:
```Computed Solution:
idx   Lower bound        Value      Upper bound
1       -inf       -1.00000E+00        inf
2       -inf       -1.00000E+00        inf
3       -inf       -1.00000E+00        inf
4       -inf       -1.00000E+00        inf```

10Example

In this example, we minimize the two-dimension Rosenbrock function under some bound constraints. In this problem, the number of variables $n=2$ and the number of residuals ${m}_{r}=2$
 $r1x = 1.0-x1 r2x = 10.0*x2-x12 .$
The following bounds are defined on the variables
 $-1.5989 ≤ x1 ≤ 2.0 -2.0 ≤ x2.$
The initial guess is
 $x0= -1.2,1.0.$
The expected solution is
 $x*= 1.0,1.0.$

10.1Program Text

Program Text (e04fgce.c)

None.

10.3Program Results

Program Results (e04fgce.r)

11Algorithmic Details

This section contains a short description of the algorithm used in e04fgc which is based on the collaborative work between NAG and the University of Oxford (Cartis and Roberts (2017) and Cartis et al. (2018)). It uses a model-based derivative-free trust region framework adapted to exploit least squares problems structure.

11.1Derivative-free Trust Region Algorithm

In this section, we are interested in generic problems of the form
 $minimize x∈ℝn fx$
where the derivatives of the objective function $f$ are not easily available. A model-based DFO algorithm maintains a set of points ${Y}_{k}$ centred on an iterate ${x}_{k}$ to build quadratic interpolation models of the objective
 $fxk+s ≈ ϕks= fxk+ gkTs+ 12 sT Hk s ,$
where ${g}_{k}$ and ${H}_{k}$ are built with the interpolation conditions
 $∀y∈Yk , ​ϕk y-xk =fy .$ (1)
Note that if the number of interpolation points $\mathit{npt}$ is smaller than $\frac{\left({n}_{r}+1\right)×\left({n}_{r}+2\right)}{2}$, the model chosen is the one for which the Hessian ${H}_{k}$ is the closest to ${H}_{k-1}$ in the Frobenius norm sense. This model is iteratively optimized over a trust region, updated and moved around the new computed points. More precisely, it can be described as:
• DFO Algorithm
1. 1.Initialization
Choose an initial interpolation set ${Y}_{0}$, trust region radius ${\rho }_{\mathrm{beg}}$ and build the first quadratic model ${\varphi }_{0}$.
2. 2.Iteration k
1. (i)Minimize the model in the trust region to obtain a step ${s}_{k}$.
2. (ii)If the step is too small, adjust the geometry of the interpolation set and the trust region size ${\rho }_{k}$ and restart the iteration.
3. (iii)Evaluate the objective at the new point ${x}_{k}+{s}_{k}$.
4. (iv)Replace a far away point from ${Y}_{k}$ by ${x}_{k}+{s}_{k}$ to obtain ${Y}_{k+1}$.
5. (v)If the decrease of the objective is sufficient (successful step), choose ${x}_{k+1}={x}_{k}+{s}_{k}$, else choose ${x}_{k+1}={x}_{k}$.
6. (vi)Choose ${\rho }_{k+1}$ and adjust the geometry of ${Y}_{k+1}$, if necessary.
7. (vii)Build ${\varphi }_{k+1}$ using the new interpolation set.
8. (viii)Stop the algorithm if ${\rho }_{k+1}$is below the chosen tolerance ${\rho }_{\mathrm{end}}$.
In the following sections, we call an iteration ‘successful’ when the trial point ${x}_{k}+{s}_{k}$ is accepted as the next iterate.

11.2Bounds on the Variables

The bounds on the variables are handled during the model optimization step (step 2(i) of DFO Algorithm) with an active set method. If a bound is hit, it is fixed and step 2(i) is restarted.

