e04jyf is an easy-to-use quasi-Newton algorithm for finding a minimum of a function $F({x}_{1},{x}_{2},\dots ,{x}_{n})$, subject to fixed upper and lower bounds of the independent variables ${x}_{1},{x}_{2},\dots ,{x}_{n}$, using function values only.
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 e04jyf or nagf_opt_bounds_quasi_func_easy.
3Description
e04jyf 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 derivatives of $F\left(x\right)$ are unavailable.
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 value of $F\left(x\right)$ at any point $x$.
From a starting point you supplied there is generated, on the basis of estimates of the gradient and 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. An attempt is made to verify that the final point is a minimum.
A typical iteration starts at the current point $x$ where ${n}_{z}$ (say) variables are free from both their bounds. The projected gradient vector ${g}_{z}$, whose elements are finite difference approximations to the derivatives of $F\left(x\right)$ with respect to the free variables, is known. A unit lower triangular matrix $L$ and a diagonal matrix $D$ (both of dimension ${n}_{z}$), such that $LD{L}^{\mathrm{T}}$ is a positive definite approximation of the matrix of second derivatives with respect to the free variables (i.e., the projected Hessian) are also held. The equations
$$LD{L}^{\mathrm{T}}{p}_{z}=-{g}_{z}$$
are solved to give a search direction ${p}_{z}$, which is expanded to an $n$-vector $p$ by an insertion of appropriate zero elements. Then $\alpha $ is found such that $F(x+\alpha p)$ is approximately a minimum (subject to the fixed bounds) with respect to $\alpha $; $x$ is replaced by $x+\alpha p$, and the matrices $L$ and $D$ are updated so as to be consistent with the change produced in the estimated gradient by the step $\alpha p$. If any variable actually reaches a bound during the search along $p$, it is fixed and ${n}_{z}$ is reduced for the next iteration. Most iterations calculate ${g}_{z}$ using forward differences, but central differences are used when they seem necessary.
There are two sets of convergence criteria – a weaker and a stronger. Whenever the weaker criteria are satisfied, the Lagrange multipliers are estimated for all the active constraints. If any Lagrange multiplier estimate is significantly negative, then one of the variables associated with a negative Lagrange multiplier estimate is released from its bound and the next search direction is computed in the extended subspace (i.e., ${n}_{z}$ is increased). Otherwise minimization continues in the current subspace provided that this is practicable. When it is not, or when the stronger convergence criteria are already satisfied, then, if one or more Lagrange multiplier estimates are close to zero, a slight perturbation is made in the values of the corresponding variables in turn until a lower function value is obtained. The normal algorithm is then resumed from the perturbed point.
If a saddle point is suspected, a local search is carried out with a view to moving away from the saddle point. A local search is also performed when a point is found which is thought to be a constrained minimum.
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 ${u}_{1}={u}_{2}=\cdots ={u}_{n}$.
3: $\mathbf{funct1}$ – Subroutine, supplied by the user.External Procedure
You must supply funct1 to calculate the value of the function $F\left(x\right)$ at any point $x$. It should be tested separately before being used with e04jyf (see the E04 Chapter Introduction).
5: $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace
funct1 is called with the arguments iuser and ruser as supplied to e04jyf. You should use the arrays iuser and ruser to supply information to funct1.
funct1 must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which e04jyf is called. Arguments denoted as Input must not be changed by this procedure.
Note:funct1 should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by e04jyf. If your code inadvertently does return any NaNs or infinities, e04jyf is likely to produce unexpected results.
4: $\mathbf{bl}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: the lower bounds ${l}_{j}$.
If ibound is set to $0$, you must set
${\mathbf{bl}}\left(\mathit{j}\right)$ to ${l}_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,n$. (If a lower bound is not specified for a particular ${x}_{\mathit{j}}$, the corresponding ${\mathbf{bl}}\left(\mathit{j}\right)$ should be set to $-{10}^{6}$.)
If ibound is set to $3$, you must set ${\mathbf{bl}}\left(1\right)$ to ${l}_{1}$; e04jyf will then set the remaining elements of bl equal to ${\mathbf{bl}}\left(1\right)$.
On exit: the lower bounds actually used by e04jyf.
5: $\mathbf{bu}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: the upper bounds ${u}_{j}$.
If ibound is set to $0$, you must set
${\mathbf{bu}}\left(\mathit{j}\right)$ to ${u}_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,n$. (If an upper bound is not specified for a particular ${x}_{j}$, 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 ${u}_{1}$; e04jyf will then set the remaining elements of bu equal to ${\mathbf{bu}}\left(1\right)$.
On exit: the upper bounds actually used by e04jyf.
6: $\mathbf{x}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: ${\mathbf{x}}\left(\mathit{j}\right)$ must be set to an estimate of the $\mathit{j}$th component of the position of the minimum, for $\mathit{j}=1,2,\dots ,n$.
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.
7: $\mathbf{f}$ – Real (Kind=nag_wp)Output
On exit: the value of $F\left(x\right)$ corresponding to the final point stored in x.
On exit: if ${\mathbf{ifail}}={\mathbf{0}}$, ${\mathbf{3}}$ or ${\mathbf{5}}$, the first n elements of iw contain information about which variables are currently on their bounds and which are free. Specifically, if ${x}_{i}$ is:
–fixed on its upper bound, ${\mathbf{iw}}\left(i\right)$ is $-1$;
–fixed on its lower bound, ${\mathbf{iw}}\left(i\right)$ is $-2$;
–effectively a constant (i.e., ${l}_{j}={u}_{j}$), ${\mathbf{iw}}\left(i\right)$ is $-3$;
–free, ${\mathbf{iw}}\left(i\right)$ gives its position in the sequence of free variables.
