Control Options for FOCEi

## Usage

```
foceiControl(
sigdig = 3,
...,
epsilon = NULL,
maxInnerIterations = 1000,
maxOuterIterations = 5000,
n1qn1nsim = NULL,
print = 1L,
printNcol = floor((getOption("width") - 23)/12),
scaleTo = 1,
scaleObjective = 0,
normType = c("rescale2", "mean", "rescale", "std", "len", "constant"),
scaleType = c("nlmixr2", "norm", "mult", "multAdd"),
scaleCmax = 1e+05,
scaleCmin = 1e-05,
scaleC = NULL,
scaleC0 = 1e+05,
derivEps = rep(20 * sqrt(.Machine$double.eps), 2),
derivMethod = c("switch", "forward", "central"),
derivSwitchTol = NULL,
covDerivMethod = c("central", "forward"),
covMethod = c("r,s", "r", "s", ""),
hessEps = (.Machine$double.eps)^(1/3),
hessEpsLlik = (.Machine$double.eps)^(1/3),
optimHessType = c("central", "forward"),
optimHessCovType = c("central", "forward"),
eventType = c("central", "forward"),
centralDerivEps = rep(20 * sqrt(.Machine$double.eps), 2),
lbfgsLmm = 7L,
lbfgsPgtol = 0,
lbfgsFactr = NULL,
eigen = TRUE,
addPosthoc = TRUE,
diagXform = c("sqrt", "log", "identity"),
sumProd = FALSE,
optExpression = TRUE,
literalFix = TRUE,
ci = 0.95,
useColor = crayon::has_color(),
boundTol = NULL,
calcTables = TRUE,
noAbort = TRUE,
interaction = TRUE,
cholSEtol = (.Machine$double.eps)^(1/3),
cholAccept = 0.001,
resetEtaP = 0.15,
resetThetaP = 0.05,
resetThetaFinalP = 0.15,
diagOmegaBoundUpper = 5,
diagOmegaBoundLower = 100,
cholSEOpt = FALSE,
cholSECov = FALSE,
fo = FALSE,
covTryHarder = FALSE,
outerOpt = c("nlminb", "bobyqa", "lbfgsb3c", "L-BFGS-B", "mma", "lbfgsbLG", "slsqp",
"Rvmmin"),
innerOpt = c("n1qn1", "BFGS"),
rhobeg = 0.2,
rhoend = NULL,
npt = NULL,
rel.tol = NULL,
x.tol = NULL,
eval.max = 4000,
iter.max = 2000,
abstol = NULL,
reltol = NULL,
resetHessianAndEta = FALSE,
stateTrim = Inf,
shi21maxOuter = 0L,
shi21maxInner = 20L,
shi21maxInnerCov = 20L,
shi21maxFD = 20L,
gillK = 10L,
gillStep = 4,
gillFtol = 0,
gillRtol = sqrt(.Machine$double.eps),
gillKcov = 10L,
gillKcovLlik = 10L,
gillStepCovLlik = 4.5,
gillStepCov = 2,
gillFtolCov = 0,
gillFtolCovLlik = 0,
rmatNorm = TRUE,
rmatNormLlik = TRUE,
smatNorm = TRUE,
smatNormLlik = TRUE,
covGillF = TRUE,
optGillF = TRUE,
covSmall = 1e-05,
adjLik = TRUE,
gradTrim = Inf,
maxOdeRecalc = 5,
odeRecalcFactor = 10^(0.5),
gradCalcCentralSmall = 1e-04,
gradCalcCentralLarge = 10000,
etaNudge = qnorm(1 - 0.05/2)/sqrt(3),
etaNudge2 = qnorm(1 - 0.05/2) * sqrt(3/5),
nRetries = 3,
seed = 42,
resetThetaCheckPer = 0.1,
etaMat = NULL,
repeatGillMax = 1,
stickyRecalcN = 4,
gradProgressOfvTime = 10,
addProp = c("combined2", "combined1"),
badSolveObjfAdj = 100,
compress = TRUE,
rxControl = NULL,
sigdigTable = NULL,
fallbackFD = FALSE,
smatPer = 0.6,
sdLowerFact = 0.001
)
```

## Arguments

- sigdig
Optimization significant digits. This controls:

The tolerance of the inner and outer optimization is

`10^-sigdig`

The tolerance of the ODE solvers is

`0.5*10^(-sigdig-2)`

; For the sensitivity equations and steady-state solutions the default is`0.5*10^(-sigdig-1.5)`

(sensitivity changes only applicable for liblsoda)The tolerance of the boundary check is

`5 * 10 ^ (-sigdig + 1)`

- ...
Additional arguments passed to

`nlmixr2est::foceiControl()`

.- epsilon
Precision of estimate for n1qn1 optimization.