11.3Adaptation to Nonlinear Least Squares Problems

In the specific case where $f$ is a sum of square $f\left(x\right)=\sum _{i=1}^{{m}_{r}}{{r}_{i}\left(x\right)}^{2}$, a good approximation of the Hessian of the objective can be
 $∇2fx ≈ JxT Jx ,$
where $J$ is the ${m}_{r}$ by $n$ first derivative matrix of $f$. This approximation is the main idea behind the Gauss–Newton and Levenberg–Marquardt methods. Following the work of Zhang et al. (2010), it is possible to adapt it to the DFO framework. In e04fgc, one linear model is built for each residual ${r}_{i}$
 $rix+s ≈ rix+ giTs.$
Let $J={\left({g}^{\left(1\right)},{g}^{\left(2\right)},\dots \right)}^{\mathrm{T}}$. To build the model of the objective $f$, we then choose
 $fx+s ≈ ϕs= fx+ gfTs+ 12sTHfs,$
where ${g}_{f}$ is chosen as
 $gf= JTfx,$
and ${H}_{f}$ as
 $Hf= JTJ + 0 if gf≥κ1 κ3 fxI if gf<κ1 and 12 fx< κ2gf.$
The first expression amounts to making a Gauss–Newton approximation when we are far from a stationary point and the second to a Levenberg–Marquardt approximation when we are close to a stationary point with small residuals.
e04fgc integrates this method of building models into the framework presented in the DFO Algorithm.

11.4Growing the Interpolation Set

In the case where the function is very expensive, it might be desirable for the solver to make some progress before the ${n}_{r}+1$ evaluations necessary to build the first interpolation model are done. To get that behaviour, you can set the optional parameter ${\mathbf{DFO Number Initial Points}}$, controlling the number of initial interpolation points, to a value that is lower than ${n}_{r}+1$. The solver will then start its iteration earlier while adding random perturbations to the interpolation models to ensure that the full space is explored.
It is to be noted that this mode will typically not lead to a faster convergence to the solution and should only be used if early progress is desirable.

11.5Dealing with Noisy Problems

If the problem solved is known to be noisy, declaring it as such to the solver with the optional parameter ${\mathbf{DFO Noisy Problem}}$ will modify the behaviour of the solver to take into account the uncertainty of the function evaluations. The two main features implemented to handle noisy objective functions are:
1. (i)slow update of the trust regions;
2. (ii)soft restarts of the algorithm can be performed instead of declaring convergence to ensure the solver did not get stuck in a flat region due to the noise.
A soft restart consists of a reset of the trust region's values to the starting ones and a few objective evaluations to improve the geometry of the interpolation set in the new trust region. It is possible to control the number of objective evaluations performed during a soft restart with the optional parameter ${\mathbf{DFO Number Soft Restarts Pts}}$. After a set maximum number of restarts (${\mathbf{DFO Max Soft Restarts}}$) or maximum number of unsuccessful restarts (${\mathbf{DFO Max Unsucc Soft Restarts}}$), the solver will declare convergence in the usual way.

12Optional Parameters

Several optional parameters in e04fgc define choices in the problem specification or the algorithm logic. In order to reduce the number of formal arguments of e04fgc these optional parameters have associated default values that are appropriate for most problems. Therefore, you need only specify those optional parameters whose values are to be different from their default values.
The remainder of this section can be skipped if you wish to use the default values for all optional parameters.
The optional parameters can be changed by calling e04zmc anytime between the initialization of the handle and the call to the solver. Modification of the optional parameters during intermediate monitoring stops is not allowed. Once the solver finishes, the optional parameters can be altered again for the next solve.
The option values may be retrieved by e04znc.
The following is a list of the optional parameters available. A full description of each optional parameter is provided in Section 12.1.

12.1Description 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, where the minimum abbreviation of each keyword is underlined;
• 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 X02AJC).
All options accept the value $\mathrm{DEFAULT}$ to return single options to their default states.
Keywords and character values are case and white space insensitive.
 Defaults
This special keyword may be used to reset all optional parameters to their default values. Any value given with this keyword will be ignored.
 DFLS Small Residuals Tol $r$ Default $={\epsilon }^{0.75}$
This option defines the tolerance on the value of the residuals. Namely, the solver declares convergence if
 $f x = ∑ i=1 mr ri x 2 < DFLS Small Residuals Tol .$
Constraint: ${\mathbf{DFLS Small Residuals Tol}}>{\epsilon }^{2}$.
