NAG Library Routine Document

e05usf  (nlp_multistart_sqp_lsq)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{m}$ – IntegerInput

On entry: $m$, the number of subfunctions associated with $F\left(x\right)$.

Constraint: $\mathbf{m}>0$.

2:     $\mathbf{n}$ – IntegerInput

On entry: $n$, the number of variables.

Constraint: $\mathbf{n}>0$.

3:     $\mathbf{nclin}$ – IntegerInput

On entry: $n_L$, the number of general linear constraints.

Constraint: $\mathbf{nclin}\ge 0$.

4:     $\mathbf{ncnln}$ – IntegerInput

On entry: $n_N$, the number of nonlinear constraints.

Constraint: $\mathbf{ncnln}\ge 0$.

5:     $\mathbf{a}\left(\mathbf{lda},*\right)$ – Real (Kind=nag_wp) arrayInput

Note: the second dimension of the array a must be at least $\mathbf{n}$ if $\mathbf{nclin}>0$, and at least $1$ otherwise.

On entry: the matrix $A_L$ of general linear constraints in (1). That is, the $\mathit{i}$th row contains the coefficients of the $\mathit{i}$th general linear constraint, for $\mathit{i}=1,2,\dots ,\mathbf{nclin}$.

If $\mathbf{nclin}=0$, the array a is not referenced.

6:     $\mathbf{lda}$ – IntegerInput

On entry: the first dimension of the array a as declared in the (sub)program from which e05usf is called.

Constraint: $\mathbf{lda}\ge \mathbf{nclin}$.

7:     $\mathbf{bl}\left(\mathbf{n}+\mathbf{nclin}+\mathbf{ncnln}\right)$ – Real (Kind=nag_wp) arrayInput

8:     $\mathbf{bu}\left(\mathbf{n}+\mathbf{nclin}+\mathbf{ncnln}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: bl must contain the lower bounds and bu the upper bounds for all the constraints in the following order. The first $n$ elements of each array must contain the bounds on the variables, the next $n_L$ elements the bounds for the general linear constraints (if any) and the next $n_N$ elements the bounds for the general nonlinear constraints (if any). To specify a nonexistent lower bound (i.e., $l_j=-\infty$), set $\mathbf{bl}\left(j\right)\le -\mathit{bigbnd}$, and to specify a nonexistent upper bound (i.e., $u_j=+\infty$), set $\mathbf{bu}\left(j\right)\ge \mathit{bigbnd}$; the default value of $\mathit{bigbnd}$ is ${10}^{20}$, but this may be changed by the optional parameter Infinite Bound Size. To specify the $j$th constraint as an equality, set $\mathbf{bl}\left(j\right)=\mathbf{bu}\left(j\right)=\beta$, say, where $\left|\beta \right|<\mathit{bigbnd}$.

Constraints:

• $\mathbf{bl}\left(\mathit{j}\right)\le \mathbf{bu}\left(\mathit{j}\right)$, for $\mathit{j}=1,2,\dots ,\mathbf{n}+\mathbf{nclin}+\mathbf{ncnln}$;
• if $\mathbf{bl}\left(j\right)=\mathbf{bu}\left(j\right)=\beta$, $\left|\beta \right|<\mathit{bigbnd}$.

9:     $\mathbf{y}\left(\mathbf{m}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: the coefficients of the constant vector $y$ of the objective function.

10:   $\mathbf{confun}$ – Subroutine, supplied by the NAG Library or the user.External Procedure

confun must calculate the vector $c\left(x\right)$ of nonlinear constraint functions and (optionally) its Jacobian ($=\frac{\partial c}{\partial x}$) for a specified $n$-element vector $x$. If there are no nonlinear constraints (i.e., $\mathbf{ncnln}=0$), confun will never be called by e05usf and confun may be the dummy routine e05udm. (e05udm is included in the NAG Library.) If there are nonlinear constraints, the first call to confun will occur before the first call to objfun.