- maxInnerIterations
Number of iterations for n1qn1 optimization.

- maxOuterIterations
Maximum number of L-BFGS-B optimization for outer problem.

- n1qn1nsim
Number of function evaluations for n1qn1 optimization.

Integer representing when the outer step is printed. When this is 0 or do not print the iterations. 1 is print every function evaluation (default), 5 is print every 5 evaluations.

- printNcol
Number of columns to printout before wrapping parameter estimates/gradient

- scaleTo
Scale the initial parameter estimate to this value. By default this is 1. When zero or below, no scaling is performed.

- scaleObjective
Scale the initial objective function to this value. By default this is 0 (meaning do not scale)

- normType
This is the type of parameter normalization/scaling used to get the scaled initial values for nlmixr2. These are used with

`scaleType`

of.With the exception of

`rescale2`

, these come from Feature Scaling. The`rescale2`

The rescaling is the same type described in the OptdesX software manual.In general, all all scaling formula can be described by:

$$v_{scaled}$$ = ($$v_{unscaled}-C_{1}$$)/$$C_{2}$$

Where

The other data normalization approaches follow the following formula

$$v_{scaled}$$ = ($$v_{unscaled}-C_{1}$$)/$$C_{2}$$

`rescale2`

This scales all parameters from (-1 to 1). The relative differences between the parameters are preserved with this approach and the constants are:$$C_{1}$$ = (max(all unscaled values)+min(all unscaled values))/2

$$C_{2}$$ = (max(all unscaled values) - min(all unscaled values))/2

`rescale`

or min-max normalization. This rescales all parameters from (0 to 1). As in the`rescale2`

the relative differences are preserved. In this approach:$$C_{1}$$ = min(all unscaled values)

$$C_{2}$$ = max(all unscaled values) - min(all unscaled values)

`mean`

or mean normalization. This rescales to center the parameters around the mean but the parameters are from 0 to 1. In this approach:$$C_{1}$$ = mean(all unscaled values)

$$C_{2}$$ = max(all unscaled values) - min(all unscaled values)

`std`

or standardization. This standardizes by the mean and standard deviation. In this approach:$$C_{1}$$ = mean(all unscaled values)

$$C_{2}$$ = sd(all unscaled values)

`len`

or unit length scaling. This scales the parameters to the unit length. For this approach we use the Euclidean length, that is:$$C_{1}$$ = 0

$$C_{2}$$ = $$\sqrt(v_1^2 + v_2^2 + \cdots + v_n^2)$$

`constant`

which does not perform data normalization. That is$$C_{1}$$ = 0

$$C_{2}$$ = 1

- scaleType
The scaling scheme for nlmixr2. The supported types are:

`nlmixr2`

In this approach the scaling is performed by the following equation:$$v_{scaled}$$ = ($$v_{current} - v_{init}$$)*scaleC[i] + scaleTo

The

`scaleTo`

parameter is specified by the`normType`

, and the scales are specified by`scaleC`

.`norm`

This approach uses the simple scaling provided by the`normType`

argument.`mult`

This approach does not use the data normalization provided by`normType`

, but rather uses multiplicative scaling to a constant provided by the`scaleTo`

argument.In this case:

$$v_{scaled}$$ = $$v_{current}$$/$$v_{init}$$*scaleTo

`multAdd`

This approach changes the scaling based on the parameter being specified. If a parameter is defined in an exponential block (ie exp(theta)), then it is scaled on a linearly, that is:$$v_{scaled}$$ = ($$v_{current}-v_{init}$$) + scaleTo

Otherwise the parameter is scaled multiplicatively.

$$v_{scaled}$$ = $$v_{current}$$/$$v_{init}$$*scaleTo

- scaleCmax
Maximum value of the scaleC to prevent overflow.

- scaleCmin
Minimum value of the scaleC to prevent underflow.