 DFO Initial Interp Points $a$ Default $=\mathrm{Coordinate}$
Determines how the initial interpolation points are chosen. If ${\mathbf{DFO Initial Interp Points}}=\mathrm{Coordinate}$, the interpolation points are chosen along the coordinate directions around the initial point. If ${\mathbf{DFO Initial Interp Points}}=\mathrm{Random}$, the initial interpolation points are chosen along random orthogonal directions around the initial point. Set ${\mathbf{DFO Random Seed}}$ to a positive value to fix the random seed and get reproducible results.
Constraint: ${\mathbf{DFO Initial Interp Points}}=\mathrm{Coordinate}$ or $\mathrm{Random}$.
 DFO Maximum Slow Steps $i$ Default $=20$
If ${\mathbf{DFO Maximum Slow Steps}}>0$, this parameter defines the maximum number of consecutive slow iterations ${n}_{\mathrm{slow}}$ allowed. Set ${\mathbf{DFO Maximum Slow Steps}}=0$ to deactivate the slow iteration detection. The algorithm can stop in two situations:
1. (i)${n}_{\mathrm{slow}}>{\mathbf{DFO Maximum Slow Steps}}$ and $\rho <{\mathbf{DFO Trust Region Slow Tol}}$ with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NW_NOT_CONVERGED,
2. (ii)${n}_{\mathrm{slow}}>5×{\mathbf{DFO Maximum Slow Steps}}$ with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NO_IMPROVEMENT.
Constraint: ${\mathbf{DFO Maximum Slow Steps}}\ge 0$.
 DFO Max Objective Calls $i$ Default $=500$
A limit on the number of objective function evaluations the solver is allowed to compute. If the limit is reached, the solver stops with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_TOO_MANY_ITER.
Constraint: ${\mathbf{DFO Max Objective Calls}}\ge 1$.
 DFO Max Soft Restarts $i$ Default $=5$
The maximum total number of soft restarts that can be performed if the objective function is declared as noisy (${\mathbf{DFO Noisy Problem}}=\mathrm{YES}$).
Constraint: ${\mathbf{DFO Max Soft Restarts}}\ge 1$.
 DFO Max Unsucc Soft Restarts $i$ Default $=3$
The maximum number of consecutive unsuccessful soft restarts that can be performed if the objective function is declared as noisy (${\mathbf{DFO Noisy Problem}}=\mathrm{YES}$).
Constraint: ${\mathbf{DFO Max Unsucc Soft Restarts}}\ge 1$.
 DFO Monitor Frequency $i$ Default $=0$
If ${\mathbf{DFO Monitor Frequency}}>0$, the solver will stop at the end of every $i$th step for monitoring purposes. e04fgc needs to be called again to continue the optimization.
Constraint: ${\mathbf{DFO Monitor Frequency}}\ge 0$.
 DFO Noise Level $r$ Default $=0.0$
Indicates the noise level expected when evaluating the objective function. If ${\mathbf{DFO Noisy Problem}}=\mathrm{YES}$.
Constraint: ${\mathbf{DFO Noise Level}}\ge 0.0$.
 DFO Noisy Problem $a$ Default $=\mathrm{NO}$
Indicates if the function evaluations provided to the solver are noisy. If ${\mathbf{DFO Noisy Problem}}=\mathrm{YES}$, some algorithmic features will be activated:
1. (i)The trust region update becomes slower to reflect the decreased confidence in the objective values.
2. (ii)Soft restarts of the algorithm can be performed to ensure the algorithm did not get stuck because of the noise (see ${\mathbf{DFO Max Soft Restarts}}$, ${\mathbf{DFO Max Unsucc Soft Restarts}}$ and ${\mathbf{DFO Number Soft Restarts Pts}}$ to control the restart characteristics).
3. (iii)In addition, if ${\mathbf{DFO Noise Level}}>0.0$, the solver will trigger a soft restart if all the function values are within the noise level.
 DFO Number Initial Points $i$ Default $=0$
The initial number of interpolation points in ${Y}_{0}$ (1) used to build the linear models of the residuals. If ${\mathbf{DFO Number Initial Points}}=0$, the number of points is chosen to be equal to the total number of interpolation points set by ${\mathbf{DFO Number Interp Points}}$.
