e04lyf 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).
The routine may be called by the names e04lyf or nagf_opt_bounds_mod_deriv2_easy.
3Description
e04lyf is applicable to problems of the form:
$$\mathrm{Minimize}F({x}_{1},{x}_{2},\dots ,{x}_{n})\text{\hspace{1em} subject to \hspace{1em}}{l}_{j}\le {x}_{j}\le {u}_{j}\text{, \hspace{1em}}j=1,2,\dots ,n$$
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 subroutine to calculate the values of $F\left(x\right)$ and its first derivatives at any point $x$ and a subroutine 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.
4References
Gill P E and Murray W (1976) Minimization subject to bounds on the variables NPL Report NAC 72 National Physical Laboratory
5Arguments
1: $\mathbf{n}$ – IntegerInput
On entry: the number $n$ of independent variables.
Constraint:
${\mathbf{n}}\ge 1$.
2: $\mathbf{ibound}$ – IntegerInput
On entry: 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:
${\mathbf{ibound}}=0$
If you are supplying all the ${l}_{j}$ and ${u}_{j}$ individually.
${\mathbf{ibound}}=1$
If there are no bounds on any ${x}_{j}$.
${\mathbf{ibound}}=2$
If all the bounds are of the form $0\le {x}_{j}$.
${\mathbf{ibound}}=3$
If ${l}_{1}={l}_{2}=\cdots ={l}_{n}$ and ${u}_{1}={u}_{2}=\cdots ={u}_{n}$.
Constraint:
$0\le {\mathbf{ibound}}\le 3$.
3: $\mathbf{funct2}$ – Subroutine, supplied by the user.External Procedure
You must supply this routine 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 e04lyf (see the E04 Chapter Introduction).
2: $\mathbf{xc}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the point $x$ at which the function and its derivatives are required.
3: $\mathbf{fc}$ – Real (Kind=nag_wp)Output
On exit: the value of the function $F$ at the current point $x$.
4: $\mathbf{gc}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{gc}}\left(\mathit{j}\right)$ must be set to the value of the first derivative $\frac{\partial F}{\partial {x}_{\mathit{j}}}$ at the point $x$, for $\mathit{j}=1,2,\dots ,n$.
6: $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace
funct2 is called with the arguments iuser and ruser as supplied to e04lyf. You should use the arrays iuser and ruser to supply information to funct2.
funct2 must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which e04lyf is called. Arguments denoted as Input must not be changed by this procedure.
Note:funct2 should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by e04lyf. If your code inadvertently does return any NaNs or infinities, e04lyf is likely to produce unexpected results.
4: $\mathbf{hess2}$ – Subroutine, supplied by the user.External Procedure
You must supply this routine 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 e04lyf (see the E04 Chapter Introduction).
2: $\mathbf{xc}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the point $x$ at which the derivatives are required.
3: $\mathbf{heslc}\left({\mathbf{lh}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: hess2 must place the strict lower triangle of the second derivative matrix $H$ in heslc, stored by rows, i.e., set
${\mathbf{heslc}}\left((\mathit{i}-1)(\mathit{i}-2)/2+\mathit{j}\right)=\frac{{\partial}^{2}F}{\partial {x}_{\mathit{i}}\partial {x}_{\mathit{j}}}$, for $\mathit{i}=2,3,\dots ,n$ and $\mathit{j}=1,2,\dots ,\mathit{i}-1$. (The upper triangle is not required because the matrix is symmetric.)
5: $\mathbf{hesdc}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: must contain the diagonal elements of the second derivative matrix, i.e., set
${\mathbf{hesdc}}\left(\mathit{j}\right)=\frac{{\partial}^{2}F}{\partial {x}_{\mathit{j}}^{2}}$, for $\mathit{j}=1,2,\dots ,n$.
7: $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace
hess2 is called with the arguments iuser and ruser as supplied to e04lyf. You should use the arrays iuser and ruser to supply information to hess2.
hess2 must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which e04lyf is called. Arguments denoted as Input must not be changed by this procedure.
Note:hess2 should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by e04lyf. If your code inadvertently does return any NaNs or infinities, e04lyf is likely to produce unexpected results.
5: $\mathbf{bl}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: the lower bounds ${l}_{j}$.
If ibound is set to $0$,
${\mathbf{bl}}\left(\mathit{j}\right)$ must be set to ${l}_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,n$. (If a lower bound is not specified for any ${x}_{j}$, the corresponding ${\mathbf{bl}}\left(j\right)$ should be set to $-{10}^{6}$.)
If ibound is set to $3$, you must set ${\mathbf{bl}}\left(1\right)$ to ${l}_{1}$; e04lyf will then set the remaining elements of bl equal to ${\mathbf{bl}}\left(1\right)$.
On exit: the lower bounds actually used by e04lyf.
6: $\mathbf{bu}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: the upper bounds ${u}_{j}$.
If ibound is set to $0$,
${\mathbf{bu}}\left(\mathit{j}\right)$ must be set to ${u}_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,n$. (If an upper bound is not specified for any ${x}_{j}$ the corresponding ${\mathbf{bu}}\left(j\right)$ should be set to ${10}^{6}$.)
If ibound is set to $3$, you must set ${\mathbf{bu}}\left(1\right)$ to ${u}_{1}$; e04lyf will then set the remaining elements of bu equal to ${\mathbf{bu}}\left(1\right)$.