- scaleC
The scaling constant used with

`scaleType=nlmixr2`

. When not specified, it is based on the type of parameter that is estimated. The idea is to keep the derivatives similar on a log scale to have similar gradient sizes. Hence parameters like log(exp(theta)) would have a scaling factor of 1 and log(theta) would have a scaling factor of ini_value (to scale by 1/value; ie d/dt(log(ini_value)) = 1/ini_value or scaleC=ini_value)For parameters in an exponential (ie exp(theta)) or parameters specifying powers, boxCox or yeoJohnson transformations , this is 1.

For additive, proportional, lognormal error structures, these are given by 0.5*abs(initial_estimate)

Factorials are scaled by abs(1/digamma(initial_estimate+1))

parameters in a log scale (ie log(theta)) are transformed by log(abs(initial_estimate))*abs(initial_estimate)

These parameter scaling coefficients are chose to try to keep similar slopes among parameters. That is they all follow the slopes approximately on a log-scale.

While these are chosen in a logical manner, they may not always apply. You can specify each parameters scaling factor by this parameter if you wish.

- scaleC0
Number to adjust the scaling factor by if the initial gradient is zero.

- derivEps
Forward difference tolerances, which is a vector of relative difference and absolute difference. The central/forward difference step size h is calculated as:

`h = abs(x)*derivEps[1] + derivEps[2]`

- derivMethod
indicates the method for calculating derivatives of the outer problem. Currently supports "switch", "central" and "forward" difference methods. Switch starts with forward differences. This will switch to central differences when abs(delta(OFV)) <= derivSwitchTol and switch back to forward differences when abs(delta(OFV)) > derivSwitchTol.

- derivSwitchTol
The tolerance to switch forward to central differences.

- covDerivMethod
indicates the method for calculating the derivatives while calculating the covariance components (Hessian and S).

- covMethod
Method for calculating covariance. In this discussion, R is the Hessian matrix of the objective function. The S matrix is the sum of individual gradient cross-product (evaluated at the individual empirical Bayes estimates).

"

`r,s`

" Uses the sandwich matrix to calculate the covariance, that is:`solve(R) %*% S %*% solve(R)`

"

`r`

" Uses the Hessian matrix to calculate the covariance as`2 %*% solve(R)`

"

`s`

" Uses the cross-product matrix to calculate the covariance as`4 %*% solve(S)`

"" Does not calculate the covariance step.

- hessEps
is a double value representing the epsilon for the Hessian calculation. This is used for the R matrix calculation.

- hessEpsLlik
is a double value representing the epsilon for the Hessian calculation when doing focei generalized log-likelihood estimation. This is used for the R matrix calculation.

- optimHessType
The hessian type for when calculating the individual hessian by numeric differences (in generalized log-likelihood estimation). The options are "central", and "forward". The central differences is what R's `optimHess()` uses and is the default for this method. (Though the "forward" is faster and still reasonable for most cases). The Shi21 cannot be changed for the Gill83 algorithm with the optimHess in a generalized likelihood problem.

- optimHessCovType
The hessian type for when calculating the individual hessian by numeric differences (in generalized log-likelihood estimation). The options are "central", and "forward". The central differences is what R's `optimHess()` uses. While this takes longer in optimization, it is more accurate, so for calculating the covariance and final likelihood, the central differences are used. This also uses the modified Shi21 method

- eventType
Event gradient type for dosing events; Can be "central" or "forward"

- centralDerivEps
Central difference tolerances. This is a numeric vector of relative difference and absolute difference. The central/forward difference step size h is calculated as:

`h = abs(x)*derivEps[1] + derivEps[2]`

- lbfgsLmm
An integer giving the number of BFGS updates retained in the "L-BFGS-B" method, It defaults to 7.

- lbfgsPgtol
is a double precision variable.

On entry pgtol >= 0 is specified by the user. The iteration will stop when:

`max(\| proj g_i \| i = 1, ..., n) <= lbfgsPgtol`

where pg_i is the ith component of the projected gradient.

On exit pgtol is unchanged. This defaults to zero, when the check is suppressed.

- lbfgsFactr
Controls the convergence of the "L-BFGS-B" method. Convergence occurs when the reduction in the objective is within this factor of the machine tolerance. Default is 1e10, which gives a tolerance of about

`2e-6`

, approximately 4 sigdigs. You can check your exact tolerance by multiplying this value by`.Machine$double.eps`

- eigen
A boolean indicating if eigenvectors are calculated to include a condition number calculation.

