naginterfaces.library.opt.estimate_​deriv

naginterfaces.library.opt.estimate_deriv(msglvl, epsrf, x, mode, objfun, hforw, data=None, io_manager=None)

estimate_deriv computes an approximation to the gradient vector and/or the Hessian matrix for use in conjunction with, or following the use of an optimization function (such as nlp1_rcomm()).

For full information please refer to the NAG Library document for e04xa

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/e04/e04xaf.html

Parameters

msglvlint

Must indicate the amount of intermediate output desired (see Further Comments for a description of the printed output). All output is written on the file object associated with the advisory I/O unit (see FileObjManager).

Value	Definition
0	No printout
1	A summary is printed out for each variable plus any warning messages.
Other	Values other than $0$ and $1$ should normally be used only at the direction of NAG.

epsrffloat

Must define $e_R$, which is intended to be a measure of the accuracy with which the problem function $F$ can be computed. The value of $e_R$ should reflect the relative precision of $1+\left|F\left(x\right)\right|$, i.e., acts as a relative precision when $\left|F\right|$ is large, and as an absolute precision when $\left|F\right|$ is small. For example, if $F\left(x\right)$ is typically of order $1000$ and the first six significant digits are known to be correct, an appropriate value for $e_R$ would be $1.0e-6$.

A discussion of $epsrf$ is given in Module 8 of Gill et al. (1981).

If $epsrf$ is either too small or too large on entry a warning will be printed if $msglvl=1$, the argument $iwarn$ set to the appropriate value on exit and estimate_deriv will use a default value of $e_M^{0.9}$, where $e_M$ is the machine precision.

If $epsrf\le 0.0$ on entry, then estimate_deriv will use the default value internally.

The default value will be appropriate for most simple functions that are computed with full accuracy.

xfloat, array-like, shape $\left(n\right)$

The point $x$ at which the derivatives are to be computed.

modeint

Indicates which derivatives are required.

$mode=0$

The gradient and Hessian diagonal values having supplied the objective function via $objfun$.

$mode=1$

The Hessian matrix having supplied both the objective function and gradients via $objfun$.

$mode=2$

The gradient values and Hessian matrix having supplied the objective function via $objfun$.

objfuncallable (objf, objgrd) = objfun(mode, x, nstate, data=None)

If $mode=0$ or $2$, $objfun$ must calculate the objective function; otherwise, if $mode=1$, $objfun$ must calculate the objective function and the gradients.

Parameters

modeint

$mode$ indicates which argument values within $objfun$ need to be set.

To $objfun$, $mode$ is always set to the value that you set it to before the call to estimate_deriv.

xfloat, ndarray, shape $\left(n\right)$

The point $x$ at which the objective function (and gradients if $mode=1$) is to be evaluated.

nstateint

Will be set to $1$ on the first call of $objfun$ by estimate_deriv, and is $0$ for all subsequent calls. Thus, if you wish, $nstate$ may be tested within $objfun$ in order to perform certain calculations once only. For example you may read data.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns

objffloat

Must be set to the value of the objective function.

objgrdfloat, array-like, shape $\left(n\right)$

If $mode=1$, $objgrd[j-1]$ must contain the value of the first derivative with respect to $x$.

If $mode\ne 1$, $objgrd$ need not be set.

hforwfloat, array-like, shape $\left(n\right)$

The initial trial interval for computing the appropriate partial derivative to the $j$th variable.

If $hforw[j-1]\le 0.0$, the initial trial interval is computed by estimate_deriv (see Notes).

dataarbitrary, optional

User-communication data for callback functions.

io_managerFileObjManager, optional

Manager for I/O in this routine.

Returns

modeint

Is changed only if you set $mode$ negative in $objfun$, i.e., you have requested termination of estimate_deriv.

hforwfloat, ndarray, shape $\left(n\right)$

$hforw[j-1]$ is the best interval found for computing a forward-difference approximation to the appropriate partial derivative for the $j$th variable.

objffloat

The value of the objective function evaluated at the input vector in $x$.

objgrdfloat, ndarray, shape $\left(n\right)$

If $mode=0$ or $2$, $objgrd[j-1]$ contains the best estimate of the first partial derivative for the $j$th variable.

If $mode=1$, $objgrd[j-1]$ contains the first partial derivative for the $j$th variable evaluated at the input vector in $x$.

hcntrlfloat, ndarray, shape $\left(n\right)$

$hcntrl[j-1]$ is the best interval found for computing a central-difference approximation to the appropriate partial derivative for the $j$th variable.

hfloat, ndarray, shape $(n,:)$

If $mode=0$, the estimated Hessian diagonal elements are contained in the first column of this array.

