naginterfaces.library.opt.bounds_​mod_​deriv2_​easy

naginterfaces.library.opt.bounds_mod_deriv2_easy(ibound, funct2, hess2, bl, bu, x, data=None)

bounds_mod_deriv2_easy is an easy-to-use modified-Newton algorithm for finding a minimum of a function, $F(x_1,x_2,\dots ,x_n)$ subject to fixed upper and lower bounds on the independent variables, $x_1,x_2,\dots ,x_n$ when first and second derivatives of $F$ are available. It is intended for functions which are continuous and which have continuous first and second derivatives (although it will usually work even if the derivatives have occasional discontinuities).

For full information please refer to the NAG Library document for e04ly

https://support.nag.com/numeric/nl/nagdoc_30.1/flhtml/e04/e04lyf.html

Parameters

iboundint

Indicates whether the facility for dealing with bounds of special forms is to be used. It must be set to one of the following values:

$ibound=0$

If you are supplying all the $l_j$ and $u_j$ individually.

$ibound=1$

If there are no bounds on any $x_j$.

$ibound=2$

If all the bounds are of the form $0\le x_j$.

$ibound=3$

If $l_1=l_2=\cdots =l_n$ and $u_1=u_2=\cdots =u_n$.

funct2callable (fc, gc) = funct2(xc, data=None)

You must supply this function to calculate the values of the function $F\left(x\right)$ and its first derivatives $\frac{\partial F}{\partial x_j}$ at any point $x$.

It should be tested separately before being used in conjunction with bounds_mod_deriv2_easy (see the E04 Introduction).

Parameters

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

The point $x$ at which the function and its derivatives are required.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns

fcfloat

The value of the function $F$ at the current point $x$.

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

$gc[\text{j}-1]$ must be set to the value of the first derivative $\frac{\partial F}{\partial x_{\text{j}}}$ at the point $x$, for $\text{j}=1,2,\dots ,n$.

hess2callable (heslc, hesdc) = hess2(xc, lh, data=None)

You must supply this function to evaluate the elements $H_{ij}=\frac{{\partial }^{2}F}{\partial x_i\partial x_j}$ of the matrix of second derivatives of $F\left(x\right)$ at any point $x$.

It should be tested separately before being used in conjunction with bounds_mod_deriv2_easy (see the E04 Introduction).

Parameters

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

The point $x$ at which the derivatives are required.

lhint

The length of the array $heslc$.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns

heslcfloat, array-like, shape $\left(lh\right)$

$hess2$ must place the strict lower triangle of the second derivative matrix $H$ in $heslc$, stored by rows, i.e., set $heslc[(\text{i}-1)(\text{i}-2)/2+\text{j}-1]=\frac{{\partial }^{2}F}{\partial x_{\text{i}}\partial x_{\text{j}}}$, for $\text{j}=1,2,\dots ,\text{i}-1$, for $\text{i}=2,3,\dots ,n$. (The upper triangle is not required because the matrix is symmetric.)

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

Must contain the diagonal elements of the second derivative matrix, i.e., set $hesdc[\text{j}-1]=\frac{{\partial }^{2}F}{\partial x_{\text{j}}^2}$, for $\text{j}=1,2,\dots ,n$.

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

The lower bounds $l_j$.

If $ibound$ is set to $0$, $bl[\text{j}-1]$ must be set to $l_{\text{j}}$, for $\text{j}=1,2,\dots ,n$. (If a lower bound is not specified for any $x_j$, the corresponding $bl[j-1]$ should be set to $-{10}^6$.)

If $ibound$ is set to $3$, you must set $bl\left[0\right]$ to $l_1$; bounds_mod_deriv2_easy will then set the remaining elements of $bl$ equal to $bl\left[0\right]$.

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

The upper bounds $u_j$.

If $ibound$ is set to $0$, $bu[\text{j}-1]$ must be set to $u_{\text{j}}$, for $\text{j}=1,2,\dots ,n$. (If an upper bound is not specified for any $x_j$ the corresponding $bu[j-1]$ should be set to ${10}^6$.)

If $ibound$ is set to $3$, you must set $bu\left[0\right]$ to $u_1$; bounds_mod_deriv2_easy will then set the remaining elements of $bu$ equal to $bu\left[0\right]$.

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

$x[\text{j}-1]$ must be set to a guess at the $\text{j}$th component of the position of the minimum, for $\text{j}=1,2,\dots ,n$. The function checks the gradient and the Hessian matrix at the starting point, and is more likely to detect any error in your programming if the initial $x[j-1]$ are nonzero and mutually distinct.

dataarbitrary, optional

User-communication data for callback functions.

Returns

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

The lower bounds actually used by bounds_mod_deriv2_easy.

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

The upper bounds actually used by bounds_mod_deriv2_easy.

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

The lowest point found during the calculations. Thus, if no exception or warning is raised on exit, $x[j-1]$ is the $j$th component of the position of the minimum.

ffloat

The value of $F\left(x\right)$ corresponding to the final point stored in $x$.

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

The value of $\frac{\partial F}{\partial x_{\text{j}}}$ corresponding to the final point stored in $x$, for $\text{j}=1,2,\dots ,n$; the value of $g[j-1]$ for variables not on a bound should normally be close to zero.

Raises

NagValueError

(errno $1$)

On entry, $ibound=0$ and $bl[\text{j}-1]>bu[\text{j}-1]$ for some $j$.

(errno $1$)

On entry, $ibound=3$ and $bl\left[0\right]>bu\left[0\right]$.

(errno $1$)

On entry, $ibound=⟨value⟩$.

Constraint: $0\le ibound\le 3$.

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 1$.

(errno $2$)

There have been $50\times n$ function evaluations.

(errno $9$)

The modulus of a variable has become very large. There may be a mistake in your supplied functions, your problem has no finite solution, or the problem needs rescaling.

Warns

NagAlgorithmicWarning

(errno $3$)

The conditions for a minimum have not all been satisfied, but a lower point could not be found.

(errno $5$)

It is probable that a local minimum has been found, but it cannot be guaranteed.

(errno $6$)

It is possible that a local minimum has been found, but it cannot be guaranteed.

(errno $7$)

It is unlikely that a local minimum has been found.

(errno $8$)

It is very unlikely that a local minimum has been found.

(errno $10$)

It is very likely that you have made an error forming the gradient.

(errno $11$)

It is very likely that you have made an error forming the 2nd derivatives.

Notes

No equivalent traditional C interface for this routine exists in the NAG Library.

bounds_mod_deriv2_easy is applicable to problems of the form:

$$Minimize\left(F\right)(x_1,x_2,\dots ,x_n)\text{subject to}{l}_j\le x_j\le u_j\text{,}j=1,2,\dots ,n$$

when first and second derivatives of $F\left(x\right)$ are available.

Special provision is made for problems which actually have no bounds on the $x_j$, problems which have only non-negativity bounds and problems in which $l_1=l_2=\cdots =l_n$ and $u_1=u_2=\cdots =u_n$. You must supply a function to calculate the values of $F\left(x\right)$ and its first derivatives at any point $x$ and a function to calculate the second derivatives.

From a starting point you supplied there is generated, on the basis of estimates of the curvature of $F\left(x\right)$, a sequence of feasible points which is intended to converge to a local minimum of the constrained function.

References

Gill, P E and Murray, W, 1976, Minimization subject to bounds on the variables, NPL Report NAC 72, National Physical Laboratory