The specification of confun is:

Fortran Interface

Subroutine confun ( mode, ncnln, n, ldcjsl, needc, x, c, cjsl, nstate, iuser, ruser) Integer, Intent (In) :: ncnln, n, ldcjsl, needc(ncnln), nstate Integer, Intent (Inout) :: mode, iuser(*) Real (Kind=nag_wp), Intent (In) :: x(n) Real (Kind=nag_wp), Intent (Inout) :: cjsl(ldcjsl,*), ruser(*) Real (Kind=nag_wp), Intent (Out) :: c(ncnln)

C Header Interface

#include nagmk26.h void  confun ( Integer *mode, const Integer *ncnln, const Integer *n, const Integer *ldcjsl, const Integer needc[], const double x[], double c[], double cjsl[], const Integer *nstate, Integer iuser[], double ruser[])

1:     $\mathbf{mode}$ – IntegerInput/Output

On entry: indicates which values must be assigned during each call of confun. Only the following values need be assigned, for each value of $i$ such that $\mathbf{needc}\left(i\right)>0$:

$\mathbf{mode}=0$

$\mathbf{c}\left(i\right)$, the $i$th nonlinear constraint.

$\mathbf{mode}=1$

All available elements in the $i$th row of cjsl.

$\mathbf{mode}=2$

$\mathbf{c}\left(i\right)$ and all available elements in the $i$th row of cjsl.

On exit: may be set to a negative value if you wish to abandon the solution to the current local minimization problem. In this case e05usf will move to the next local minimization problem.

2:     $\mathbf{ncnln}$ – IntegerInput

On entry: $n_N$, the number of nonlinear constraints.

3:     $\mathbf{n}$ – IntegerInput

On entry: $n$, the number of variables.

4:     $\mathbf{ldcjsl}$ – IntegerInput

On entry: ldcjsl is the first dimension of the array cjsl.

5:     $\mathbf{needc}\left(\mathbf{ncnln}\right)$ – Integer arrayInput

On entry: the indices of the elements of c and/or cjsl that must be evaluated by confun. If $\mathbf{needc}\left(i\right)>0$, $\mathbf{c}\left(i\right)$ and/or the available elements of the $i$th row of cjsl (see argument mode) must be evaluated at $x$.

6:     $\mathbf{x}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: $x$, the vector of variables at which the constraint functions and/or the available elements of the constraint Jacobian are to be evaluated.

7:     $\mathbf{c}\left(\mathbf{ncnln}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: if $\mathbf{needc}\left(i\right)>0$ and $\mathbf{mode}=0$ or $2$, $\mathbf{c}\left(i\right)$ must contain the value of $c_i\left(x\right)$. The remaining elements of c, corresponding to the non-positive elements of needc, need not be set.

8:     $\mathbf{cjsl}\left(\mathbf{ldcjsl},*\right)$ – Real (Kind=nag_wp) arrayInput/Output

cjsl may be regarded as a two-dimensional ‘slice’ of the three-dimensional array cjac of e05usf.

On entry: unless $\mathbf{Derivative\; Level}=2$ or $3$, the elements of cjsl are set to special values which enable e05usf to detect whether they are changed by confun.

On exit: if $\mathbf{needc}\left(i\right)>0$ and $\mathbf{mode}=1$ or $2$, the $i$th row of cjsl must contain the available elements of the vector $\nabla c_i$ given by

$$\nabla c_i={\left(\frac{\partial c_i}{\partial x_1},\frac{\partial c_i}{\partial x_2},\dots ,\frac{\partial c_i}{\partial x_n}\right)}^{\mathrm{T}}\text{,}$$

where $\frac{\partial c_i}{\partial x_j}$ is the partial derivative of the $i$th constraint with respect to the $j$th variable, evaluated at the point $x$. See also the argument nstate. The remaining rows of cjsl, corresponding to non-positive elements of needc, need not be set.

If all elements of the constraint Jacobian are known (i.e., $\mathbf{Derivative\; Level}=2$ or $3$; note the default is $\mathbf{Derivative\; Level}=3$), any constant elements may be assigned to cjsl one time only at the start of each local optimization. An element of cjsl that is not subsequently assigned in confun will retain its initial value throughout the local optimization. Constant elements may be loaded into cjsl during the first call to confun for the local optimization (signalled by the value $\mathbf{nstate}=1$). The ability to preload constants is useful when many Jacobian elements are identically zero, in which case cjsl may be initialized to zero and nonzero elements may be reset by confun.