In addition, ${\mathbf{iw}}\left({\mathbf{n}}+1\right)$ contains the number of free variables (i.e., ${n}_{z}$). The rest of the array is used as workspace.
9: $\mathbf{liw}$ – IntegerInput
On entry: the dimension of the array iw as declared in the (sub)program from which e04jyf is called.
Constraint:
${\mathbf{liw}}\ge {\mathbf{n}}+2$.
10: $\mathbf{w}\left({\mathbf{lw}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: if ${\mathbf{ifail}}={\mathbf{0}}$, ${\mathbf{3}}$ or ${\mathbf{5}}$, ${\mathbf{w}}\left(i\right)$ contains a finite difference approximation to the
$\mathit{i}$th element of the projected gradient vector ${g}_{z}$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$. In addition, ${\mathbf{w}}\left({\mathbf{n}}+1\right)$ contains an estimate of the condition number of the projected Hessian matrix (i.e., $k$). The rest of the array is used as workspace.
11: $\mathbf{lw}$ – IntegerInput
On entry: the dimension of the array w as declared in the (sub)program from which e04jyf is called.
13: $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace
iuser and ruser are not used by e04jyf, but are passed directly to funct1 and may be used to pass information to this routine.
14: $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1$ 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 $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $-1$ 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 $-1$, 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 e04jyf 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. If persistent trouble occurs and the gradient can be calculated, it may be advisable to change to a routine which uses gradients (see the E04 Chapter Introduction).
${\mathbf{ifail}}=1$
On entry, either ${\mathbf{ibound}}=0$ and ${\mathbf{bl}}\left(\mathit{j}\right)>{\mathbf{bu}}\left(\mathit{j}\right)$ for some $j$, or ${\mathbf{ibound}}=3$ and ${\mathbf{bl}}\left(1\right)>{\mathbf{bu}}\left(1\right)$.
On entry, ${\mathbf{ibound}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: $0\le {\mathbf{ibound}}\le 3$.
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 $400\times {\mathbf{n}}$ calls to funct1.
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 met but a lower point could not be found and the algorithm has failed.
${\mathbf{ifail}}=4$
An overflow occurred during computation.
This is an unlikely failure, but if it occurs you should restart at the latest point given in x.
${\mathbf{ifail}}=5$
It is probable that a local minimum has been found, but it cannot be guaranteed.
${\mathbf{ifail}}=6$
It is probable 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 funct1, your problem has no finite solution, or the problem needs rescaling.
${\mathbf{ifail}}=10$
One of the forward differences was negative.
The computed set of forward-difference intervals (stored in ${\mathbf{w}}\left(9\times {\mathbf{n}}+1\right),{\mathbf{w}}\left(9\times {\mathbf{n}}+2\right),\dots ,\phantom{\rule{0ex}{0ex}}{\mathbf{w}}\left(10\times {\mathbf{n}}\right)$) is such that ${\mathbf{x}}\left(i\right)+{\mathbf{w}}\left(9\times {\mathbf{n}}+i\right)\le {\mathbf{x}}\left(i\right)$ for some $i$.
This is an unlikely failure, but if it occurs you should attempt to select another starting point.
${\mathbf{ifail}}=-99$
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
A successful exit (${\mathbf{ifail}}={\mathbf{0}}$) is made from e04jyf when ($\mathrm{B1}$, $\mathrm{B2}$ and $\mathrm{B3}$) or $\mathrm{B4}$ hold, and the local search confirms a minimum, where
(Quantities with superscript $k$ are the values at the $k$th iteration of the quantities mentioned in Section 3, ${x}_{\mathit{tol}}=100\sqrt{\epsilon}$, $\epsilon $ is the machine precision and $\Vert .\Vert $ denotes the Euclidean norm. The vector ${g}_{z}$ is returned in the array w.)
If ${\mathbf{ifail}}={\mathbf{0}}$, then the vector in x on exit, ${x}_{\mathrm{sol}}$, is almost certainly an estimate of the position of the minimum, ${x}_{\mathrm{true}}$, to the accuracy specified by ${x}_{\mathit{tol}}$.
If ${\mathbf{ifail}}={\mathbf{3}}$ or ${\mathbf{5}}$, ${x}_{\mathrm{sol}}$ may still be a good estimate of ${x}_{\mathrm{true}}$, but the following checks should be made. Let $k$ denote an estimate of the condition number of the projected Hessian matrix at ${x}_{\mathrm{sol}}$. (The value of $k$ is returned in ${\mathbf{w}}\left({\mathbf{n}}+1\right)$). If
(i)the sequence $\left\{F\left({x}^{\left(k\right)}\right)\right\}$ converges to $F\left({x}_{\mathrm{sol}}\right)$ at a superlinear or a fast linear rate,
(ii)${\Vert {g}_{z}\left({x}_{\mathrm{xol}}\right)\Vert}^{2}<10.0\times \epsilon $, and
then it is almost certain that ${x}_{\mathrm{sol}}$ is a close approximation to the position of a minimum. When (ii) is true, then usually $F\left({x}_{\mathrm{sol}}\right)$ is a close approximation to $F\left({x}_{\mathrm{true}}\right)$.
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
e04jyf 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 e04jyf is roughly proportional to ${n}^{2}$. In addition, each iteration makes at least $m+1$ calls of funct1, where $m$ is the number of variables not fixed on bounds. So, unless $F\left(x\right)$ can be evaluated very quickly, the run time will be dominated by the time spent in funct1.
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 $(-1,+1)$, 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 e04jyf will take less computer time.