If this parameter is chosen to be lower than the maximum set by ${\mathbf{DFO Number Interp Points}}$, the solver will progressively increase the number of interpolation points until it reaches that value. In this release, it is only possible to grow the interpolation set if ${\mathbf{DFO Number Interp Points}}$ is set to the default value.
Constraint: ${\mathbf{DFO Number Initial Points}}\ge 0$.
Consistency constraints, the solver stops with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_INT if not met:
• $0\le {\mathbf{DFO Number Initial Points}}\le {\mathbf{DFO Number Interp Points}}$.
• If ${\mathbf{DFO Number Initial Points}}<{\mathbf{DFO Number Interp Points}}$, ${\mathbf{DFO Number Interp Points}}$ must be set to the default value.
 DFO Number Interp Points $i$ Default $=0$
The maximum number of interpolation points in ${Y}_{k}$ (1) used to build the linear models of the residuals. If ${\mathbf{DFO Number Interp Points}}=0$, the number of points is chosen to be ${n}_{r}+1$ where ${n}_{r}$ is the number of non-fixed variables.
Constraint: ${\mathbf{DFO Number Interp Points}}\ge 0$.
Consistency constraint, the solver stops with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_INT if not met:
• ${n}_{r}+1\le {\mathbf{DFO Number Interp Points}}\le \frac{\left({n}_{r}+1\right)×\left({n}_{r}+2\right)}{2}$.
 DFO Number Soft Restarts Pts $i$ Default $=3$
The number of interpolation points that are replaced during a soft restart.
Constraint: ${\mathbf{DFO Number Soft Restarts Pts}}\ge 1$.
 DFO Print Frequency $i$ Default $=1$
If ${\mathbf{DFO Print Frequency}}>0$, the solver prints the iteration log to the appropriate units at the end of every $i$th step.
Constraint: ${\mathbf{DFO Print Frequency}}\ge 0$.
 DFO Random Seed $i$ Default $\text{}=-1$
The random seed used to generate the random points used to build the initial model or build the underdetermined models when the interpolation set has not fully grown (${\mathbf{DFO Number Initial Points}}<{\mathbf{DFO Number Interp Points}}$). If ${\mathbf{DFO Random Seed}}<0$, the random seed will be based on values taken from the real-time clock, potentially resulting in the solver taking a different path each time it is run. Set it to a positive value to get fully reproducible runs.
Constraint: ${\mathbf{DFO Print Frequency}}\ge -1$.
 DFO Starting Trust Region $r$ Default $=0.1$
${\rho }_{\mathrm{beg}}$, the initial trust region radius. This parameter should be set to about one tenth of the greatest expected overall change to a variable: the initial quadratic model will be constructed by taking steps from the initial $x$ of length ${\rho }_{\mathrm{beg}}$ along each coordinate direction. The default value assumes that the variables have an order of magnitude $1$.
Constraint: ${\mathbf{DFO Starting Trust Region}}>\epsilon$.
Consistency constraints, the solver stops with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_REAL_2 if not met:
• ${\mathbf{DFO Starting Trust Region}}\le {\mathbf{DFO Trust Region Tolerance}}$.
• ${\mathbf{DFO Starting Trust Region}}\le \frac{1}{2}\underset{i}{\mathrm{min}}\phantom{\rule{0.25em}{0ex}}\left({u}_{x}\left(i\right)-{l}_{x}\left(i\right)\right)$.
 DFO Trust Region Slow Tol $r$ Default $\text{}={\epsilon }^{0.25}$
The minimal acceptable trust region radius for the solution to be declared as acceptable. The solver stops if:
• ${n}_{\mathrm{slow}}>{\mathbf{DFO Maximum Slow Steps}}$ and ${\rho }_{k}<{\mathbf{DFO Trust Region Slow Tol}}$.
Constraint: ${\mathbf{DFO Trust Region Slow Tol}}>\epsilon$.
Consistency constraint, the solver stops with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_REAL_2 if not met:
• ${\mathbf{DFO Trust Region Slow Tol}}>{\mathbf{DFO Trust Region Tolerance}}$.
 DFO Trust Region Tolerance $r$ Default $={\epsilon }^{0.37}$
${\rho }_{\mathrm{end}}$, the requested trust region radius. The algorithm declares convergence when the trust region radius reaches this limit. It should indicate the absolute accuracy that is required in the final values of the variables.