Note that constant nonzero elements do affect the values of the constraints. Thus, if $\mathbf{cjsl}\left(i,j\right)$ is set to a constant value, it need not be reset in subsequent calls to confun, but the value $\mathbf{cjsl}\left(i,j\right)\times \mathbf{x}\left(j\right)$ must nonetheless be added to $\mathbf{c}\left(i\right)$. For example, if $\mathbf{cjsl}\left(1,1\right)=2$ and $\mathbf{cjsl}\left(1,2\right)=-5$ then the term $2\times \mathbf{x}\left(1\right)-5\times \mathbf{x}\left(2\right)$ must be included in the definition of $\mathbf{c}\left(1\right)$.

It must be emphasized that, if $\mathbf{Derivative\; Level}=0$ or $1$, unassigned elements of cjsl are not treated as constant; they are estimated by finite differences, at nontrivial expense. If you do not supply a value for the optional parameter Difference Interval, an interval for each element of $x$ is computed automatically at the start of each local optimization. The automatic procedure can usually identify constant elements of cjsl, which are then computed once only by finite differences.

9:     $\mathbf{nstate}$ – IntegerInput

On entry: if $\mathbf{nstate}=1$ then e05usf is calling confun for the first time on the current local optimization problem. This argument setting allows you to save computation time if certain data must be read or calculated only once.

10:   $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace

11:   $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace

confun is called with the arguments iuser and ruser as supplied to e05usf. You should use the arrays iuser and ruser to supply information to confun.

confun must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which e05usf is called. Arguments denoted as Input must not be changed by this procedure.

Note: confun should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by e05usf. If your code inadvertently does return any NaNs or infinities, e05usf is likely to produce unexpected results.

confun should be tested separately before being used in conjunction with e05usf. See also the description of the optional parameter Verify.

11:   $\mathbf{objfun}$ – Subroutine, supplied by the user.External Procedure

objfun must calculate either the $i$th element of the vector $f\left(x\right)={\left(f_1\left(x\right),f_2\left(x\right),\dots ,f_m\left(x\right)\right)}^{\mathrm{T}}$ or all $m$ elements of $f\left(x\right)$ and (optionally) its Jacobian ($=\frac{\partial f}{\partial x}$) for a specified $n$-element vector $x$.

The specification of objfun is:

Fortran Interface

Subroutine objfun ( mode, m, n, ldfjsl, needfi, x, f, fjsl, nstate, iuser, ruser) Integer, Intent (In) :: m, n, ldfjsl, needfi, nstate Integer, Intent (Inout) :: mode, iuser(*) Real (Kind=nag_wp), Intent (In) :: x(n) Real (Kind=nag_wp), Intent (Inout) :: fjsl(ldfjsl,*), ruser(*) Real (Kind=nag_wp), Intent (Out) :: f(m)

C Header Interface

#include nagmk26.h void  objfun ( Integer *mode, const Integer *m, const Integer *n, const Integer *ldfjsl, const Integer *needfi, const double x[], double f[], double fjsl[], const Integer *nstate, Integer iuser[], double ruser[])

1:     $\mathbf{mode}$ – IntegerInput/Output

On entry: indicates which values must be assigned during each call of objfun. Only the following values need be assigned:

$\mathbf{mode}=0$ and $\mathbf{needfi}=i$, where $i>0$

$\mathbf{f}\left(i\right)$.

$\mathbf{mode}=0$ and $\mathbf{needfi}<0$

f.

$\mathbf{mode}=1$ and $\mathbf{needfi}<0$

All available elements of fjsl.

$\mathbf{mode}=2$ and $\mathbf{needfi}<0$

f and all available elements of fjsl.

On exit: may be set to a negative value if you wish to abandon the solution to the current local minimization problem. In this case e05usf will move to the next local minimization problem.

2:     $\mathbf{m}$ – IntegerInput

On entry: $m$, the number of subfunctions.

3:     $\mathbf{n}$ – IntegerInput

On entry: $n$, the number of variables.

4:     $\mathbf{ldfjsl}$ – IntegerInput

On entry: ldfjsl is the first dimension of the array fjsl.

5:     $\mathbf{needfi}$ – IntegerInput

On entry: if $\mathbf{needfi}=i>0$, only the $i$th element of $f\left(x\right)$ needs to be evaluated at $x$; the remaining elements need not be set. This can result in significant computational savings when $m\gg n$.

6:     $\mathbf{x}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: $x$, the vector of variables at which the objective function and/or all available elements of its gradient are to be evaluated.

7:     $\mathbf{f}\left(\mathbf{m}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: if $\mathbf{mode}=0$ and $\mathbf{needfi}=i>0$, $\mathbf{f}\left(i\right)$ must contain the value of $f_i$ at $x$.