If $mode=1$ or $2$, the estimated Hessian matrix is contained in the leading $n\times n$ part of this array.

iwarnint

$iwarn=0$ on successful exit.

If the value of $epsrf$ on entry is too small or too large then $iwarn$ is set to $1$ or $2$ respectively on exit and the default value for $epsrf$ is used within estimate_deriv.

If $msglvl>0$ then warnings will be printed if $epsrf$ is too small or too large.

infoint, ndarray, shape $\left(n\right)$

$info[j-1]$ represents diagnostic information on variable $j$ as follows:

$info[j-1]=1$

The appropriate function appears to be constant. $hforw[i-1]$ is set to the initial trial interval value (see Notes) corresponding to a well-scaled problem and Error est. in the printed output is set to zero. This value occurs when the estimated relative condition error in the first derivative approximation is unacceptably large for every value of the finite difference interval. If this happens when the function is not constant the initial interval may be too small; in this case, it may be worthwhile to rerun estimate_deriv with larger initial trial interval values supplied in $hforw$ (see Notes). This error may also occur if the function evaluation includes an inordinately large constant term or if $epsrf$ is too large.

$info[j-1]=2$

The appropriate function appears to be linear or odd. $hforw[i-1]$ is set to the smallest interval with acceptable bounds on the relative condition error in the forward - and backward-difference estimates. In this case, the estimated relative condition error in the second derivative approximation remained large for every trial interval, but the estimated error in the first derivative approximation was acceptable for at least one interval. If the function is not linear or odd the relative condition error in the second derivative may be decreasing very slowly, it may be worthwhile to rerun estimate_deriv with larger initial trial interval values supplied in $hforw$ (see Notes).

$info[j-1]=3$

The second derivative of the appropriate function appears to be so large that it cannot be reliably estimated (i.e., near a singularity). $hforw[i-1]$ is set to the smallest trial interval.

This value occurs when the relative condition error estimate in the second derivative remained very small for every trial interval.

If the second derivative is not large the relative condition error in the second derivative may be increasing very slowly.

It may be worthwhile to rerun estimate_deriv with smaller initial trial interval values supplied in $hforw$ (see Notes).

This error may also occur when the given value of $epsrf$ is not a good estimate of a bound on the absolute error in the appropriate function (i.e., $epsrf$ is too small).

$info[j-1]=4$

The algorithm terminated with an apparently acceptable estimate of the second derivative. However the forward-difference estimates of the appropriate first derivatives (computed with the final estimate of the ‘optimal’ forward-difference interval) and the central difference estimates (computed with the interval used to compute the final estimate of the second derivative) do not agree to half a decimal place. The usual reason that the forward - and central-difference estimates fail to agree is that the first derivative is small.

If the first derivative is not small, it may be helpful to execute the procedure at a different point.

Raises

NagValueError

(errno $1$)

On entry, $mode=⟨value⟩$.

Constraint: $0\le mode\le 2$.

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 1$.

Warns

NagAlgorithmicWarning

(errno $2$)

One or more variables have a nonzero $info$ value.

NagCallbackTerminateWarning

(errno $i<0$)

User requested termination by setting $mode$ negative in $objfun$.

Notes

In the NAG Library the traditional C interface for this routine uses a different algorithmic base. Please contact NAG if you have any questions about compatibility.

estimate_deriv is similar to routine FDCALC described in Gill et al. (1983a). It should be noted that this function aims to compute sufficiently accurate estimates of the derivatives for use with an optimization algorithm. If you require more accurate estimates you should refer to submodule numdiff.

estimate_deriv computes finite difference approximations to the gradient vector and the Hessian matrix for a given function. The simplest approximation involves the forward-difference formula, in which the derivative $f^{\prime }\left(x\right)$ of a univariate function $f\left(x\right)$ is approximated by the quantity

$${\rho }_F(f,h)=\frac{f(x+h)-f\left(x\right)}{h}$$

for some interval $h>0$, where the subscript ‘F’ denotes ‘forward-difference’ (see Gill et al. (1983b)).

To summarise the procedure used by estimate_deriv (for the case when the objective function is available and you require estimates of gradient values and Hessian matrix diagonal values, i.e., $mode=0$) consider a univariate function $f$ at the point $x$. (In order to obtain the gradient of a multivariate function $F\left(x\right)$, where $x$ is an $n$-vector, the procedure is applied to each component of $x$, keeping the other components fixed.) Roughly speaking, the method is based on the fact that the bound on the relative truncation error in the forward-difference approximation tends to be an increasing function of $h$, while the relative condition error bound is generally a decreasing function of $h$, hence changes in $h$ will tend to have opposite effects on these errors (see Gill et al. (1983b)).