- addPosthoc
Boolean indicating if posthoc parameters are added to the table output.

- diagXform
This is the transformation used on the diagonal of the

`chol(solve(omega))`

. This matrix and values are the parameters estimated in FOCEi. The possibilities are:`sqrt`

Estimates the sqrt of the diagonal elements of`chol(solve(omega))`

. This is the default method.`log`

Estimates the log of the diagonal elements of`chol(solve(omega))`

`identity`

Estimates the diagonal elements without any transformations

- sumProd
Is a boolean indicating if the model should change multiplication to high precision multiplication and sums to high precision sums using the PreciseSums package. By default this is

`FALSE`

.- optExpression
Optimize the rxode2 expression to speed up calculation. By default this is turned on.

- literalFix
boolean, substitute fixed population values as literals and re-adjust ui and parameter estimates after optimization; Default is `TRUE`.

- ci
Confidence level for some tables. By default this is 0.95 or 95% confidence.

- useColor
Boolean indicating if focei can use ASCII color codes

- boundTol
Tolerance for boundary issues.

- calcTables
This boolean is to determine if the foceiFit will calculate tables. By default this is

`TRUE`

- noAbort
Boolean to indicate if you should abort the FOCEi evaluation if it runs into troubles. (default TRUE)

- interaction
Boolean indicate FOCEi should be used (TRUE) instead of FOCE (FALSE)

- cholSEtol
tolerance for Generalized Cholesky Decomposition. Defaults to suggested (.Machine$double.eps)^(1/3)

- cholAccept
Tolerance to accept a Generalized Cholesky Decomposition for a R or S matrix.

- resetEtaP
represents the p-value for reseting the individual ETA to 0 during optimization (instead of the saved value). The two test statistics used in the z-test are either chol(omega^-1) %*% eta or eta/sd(allEtas). A p-value of 0 indicates the ETAs never reset. A p-value of 1 indicates the ETAs always reset.

- resetThetaP
represents the p-value for reseting the population mu-referenced THETA parameters based on ETA drift during optimization, and resetting the optimization. A p-value of 0 indicates the THETAs never reset. A p-value of 1 indicates the THETAs always reset and is not allowed. The theta reset is checked at the beginning and when nearing a local minima. The percent change in objective function where a theta reset check is initiated is controlled in

`resetThetaCheckPer`

.- resetThetaFinalP
represents the p-value for reseting the population mu-referenced THETA parameters based on ETA drift during optimization, and resetting the optimization one final time.

- diagOmegaBoundUpper
This represents the upper bound of the diagonal omega matrix. The upper bound is given by diag(omega)*diagOmegaBoundUpper. If

`diagOmegaBoundUpper`

is 1, there is no upper bound on Omega.- diagOmegaBoundLower
This represents the lower bound of the diagonal omega matrix. The lower bound is given by diag(omega)/diagOmegaBoundUpper. If

`diagOmegaBoundLower`

is 1, there is no lower bound on Omega.- cholSEOpt
Boolean indicating if the generalized Cholesky should be used while optimizing.

- cholSECov
Boolean indicating if the generalized Cholesky should be used while calculating the Covariance Matrix.

- fo
is a boolean indicating if this is a FO approximation routine.

- covTryHarder
If the R matrix is non-positive definite and cannot be corrected to be non-positive definite try estimating the Hessian on the unscaled parameter space.

- outerOpt
optimization method for the outer problem

- innerOpt
optimization method for the inner problem (not implemented yet.)

- rhobeg
Beginning change in parameters for bobyqa algorithm (trust region). By default this is 0.2 or 20 parameters when the parameters are scaled to 1. rhobeg and rhoend must be set to the initial and final values of a trust region radius, so both must be positive with 0 < rhoend < rhobeg. Typically rhobeg should be about one tenth of the greatest expected change to a variable. Note also that smallest difference abs(upper-lower) should be greater than or equal to rhobeg*2. If this is not the case then rhobeg will be adjusted. (bobyqa)

- rhoend
The smallest value of the trust region radius that is allowed. If not defined, then 10^(-sigdig-1) will be used. (bobyqa)

- npt
The number of points used to approximate the objective function via a quadratic approximation for bobyqa. The value of npt must be in the interval [n+2,(n+1)(n+2)/2] where n is the number of parameters in par. Choices that exceed 2*n+1 are not recommended. If not defined, it will be set to 2*n + 1. (bobyqa)

- rel.tol
Relative tolerance before nlminb stops (nlmimb).