If $\mathbf{mode}=0$ or $2$ and $\mathbf{needfi}<0$, $\mathbf{f}\left(\mathit{i}\right)$ must contain the value of $f_{\mathit{i}}$ at $x$, for $\mathit{i}=1,2,\dots ,m$.

8:     $\mathbf{fjsl}\left(\mathbf{ldfjsl},*\right)$ – Real (Kind=nag_wp) arrayInput/Output

fjsl may be regarded as a two-dimensional ‘slice’ of the three-dimensional array fjac of e05usf.

On entry: is set to a special value.

On exit: if $\mathbf{mode}=1$ or $2$ and $\mathbf{needfi}<0$, the $i$th row of fjsl must contain the available elements of the vector $\nabla f_i$ given by

$$\nabla f_i={\left(\partial f_i/\partial x_1,\partial f_i/\partial x_2,\dots ,\partial f_i/\partial x_n\right)}^{\mathrm{T}}\text{,}$$

evaluated at the point $x$. See also the argument nstate.

9:     $\mathbf{nstate}$ – IntegerInput

On entry: if $\mathbf{nstate}=1$ then e05usf is calling objfun for the first time on the current local optimization problem. This argument setting allows you to save computation time if certain data must be read or calculated only once.

10:   $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace

11:   $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace

objfun is called with the arguments iuser and ruser as supplied to e05usf. You should use the arrays iuser and ruser to supply information to objfun.

objfun must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which e05usf is called. Arguments denoted as Input must not be changed by this procedure.

Note: objfun should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by e05usf. If your code inadvertently does return any NaNs or infinities, e05usf is likely to produce unexpected results.

objfun should be tested separately before being used in conjunction with e05usf. See also the description of the optional parameter Verify.

12:   $\mathbf{npts}$ – IntegerInput

On entry: the number of different starting points to be generated and used. The more points used, the more likely that the best returned solution will be a global minimum.

Constraint: $1\le \mathbf{nb}\le \mathbf{npts}$.

13:   $\mathbf{x}\left(\mathbf{ldx},*\right)$ – Real (Kind=nag_wp) arrayOutput

Note: the second dimension of the array x must be at least $\mathbf{nb}$.

On exit: $\mathbf{x}\left(\mathit{j},i\right)$ contains the final estimate of the $i$th solution, for $\mathit{j}=1,2,\dots ,\mathbf{n}$.

14:   $\mathbf{ldx}$ – IntegerInput

On entry: the first dimension of the array x as declared in the (sub)program from which e05usf is called.

Constraint: $\mathbf{ldx}\ge \mathbf{n}$.

15:   $\mathbf{start}$ – Subroutine, supplied by the NAG Library or the user.External Procedure

start must calculate the npts starting points to be used by the local optimizer. If you do not wish to write a routine specific to your problem then e05ucz may be used as the actual argument. e05ucz is supplied in the NAG Library and uses the NAG quasi-random number generators to distribute starting points uniformly across the domain. It is affected by the value of repeat1.

The specification of start is:

Fortran Interface

Subroutine start ( npts, quas, n, repeat1, bl, bu, iuser, ruser, mode) Integer, Intent (In) :: npts, n Integer, Intent (Inout) :: iuser(*), mode Real (Kind=nag_wp), Intent (In) :: bl(n), bu(n) Real (Kind=nag_wp), Intent (Inout) :: quas(n,npts), ruser(*) Logical, Intent (In) :: repeat1

C Header Interface

#include nagmk26.h void  start ( const Integer *npts, double quas[], const Integer *n, const logical *repeat1, const double bl[], const double bu[], Integer iuser[], double ruser[], Integer *mode)

1:     $\mathbf{npts}$ – IntegerInput

On entry: indicates the number of starting points.

2:     $\mathbf{quas}\left(\mathbf{n},\mathbf{npts}\right)$ – Real (Kind=nag_wp) arrayInput/Output

On entry: all elements of quas will have been set to zero, so only nonzero values need be set subsequently.

On exit: must contain the starting points for the npts local minimizations, i.e., $\mathbf{quas}\left(j,i\right)$ must contain the $j$th component of the $i$th starting point.

3:     $\mathbf{n}$ – IntegerInput

On entry: the number of variables.