The ‘best’ interval $h$ is given by

$$h_F=2\sqrt{\frac{(1+\left|f\left(x\right)\right|)e_R}{\left|\Phi \right|}}$$

where $\Phi$ is an estimate of $f^{\prime \prime }\left(x\right)$, and $e_R$ is an estimate of the relative error associated with computing the function (see Module 8 of Gill et al. (1981)). Given an interval $h$, $\Phi$ is defined by the second-order approximation

$$\Phi =\frac{f(x+h)-2f\left(x\right)+f(x-h)}{h^2}\text{.}$$

The decision as to whether a given value of $\Phi$ is acceptable involves $\hat{c}\left(\Phi \right)$, the following bound on the relative condition error in $\Phi$:

$$\hat{c}\left(\Phi \right)=\frac{4e_R(1+\left|f\right|)}{h^2\left|\Phi \right|}$$

(When $\Phi$ is zero, $\hat{c}\left(\Phi \right)$ is taken as an arbitrary large number.)

The procedure selects the interval $h_{\varphi }$ (to be used in computing $\Phi$) from a sequence of trial intervals $\left(h_k\right)$. The initial trial interval is taken as $10\overline{h}$, where

$$\overline{h}=2(1+\left|x\right|)\sqrt{e_R}$$

unless you specify the initial value to be used.

The value of $\hat{c}\left(\Phi \right)$ for a trial value $h_k$ is defined as ‘acceptable’ if it lies in the interval $[0.001,0.1]$. In this case $h_{\varphi }$ is taken as $h_k$, and the current value of $\Phi$ is used to compute $h_F$ from (1). If $\hat{c}\left(\Phi \right)$ is unacceptable, the next trial interval is chosen so that the relative condition error bound will either decrease or increase, as required. If the bound on the relative condition error is too large, a larger interval is used as the next trial value in an attempt to reduce the condition error bound. On the other hand, if the relative condition error bound is too small, $h_k$ is reduced.

The procedure will fail to produce an acceptable value of $\hat{c}\left(\Phi \right)$ in two situations. Firstly, if $f^{\prime \prime }\left(x\right)$ is extremely small, then $\hat{c}\left(\Phi \right)$ may never become small, even for a very large value of the interval. Alternatively, $\hat{c}\left(\Phi \right)$ may never exceed $0.001$, even for a very small value of the interval. This usually implies that $f^{\prime \prime }\left(x\right)$ is extremely large, and occurs most often near a singularity.

As a check on the validity of the estimated first derivative, the procedure provides a comparison of the forward-difference approximation computed with $h_F$ (as above) and the central-difference approximation computed with $h_{\varphi }$. Using the central-difference formula the first derivative can be approximated by

$${\rho }_c(f,h)=\frac{f(x+h)-f(x-h)}{2h}$$

where $h>0$. If the values $h_F$ and $h_{\varphi }$ do not display some agreement, neither can be considered reliable.

When both function and gradients are available and you require the Hessian matrix (i.e., $mode=1$) estimate_deriv follows a similar procedure to the case above with the exception that the gradient function $g\left(x\right)$ is substituted for the objective function and so the forward-difference interval for the first derivative of $g\left(x\right)$ with respect to variable $x_j$ is computed. The $j$th column of the approximate Hessian matrix is then defined as in Module 2 of Gill et al. (1981), by

$$\frac{g(x+h_j{e}_j)-g\left(x\right)}{h_j}$$

where $h_j$ is the best forward-difference interval associated with the $j$th component of $g$ and $e_j$ is the vector with unity in the $j$th position and zeros elsewhere.

When only the objective function is available and you require the gradients and Hessian matrix (i.e., $mode=2$) estimate_deriv again follows the same procedure as the case for $mode=0$ except that this time the value of $\hat{c}\left(\Phi \right)$ for a trial value $h_k$ is defined as acceptable if it lies in the interval $[0.0001,0.01]$ and the initial trial interval is taken as

$$\overline{h}=2(1+\left|x\right|)\sqrt[4]{4e_R}\text{.}$$

The approximate Hessian matrix $G$ is then defined as in Module 2 of Gill et al. (1981), by

$$G_{ij}\left(x\right)=\frac{1}{h_i{h}_j}(f(x+h_i{e}_i+h_j{e}_j)-f(x+h_i{e}_i)-f(x+h_j{e}_j)+f\left(x\right))\text{.}$$

References

Gill, P E, Murray, W, Saunders, M A and Wright, M H, 1983, Documentation for FDCALC and FDCORE, Technical Report SOL, 83–6, Stanford University

Gill, P E, Murray, W, Saunders, M A and Wright, M H, 1983, Computing forward-difference intervals for numerical optimization, SIAM J. Sci. Statist. Comput. (4), 310–321

Gill, P E, Murray, W and Wright, M H, 1981, Practical Optimization, Academic Press