- x.tol
X tolerance for nlmixr2 optimizer

- eval.max
Number of maximum evaluations of the objective function (nlmimb)

- iter.max
Maximum number of iterations allowed (nlmimb)

- abstol
Absolute tolerance for nlmixr2 optimizer (BFGS)

- reltol
tolerance for nlmixr2 (BFGS)

- resetHessianAndEta
is a boolean representing if the individual Hessian is reset when ETAs are reset using the option

`resetEtaP`

.- stateTrim
Trim state amounts/concentrations to this value.

- shi21maxOuter
The maximum number of steps for the optimization of the forward-difference step size. When not zero, use this instead of Gill differences.

- shi21maxInner
The maximum number of steps for the optimization of the individual Hessian matrices in the generalized likelihood problem. When 0, un-optimized finite differences are used.

- shi21maxInnerCov
The maximum number of steps for the optimization of the individual Hessian matrices in the generalized likelihood problem for the covariance step. When 0, un-optimized finite differences are used.

- shi21maxFD
The maximum number of steps for the optimization of the forward difference step size when using dosing events (lag time, modeled duration/rate and bioavailability)

- gillK
The total number of possible steps to determine the optimal forward/central difference step size per parameter (by the Gill 1983 method). If 0, no optimal step size is determined. Otherwise this is the optimal step size determined.

- gillStep
When looking for the optimal forward difference step size, this is This is the step size to increase the initial estimate by. So each iteration the new step size = (prior step size)*gillStep

- gillFtol
The gillFtol is the gradient error tolerance that is acceptable before issuing a warning/error about the gradient estimates.

- gillRtol
The relative tolerance used for Gill 1983 determination of optimal step size.

- gillKcov
The total number of possible steps to determine the optimal forward/central difference step size per parameter (by the Gill 1983 method) during the covariance step. If 0, no optimal step size is determined. Otherwise this is the optimal step size determined.

- gillKcovLlik
The total number of possible steps to determine the optimal forward/central difference step per parameter when using the generalized focei log-likelihood method (by the Gill 1986 method). If 0, no optimal step size is determined. Otherwise this is the optimal step size is determined

- gillStepCovLlik
Same as above but during generalized focei log-likelihood

- gillStepCov
When looking for the optimal forward difference step size, this is This is the step size to increase the initial estimate by. So each iteration during the covariance step is equal to the new step size = (prior step size)*gillStepCov

- gillFtolCov
The gillFtol is the gradient error tolerance that is acceptable before issuing a warning/error about the gradient estimates during the covariance step.

- gillFtolCovLlik
Same as above but applied during generalized log-likelihood estimation.

- rmatNorm
A parameter to normalize gradient step size by the parameter value during the calculation of the R matrix

- rmatNormLlik
A parameter to normalize gradient step size by the parameter value during the calculation of the R matrix if you are using generalized log-likelihood Hessian matrix.

- smatNorm
A parameter to normalize gradient step size by the parameter value during the calculation of the S matrix

- smatNormLlik
A parameter to normalize gradient step size by the parameter value during the calculation of the S matrix if you are using the generalized log-likelihood.

- covGillF
Use the Gill calculated optimal Forward difference step size for the instead of the central difference step size during the central difference gradient calculation.

- optGillF
Use the Gill calculated optimal Forward difference step size for the instead of the central difference step size during the central differences for optimization.

- covSmall
The covSmall is the small number to compare covariance numbers before rejecting an estimate of the covariance as the final estimate (when comparing sandwich vs R/S matrix estimates of the covariance). This number controls how small the variance is before the covariance matrix is rejected.

- adjLik
In nlmixr2, the objective function matches NONMEM's objective function, which removes a 2*pi constant from the likelihood calculation. If this is TRUE, the likelihood function is adjusted by this 2*pi factor. When adjusted this number more closely matches the likelihood approximations of nlme, and SAS approximations. Regardless of if this is turned on or off the objective function matches NONMEM's objective function.

- gradTrim
The parameter to adjust the gradient to if the |gradient| is very large.

- maxOdeRecalc
Maximum number of times to reduce the ODE tolerances and try to resolve the system if there was a bad ODE solve.