4:     $\mathbf{repeat1}$ – LogicalInput

On entry: specifies whether a repeatable or non-repeatable sequence of points are to be generated.

5:     $\mathbf{bl}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: the lower bounds on the variables. These may be used to ensure that the starting points generated in some sense ‘cover’ the region, but there is no requirement that a starting point be feasible.

6:     $\mathbf{bu}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: the upper bounds on the variables. (See bl.)

7:     $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace

8:     $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace

start is called with the arguments iuser and ruser as supplied to e05usf. You should use the arrays iuser and ruser to supply information to start.

9:     $\mathbf{mode}$ – IntegerInput/Output

On entry: mode will contain $0$.

On exit: if you set mode to a negative value then e05usf will terminate immediately with $\mathbf{ifail}=\mathbf{9}$.

start must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which e05usf is called. Arguments denoted as Input must not be changed by this procedure.

Note: start should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by e05usf. If your code inadvertently does return any NaNs or infinities, e05usf is likely to produce unexpected results.

16:   $\mathbf{repeat1}$ – LogicalInput

On entry: is passed as an argument to start and may be used to initialize a random number generator to a repeatable, or non-repeatable, sequence. See Section 9 for more detail.

17:   $\mathbf{nb}$ – IntegerInput

On entry: the number of solutions to be returned. The routine saves up to nb local minima ordered by increasing value of the final objective function. If the defining criterion for ‘best solution’ is only that the value of the objective function is as small as possible then nb should be set to $1$. However, if you want to look at other solutions that may have desirable properties then setting $\mathbf{nb}>1$ will produce nb local minima, ordered by increasing value of their objective functions at the minima.

Constraint: $1\le \mathbf{nb}\le \mathbf{npts}$.

18:   $\mathbf{objf}\left(\mathbf{nb}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: $\mathbf{objf}\left(i\right)$ contains the value of the objective function at the final iterate for the $i$th solution.

19:   $\mathbf{f}\left(\mathbf{m},*\right)$ – Real (Kind=nag_wp) arrayOutput

Note: the second dimension of the array f must be at least $\mathbf{nb}$.

On exit: $\mathbf{f}\left(\mathit{j},i\right)$ contains the value of the $\mathit{j}$th function $f_j$ at the final iterate, for $\mathit{j}=1,2,\dots ,\mathbf{m}$, for the $\mathit{i}$th solution, for $\mathit{i}=1,2,\dots ,\mathbf{nb}$.

20:   $\mathbf{fjac}\left(\mathbf{ldfjac},\mathbf{sdfjac},*\right)$ – Real (Kind=nag_wp) arrayOutput

Note: the last dimension of the array fjac must be at least $\mathbf{nb}$.

On exit: for the $i$th returned solution, the Jacobian matrix of the functions $f_1,f_2,\dots ,f_m$ at the final iterate, i.e., $\mathbf{fjac}\left(\mathit{k},\mathit{j},\mathit{i}\right)$ contains the partial derivative of the $\mathit{k}$th function with respect to the $\mathit{j}$th variable, for $\mathit{k}=1,2,\dots ,\mathbf{m}$, $\mathit{j}=1,2,\dots ,\mathbf{n}$ and $\mathit{i}=1,2,\dots ,\mathbf{nb}$. (See also the discussion of argument fjsl under objfun.)

21:   $\mathbf{ldfjac}$ – IntegerInput

On entry: the first dimension of the array fjac as declared in the (sub)program from which e05usf is called.

Constraint: $\mathbf{ldfjac}\ge \mathbf{m}$.

22:   $\mathbf{sdfjac}$ – IntegerInput

On entry: the second dimension of the array fjac as declared in the (sub)program from which e05usf is called.

Constraint: $\mathbf{sdfjac}\ge \mathbf{n}$.

23:   $\mathbf{iter}\left(\mathbf{nb}\right)$ – Integer arrayOutput

On exit: $\mathbf{iter}\left(i\right)$ contains the number of major iterations performed to obtain the $i$th solution. If less than nb solutions are returned then $\mathbf{iter}\left(\mathbf{nb}\right)$ contains the number of starting points that have resulted in a converged solution. If this is close to npts then this might be indicative that fewer than nb local minima exist.

24:   $\mathbf{c}\left(\mathbf{ldc},*\right)$ – Real (Kind=nag_wp) arrayOutput

Note: the second dimension of the array c must be at least $\mathbf{nb}$.