Constraint: ${\mathbf{DFO Trust Region Tolerance}}>\epsilon$.
Consistency constraints, the solver stops with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_REAL_2 if not met:
• ${\mathbf{DFO Starting Trust Region}}>{\mathbf{DFO Trust Region Tolerance}}$.
• ${\mathbf{DFO Trust Region Slow Tol}}>{\mathbf{DFO Trust Region Tolerance}}$.
 Infinite Bound Size $r$ Default $\text{}={10}^{20}$
This defines the ‘infinite’ bound $\mathit{bigbnd}$ in the definition of the problem constraints. Any upper bound greater than or equal to $\mathit{bigbnd}$ will be regarded as $+\infty$ (and similarly any lower bound less than or equal to $-\mathit{bigbnd}$ will be regarded as $-\infty$). Note that a modification of this optional parameter does not influence constraints which have already been defined; only the constraints formulated after the change will be affected.
Constraint: ${\mathbf{Infinite Bound Size}}\ge 1000$.
 Monitoring File $i$ Default $\text{}=-1$
(See Section 3.1.1 in the Introduction to the NAG Library CL Interface for further information on NAG data types.)
If $i\ge 0$, the Nag_FileID number (as returned from x04acc) for the secondary (monitoring) output. If ${\mathbf{Monitoring File}}=-1$, no secondary output is provided. The information output to this file ID is controlled by ${\mathbf{Monitoring Level}}$.
Constraint: ${\mathbf{Monitoring File}}\ge -1$.
 Monitoring Level $i$ Default $=4$
This parameter sets the amount of information detail that will be printed by the solver to the secondary output. The meaning of the levels is the same as with ${\mathbf{Print Level}}$.
Constraint: $0\le {\mathbf{Monitoring Level}}\le 5$.
 Print File $i$ Default $=\mathrm{Nag_FileID number associated with stdout}$
(See Section 3.1.1 in the Introduction to the NAG Library CL Interface for further information on NAG data types.)
If $i\ge 0$, the Nag_FileID number (as returned from x04acc, stdout as the default) for the primary output of the solver. If ${\mathbf{Print File}}=-1$, the primary output is completely turned off independently of other settings. The information output to this unit is controlled by ${\mathbf{Print Level}}$.
Constraint: ${\mathbf{Print File}}\ge -1$.
 Print Level $i$ Default $=2$
This parameter defines how detailed information should be printed by the solver to the primary and secondary output.
 $i$ Output $0$ No output from the solver. $1$ The Header and Summary. $2$, $3$, $4$, $5$ Additionally, the Iteration log.
Constraint: $0\le {\mathbf{Print Level}}\le 5$.
 Print Options $a$ Default $=\mathrm{YES}$
If ${\mathbf{Print Options}}=\mathrm{YES}$, a listing of optional parameters will be printed to the primary output and is always printed to the secondary output.
Constraint: ${\mathbf{Print Options}}=\mathrm{YES}$ or $\mathrm{NO}$.
 Print Solution $a$ Default $=\mathrm{NO}$
If ${\mathbf{Print Solution}}=\mathrm{YES}$, the solution will be printed to the primary and secondary output.
Constraint: ${\mathbf{Print Solution}}=\mathrm{YES}$ or $\mathrm{NO}$.
 Stats Time $a$ Default $=\mathrm{NO}$
This parameter turns on timings of various parts of the algorithm to give a better overview of where most of the time is spent. This might be helpful for a choice of different solving approaches. It is possible to choose between CPU and wall clock time. Choice $\mathrm{YES}$ is equivalent to $\mathrm{WALL CLOCK}$.
Constraint: ${\mathbf{Stats Time}}=\mathrm{YES}$, $\mathrm{NO}$, $\mathrm{CPU}$ or $\mathrm{WALL CLOCK}$.
 Time Limit $r$ Default $\text{}={10}^{6}$
A limit to the number of seconds that the solver can use to solve one problem. If during the convergence check this limit is exceeded, the solver will terminate with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_TIME_LIMIT.
Constraint: ${\mathbf{Time Limit}}>0$.