- odeRecalcFactor
The ODE recalculation factor when ODE solving goes bad, this is the factor the rtol/atol is reduced

- gradCalcCentralSmall
A small number that represents the value where |grad| < gradCalcCentralSmall where forward differences switch to central differences.

- gradCalcCentralLarge
A large number that represents the value where |grad| > gradCalcCentralLarge where forward differences switch to central differences.

- etaNudge
By default initial ETA estimates start at zero; Sometimes this doesn't optimize appropriately. If this value is non-zero, when the n1qn1 optimization didn't perform appropriately, reset the Hessian, and nudge the ETA up by this value; If the ETA still doesn't move, nudge the ETA down by this value. By default this value is qnorm(1-0.05/2)*1/sqrt(3), the first of the Gauss Quadrature numbers times by the 0.95% normal region. If this is not successful try the second eta nudge number (below). If +-etaNudge2 is not successful, then assign to zero and do not optimize any longer

- etaNudge2
This is the second eta nudge. By default it is qnorm(1-0.05/2)*sqrt(3/5), which is the n=3 quadrature point (excluding zero) times by the 0.95% normal region

- nRetries
If FOCEi doesn't fit with the current parameter estimates, randomly sample new parameter estimates and restart the problem. This is similar to 'PsN' resampling.

- seed
an object specifying if and how the random number generator should be initialized

- resetThetaCheckPer
represents objective function % percentage below which resetThetaP is checked.

- etaMat
Eta matrix for initial estimates or final estimates of the ETAs.

- repeatGillMax
If the tolerances were reduced when calculating the initial Gill differences, the Gill difference is repeated up to a maximum number of times defined by this parameter.

- stickyRecalcN
The number of bad ODE solves before reducing the atol/rtol for the rest of the problem.

- gradProgressOfvTime
This is the time for a single objective function evaluation (in seconds) to start progress bars on gradient evaluations

- addProp
specifies the type of additive plus proportional errors, the one where standard deviations add (combined1) or the type where the variances add (combined2).

The combined1 error type can be described by the following equation:

$$y = f + (a + b\times f^c) \times \varepsilon$$

The combined2 error model can be described by the following equation:

$$y = f + \sqrt{a^2 + b^2\times f^{2\times c}} \times \varepsilon$$

Where:

- y represents the observed value

- f represents the predicted value

- a is the additive standard deviation

- b is the proportional/power standard deviation

- c is the power exponent (in the proportional case c=1)

- badSolveObjfAdj
The objective function adjustment when the ODE system cannot be solved. It is based on each individual bad solve.

- compress
Should the object have compressed items

- rxControl
`rxode2` ODE solving options during fitting, created with `rxControl()`

- sigdigTable
Significant digits in the final output table. If not specified, then it matches the significant digits in the `sigdig` optimization algorithm. If `sigdig` is NULL, use 3.

- fallbackFD
Fallback to the finite differences if the sensitivity equations do not solve.

- smatPer
A percentage representing the number of failed parameter gradients for each individual (which are replaced with the overall gradient for the parameter) out of the total number of gradients parameters (ie `ntheta*nsub`) before the S matrix is considered to be a bad matrix.

- sdLowerFact
A factor for multiplying the estimate by when the lower estimate is zero and the error is known to represent a standard deviation of a parameter (like add.sd, prop.sd, pow.sd, lnorm.sd, etc). When zero, no factor is applied. If your initial estimate is 0.15 and your lower bound is zero, then the lower bound would be assumed to be 0.00015.

## Details

Note this uses the R's L-BFGS-B in `optim`

for the
outer problem and the BFGS `n1qn1`

with that
allows restoring the prior individual Hessian (for faster
optimization speed).

However the inner problem is not scaled. Since most eta estimates start near zero, scaling for these parameters do not make sense.

This process of scaling can fix some ill conditioning for the unscaled problem. The covariance step is performed on the unscaled problem, so the condition number of that matrix may not be reflective of the scaled problem's condition-number.

## References

Gill, P.E., Murray, W., Saunders, M.A., & Wright, M.H. (1983). Computing Forward-Difference Intervals for Numerical Optimization. Siam Journal on Scientific and Statistical Computing, 4, 310-321.

Shi, H.M., Xie, Y., Xuan, M.Q., & Nocedal, J. (2021). Adaptive Finite-Difference Interval Estimation for Noisy Derivative-Free Optimization.