On exit: if $\mathbf{ncnln}>0$, $\mathbf{c}\left(\mathit{j},\mathit{i}\right)$ contains the value of the $\mathit{j}$th nonlinear constraint function $c_{\mathit{j}}$ at the final iterate, for the $\mathit{i}$th solution, for $\mathit{j}=1,2,\dots ,\mathbf{ncnln}$.

If $\mathbf{ncnln}=0$, the array c is not referenced.

25:   $\mathbf{ldc}$ – IntegerInput

On entry: the first dimension of the array c as declared in the (sub)program from which e05usf is called.

Constraint: $\mathbf{ldc}\ge \mathbf{ncnln}$.

26:   $\mathbf{cjac}\left(\mathbf{ldcjac},\mathbf{sdcjac},*\right)$ – Real (Kind=nag_wp) arrayOutput

Note: the last dimension of the array cjac must be at least $\mathbf{nb}$.

On exit: if $\mathbf{ncnln}>0$, cjac contains the Jacobian matrices of the nonlinear constraint functions at the final iterate for each of the returned solutions, i.e., $\mathbf{cjac}\left(\mathit{k},\mathit{j},i\right)$ contains the partial derivative of the $\mathit{k}$th constraint function with respect to the $\mathit{j}$th variable, for $\mathit{k}=1,2,\dots ,\mathbf{ncnln}$ and $\mathit{j}=1,2,\dots ,\mathbf{n}$, for the $i$th solution. (See the discussion of argument cjsl under confun.)

If $\mathbf{ncnln}=0$, the array cjac is not referenced.

27:   $\mathbf{ldcjac}$ – IntegerInput

On entry: the first dimension of the array cjac as declared in the (sub)program from which e05usf is called.

Constraint: $\mathbf{ldcjac}\ge \mathbf{ncnln}$.

28:   $\mathbf{sdcjac}$ – IntegerInput

On entry: the second dimension of the array cjac as declared in the (sub)program from which e05usf is called.

Constraint: if $\mathbf{ncnln}>0$, $\mathbf{sdcjac}\ge \mathbf{n}$.

29:   $\mathbf{clamda}\left(\mathbf{ldclda},*\right)$ – Real (Kind=nag_wp) arrayOutput

Note: the second dimension of the array clamda must be at least $\mathbf{nb}$.

On exit: the values of the QP multipliers from the last QP subproblem solved for the $i$th solution. $\mathbf{clamda}\left(j,i\right)$ should be non-negative if $\mathbf{istate}\left(j,i\right)=1$ and non-positive if $\mathbf{istate}\left(j,i\right)=2$.

30:   $\mathbf{ldclda}$ – IntegerInput

On entry: the first dimension of the array clamda as declared in the (sub)program from which e05usf is called.

Constraint: $\mathbf{ldclda}\ge \mathbf{n}+\mathbf{nclin}+\mathbf{ncnln}$.

31:   $\mathbf{istate}\left(\mathbf{listat},*\right)$ – Integer arrayOutput

Note: the second dimension of the array istate must be at least $\mathbf{nb}$.

On exit: $\mathbf{istate}\left(j,i\right)$ contains the status of the constraints in the QP working set for the $i$th solution. The significance of each possible value of $\mathbf{istate}\left(j,i\right)$ is as follows:

$\mathbf{istate}\left(j,i\right)$	Meaning
$\phantom{-}0$	The constraint is satisfied to within the feasibility tolerance, but is not in the QP working set.
$\phantom{-}1$	This inequality constraint is included in the QP working set at its lower bound.
$\phantom{-}2$	This inequality constraint is included in the QP working set at its upper bound.
$\phantom{-}3$	This constraint is included in the QP working set as an equality. This value of istate can occur only when $\mathbf{bl}\left(j\right)=\mathbf{bu}\left(j\right)$.

32:   $\mathbf{listat}$ – IntegerInput

On entry: the first dimension of the array istate as declared in the (sub)program from which e05usf is called.

Constraint: $\mathbf{listat}\ge \mathbf{n}+\mathbf{nclin}+\mathbf{ncnln}$.

33:   $\mathbf{iopts}\left(740\right)$ – Integer arrayCommunication Array

34:   $\mathbf{opts}\left(485\right)$ – Real (Kind=nag_wp) arrayCommunication Array

The arrays iopts and opts must not be altered between calls to any of the routines e05usf and e05zkf.