On exit: the upper bounds actually used by e04lyf.
7: $\mathbf{x}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: ${\mathbf{x}}\left(\mathit{j}\right)$ must be set to a guess at the $\mathit{j}$th component of the position of the minimum, for $\mathit{j}=1,2,\dots ,n$. The routine 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 ${\mathbf{x}}\left(j\right)$ are nonzero and mutually distinct.
On exit: the lowest point found during the calculations. Thus, if ${\mathbf{ifail}}={\mathbf{0}}$ on exit, ${\mathbf{x}}\left(j\right)$ is the $j$th component of the position of the minimum.
8: $\mathbf{f}$ – Real (Kind=nag_wp)Output
On exit: the value of $F\left(x\right)$ corresponding to the final point stored in x.
9: $\mathbf{g}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the value of
$\frac{\partial F}{\partial {x}_{\mathit{j}}}$ corresponding to the final point stored in x, for $\mathit{j}=1,2,\dots ,n$; the value of ${\mathbf{g}}\left(j\right)$ for variables not on a bound should normally be close to zero.
15: $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace
iuser and ruser are not used by e04lyf, but are passed directly to funct2 and hess2 and may be used to pass information to these routines.
16: $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $\mathrm{-1}$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $\mathrm{-1}$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $\mathrm{-1}$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $\mathrm{-1}$ is recommended since useful values can be provided in some output arguments even when ${\mathbf{ifail}}\ne {\mathbf{0}}$ on exit. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry ${\mathbf{ifail}}=0$ or $\mathrm{-1}$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
Note: in some cases e04lyf may return useful information.
If you are not satisfied with the result (e.g., because ${\mathbf{ifail}}={\mathbf{5}}$, ${\mathbf{6}}$, ${\mathbf{7}}$ or ${\mathbf{8}}$), it is worth restarting the calculations from a different starting point (not the point at which the failure occurred) in order to avoid the region which caused the failure.
${\mathbf{ifail}}=1$
On entry, ${\mathbf{ibound}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: $0\le {\mathbf{ibound}}\le 3$.
On entry, ${\mathbf{ibound}}=0$ and ${\mathbf{bl}}\left(\mathit{j}\right)>{\mathbf{bu}}\left(\mathit{j}\right)$ for some $j$.
On entry, ${\mathbf{ibound}}=3$ and ${\mathbf{bl}}\left(1\right)>{\mathbf{bu}}\left(1\right)$.
On entry, ${\mathbf{liw}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{liw}}\ge \u27e8\mathit{\text{value}}\u27e9$.
On entry, ${\mathbf{lw}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{lw}}\ge \u27e8\mathit{\text{value}}\u27e9$.
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}\ge 1$.
${\mathbf{ifail}}=2$
There have been $50\times {\mathbf{n}}$ function evaluations.
The algorithm does not seem to be converging. The calculations can be restarted from the final point held in x. The error may also indicate that $F\left(x\right)$ has no minimum.
${\mathbf{ifail}}=3$
The conditions for a minimum have not all been satisfied, but a lower point could not be found. See Section 7 for further information.
${\mathbf{ifail}}=5$
It is probable that a local minimum has been found, but it cannot be guaranteed.
${\mathbf{ifail}}=6$
It is possible that a local minimum has been found, but it cannot be guaranteed.
${\mathbf{ifail}}=7$
It is unlikely that a local minimum has been found.
${\mathbf{ifail}}=8$
It is very unlikely that a local minimum has been found.
${\mathbf{ifail}}=9$
The modulus of a variable has become very large. There may be a mistake in your supplied routines, your problem has no finite solution, or the problem needs rescaling.
${\mathbf{ifail}}=10$
It is very likely that you have made an error forming the gradient.
${\mathbf{ifail}}=11$
It is very likely that you have made an error forming the 2nd derivatives.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please
contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
7Accuracy
When a successful exit is made then, for a computer with a mantissa of $t$ decimals, one would expect to get about $t/2-1$ decimals accuracy in $x$, and about $t-1$ decimals accuracy in $F$, provided the problem is reasonably well scaled.
8Parallelism and Performance
e04lyf 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9Further Comments
The number of iterations required depends on the number of variables, the behaviour of $F\left(x\right)$ and the distance of the starting point from the solution. The number of operations performed in an iteration of e04lyf is roughly proportional to ${n}^{3}+\mathit{O}\left({n}^{2}\right)$. In addition, each iteration makes one call of hess2 and at least one call of funct2. So, unless $F\left(x\right)$, the gradient vector and the matrix of second derivatives can be evaluated very quickly, the run time will be dominated by the time spent in user-supplied subroutines funct2 and hess2.
Ideally the problem should be scaled so that at the solution the value of $F\left(x\right)$ and the corresponding values of ${x}_{1},{x}_{2},\dots {x}_{n}$ are each in the range $(\mathrm{-1},+1)$, and so that at points a unit distance away from the solution, $F$ is approximately a unit value greater than at the minimum. It is unlikely that you will be able to follow these recommendations very closely, but it is worth trying (by guesswork), as sensible scaling will reduce the difficulty of the minimization problem, so that e04lyf will take less computer time.
starting from the initial guess $(3,\mathrm{-1},0,1)$. (In practice, it is worth trying to make user-supplied subroutines funct2 and hess2 as efficient as possible. This has not been done in the example program for reasons of clarity.)