35:   $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace

36:   $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace

iuser and ruser are not used by e05usf, but are passed directly to confun, objfun and start and may be used to pass information to these routines.

With SMP-enabled versions of e05usf the arrays iuser and ruser provided are classified as OpenMP shared memory. Use of iuser and ruser has to take account of this in order to preserve thread safety whenever information is written back to either of these arrays.

37:   $\mathbf{info}\left(\mathbf{nb}\right)$ – Integer arrayOutput

On exit: if $\mathbf{ifail}=\mathbf{0}$, $\mathbf{info}\left(i\right)$ does not contain an error value returned by e04usf/e04usa .

If $\mathbf{ifail}=\mathbf{8}$ on exit, then not all nb solutions have been found, and $\mathbf{info}\left(\mathbf{nb}\right)$ contains the number of solutions actually found.

38:   $\mathbf{ifail}$ – IntegerInput/Output

On entry: ifail must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{ or }1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, because for this routine the values of the output arguments may be useful even if $\mathbf{ifail}\ne \mathbf{0}$ on exit, the recommended value is $-1$. When the value $-\mathbf{1}\text{ or }1$ 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).

6

Error Indicators and Warnings

$\mathbf{ifail}=1$

An input value is incorrect. One or more of the following requirements are violated:

• On entry, $\mathbf{bl}\left(i\right)>\mathbf{bu}\left(i\right)$: $i=〈\mathit{\text{value}}〉$.
  Constraint: $\mathbf{bl}\left(i\right)\le \mathbf{bu}\left(i\right)$, for all $i$.
• On entry, $\mathbf{lda}=〈\mathit{\text{value}}〉$ and $\mathbf{nclin}=〈\mathit{\text{value}}〉$.
  Constraint: $\mathbf{lda}\ge \mathbf{nclin}$.
• On entry, $\mathbf{ldc}=〈\mathit{\text{value}}〉$ and $\mathbf{ncnln}=〈\mathit{\text{value}}〉$.
  Constraint: $\mathbf{ldc}\ge \mathbf{ncnln}$.
• On entry, $\mathbf{ldcjac}=〈\mathit{\text{value}}〉$ and $\mathbf{ncnln}=〈\mathit{\text{value}}〉$.
  Constraint: $\mathbf{ldcjac}\ge \mathbf{ncnln}$.
• On entry, $\mathbf{ldclda}=〈\mathit{\text{value}}〉$, $\mathbf{n}=〈\mathit{\text{value}}〉$, $\mathbf{nclin}=〈\mathit{\text{value}}〉$ and $\mathbf{ncnln}=〈\mathit{\text{value}}〉$.
  Constraint: $\mathbf{ldclda}\ge \mathbf{n}+\mathbf{nclin}+\mathbf{ncnln}$.
• On entry, $\mathbf{ldfjac}=〈\mathit{\text{value}}〉$ and $\mathbf{m}=〈\mathit{\text{value}}〉$.
  Constraint: $\mathbf{ldfjac}\ge \mathbf{m}$.
• On entry, $\mathbf{ldx}=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
  Constraint: $\mathbf{ldx}\ge \mathbf{n}$.
• On entry, $\mathbf{listat}=〈\mathit{\text{value}}〉$, $\mathbf{n}=〈\mathit{\text{value}}〉$, $\mathbf{nclin}=〈\mathit{\text{value}}〉$ and $\mathbf{ncnln}=〈\mathit{\text{value}}〉$.
  Constraint: $\mathbf{listat}\ge \mathbf{n}+\mathbf{nclin}+\mathbf{ncnln}$.
• On entry, $\mathbf{m}=〈\mathit{\text{value}}〉$.
  Constraint: $\mathbf{m}>0$.
• On entry, $\mathbf{n}=〈\mathit{\text{value}}〉$.
  Constraint: $\mathbf{n}>0$.
• On entry, $\mathbf{nb}=〈\mathit{\text{value}}〉$ and $\mathbf{npts}=〈\mathit{\text{value}}〉$.
  Constraint: $1\le \mathbf{nb}\le \mathbf{npts}$.
• On entry, $\mathbf{nclin}=〈\mathit{\text{value}}〉$.
  Constraint: $\mathbf{nclin}\ge 0$.
• On entry, $\mathbf{ncnln}=〈\mathit{\text{value}}〉$.
  Constraint: $\mathbf{ncnln}\ge 0$.
• On entry, $\mathbf{ncnln}>0$, $\mathbf{sdcjac}=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
  Constraint: if $\mathbf{ncnln}>0$, $\mathbf{sdcjac}\ge \mathbf{n}$.
• On entry, $\mathbf{sdfjac}=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
  Constraint: $\mathbf{sdfjac}\ge \mathbf{n}$.

$\mathbf{ifail}=2$

e05usf has terminated without finding any solutions. The majority of calls to the local optimizer have failed to find a feasible point for the linear constraints and bounds, which means that either no feasible point exists for the given value of the optional parameter Linear Feasibility Tolerance (default value $\sqrt{\mathit{macheps}}$, where $\mathit{macheps}$ is the machine precision), or no feasible point could be found in the number of iterations specified by the optional parameter Minor Iteration Limit. You should check that there are no constraint redundancies. If the data for the constraints are accurate only to an absolute precision $\sigma$, you should ensure that the value of the optional parameter Linear Feasibility Tolerance is greater than $\sigma$. For example, if all elements of $A_L$ are of order unity and are accurate to only three decimal places, Linear Feasibility Tolerance should be at least ${10}^{-3}$.

No solution obtained. Linear constraints may be infeasible.

$\mathbf{ifail}=3$

e05usf has failed to find any solutions. The majority of local optimizations could not find a feasible point for the nonlinear constraints. The problem may have no feasible solution. This behaviour will occur if there is no feasible point for the nonlinear constraints. (However, there is no general test that can determine whether a feasible point exists for a set of nonlinear constraints.)

No solution obtained. Nonlinear constraints may be infeasible.

$\mathbf{ifail}=4$

e05usf has failed to find any solutions. The majority of local optimizations have failed because the limiting number of iterations have been reached.

No solution obtained. Many potential solutions reach iteration limit.

$\mathbf{ifail}=7$

The user-supplied derivatives of the objective function and/or nonlinear constraints appear to be incorrect.

Large errors were found in the derivatives of the objective function and/or nonlinear constraints. This value of ifail will occur if the verification process indicated that at least one gradient or Jacobian element had no correct figures. You should refer to or enable the printed output to determine which elements are suspected to be in error.

As a first-step, you should check that the code for the objective and constraint values is correct – for example, by computing the function at a point where the correct value is known. However, care should be taken that the chosen point fully tests the evaluation of the function. It is remarkable how often the values $x=0$ or $x=1$ are used to test function evaluation procedures, and how often the special properties of these numbers make the test meaningless.

Gradient checking will be ineffective if the objective function uses information computed by the constraints, since they are not necessarily computed before each function evaluation.

Errors in programming the function may be quite subtle in that the function value is ‘almost’ correct. For example, the function may not be accurate to full precision because of the inaccurate calculation of a subsidiary quantity, or the limited accuracy of data upon which the function depends. A common error on machines where numerical calculations are usually performed in double precision is to include even one single precision constant in the calculation of the function; since some compilers do not convert such constants to double precision, half the correct figures may be lost by such a seemingly trivial error.

$\mathbf{ifail}=8$

Only $〈\mathit{\text{value}}〉$ solutions obtained.

Not all nb solutions have been found. $\mathbf{info}\left(\mathbf{nb}\right)$ contains the number actually found.

$\mathbf{ifail}=9$

User terminated computation from start procedure: $\mathbf{mode}=〈\mathit{\text{value}}〉$.

If e05ucz has been used as an actual argument for start then the message displayed, when $\mathbf{ifail}=\mathbf{0}$ or $-\mathbf{1}$ on entry to e05usf, will have the following meaning:

$998$	failure to allocate space, a smaller value of NPTS should be tried.
$997$	an internal error has occurred. Please contact NAG for assistance.

$\mathbf{ifail}=10$

Failed to initialize optional parameter arrays.

$\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-999$

Dynamic memory allocation failed.

See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

9.1

Description of the Printed Output

10

Example

10.1

Program Text

Program Text (e05usfe.f90)

10.2

Program Data

Program Data (e05usfe.d)

10.3

Program Results

Program Results (e05usfe.r)

11

Algorithmic Details

12

Optional Parameters

13

Description of Monitoring Information