NAG CL Interface
e04ufc (nlp1_​rcomm)

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

1: $\mathbf{irevcm}$ – Integer * Input/Output

On initial entry: must be set to $0$.

On intermediate exit: specifies what values the calling program must assign to arguments of e04ufc before re-entering the function.

$\mathbf{irevcm}=1$

Set objf to the value of the objective function $F\left(x\right)$.

$\mathbf{irevcm}=2$

Set $\mathbf{objgrd}\left[\mathit{j}-1\right]$ to the value $\frac{\partial F}{\partial x_{\mathit{j}}}$ if available, for $\mathit{j}=1,2,\dots ,n$.

$\mathbf{irevcm}=3$

Set objf and $\mathbf{objgrd}\left[j-1\right]$ as for $\mathbf{irevcm}=1$ and $\mathbf{irevcm}=2$.

$\mathbf{irevcm}=4$

Set $\mathbf{c}\left[i-1\right]$ to the value of the constraint function $c_i\left(x\right)$, for each $i$ such that $\mathbf{needc}\left[i-1\right]>0$.

$\mathbf{irevcm}=5$

Set $\mathbf{CJAC}(i,j)$ to the value $\frac{\partial c_i}{\partial x_j}$ if available, for each $i$ such that $\mathbf{needc}\left[i-1\right]>0$ and $j=1,2,\dots ,n$, where $\mathbf{CJAC}(i,j)$ is as defined in cjac.

$\mathbf{irevcm}=6$

Set $\mathbf{CJAC}(i,j)$ as for $\mathbf{irevcm}=4$ and $\mathbf{irevcm}=5$.

On intermediate re-entry: must remain unchanged, unless you wish to terminate the solution to the current problem. In this case irevcm may be set to a negative value and then e04ufc will take a final exit with $\mathbf{fail}\mathbf{.}\mathbf{code}=\mathbf{NE\_USER\_STOP}$.

On final exit: $\mathbf{irevcm}=0$.

Constraint: $\mathbf{irevcm}\le 6$.

Note: any values you return to e04ufc as part of the reverse communication procedure should not include floating-point NaN (Not a Number) or infinity values, since these are not handled by e04ufc. If your code inadvertently does return any NaNs or infinities, e04ufc is likely to produce unexpected results.

2: $\mathbf{n}$ – Integer Input

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

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

3: $\mathbf{nclin}$ – Integer Input

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

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

4: $\mathbf{ncnln}$ – Integer Input

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

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

5: $\mathbf{tda}$ – Integer Input

On entry: the stride separating matrix column elements in the array a.

Constraints:

• if $\mathbf{nclin}>0$, $\mathbf{tda}\ge \mathbf{n}$;
• otherwise $\mathbf{tda}\ge 1$.

6: $\mathbf{tdcj}$ – Integer Input

On entry: the stride separating matrix column elements in the array cjac.

Constraints:

• if $\mathbf{ncnln}>0$, $\mathbf{tdcj}\ge \mathbf{n}$;
• otherwise $\mathbf{tdcj}\ge 1$.

7: $\mathbf{tdr}$ – Integer Input

On entry: the stride separating matrix column elements in the array r.

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

8: $\mathbf{a}\left[\mathit{dim}\right]$ – const double Input

Note: the dimension, dim, of the array a must be at least $\max (1,\mathbf{nclin}\times \mathbf{tda})$.

where $\mathbf{A}(i,j)$ appears in this document, it refers to the array element $\mathbf{a}\left[(i-1)\times \mathbf{tda}+j-1\right]$.

On initial entry: the $i$th row of the matrix $A_L$ of general linear constraints in (1) must be stored in $\mathbf{A}(\mathit{i},\mathit{j})$, for $\mathit{i}=1,2,\dots ,\mathbf{nclin}$ and $\mathit{j}=1,2,\dots ,\mathbf{n}$. 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.

9: $\mathbf{bl}\left[\mathbf{n}+\mathbf{nclin}+\mathbf{ncnln}\right]$ – const double Input

10: $\mathbf{bu}\left[\mathbf{n}+\mathbf{nclin}+\mathbf{ncnln}\right]$ – const double Input

On initial 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 $\mathbf{bl}\left[j-1\right]\le -\mathit{bigbnd}$, and to specify a nonexistent upper bound (i.e., $u_j=+\infty$), set $\mathbf{bu}\left[j-1\right]\ge \mathit{bigbnd}$; the default value of $\mathit{bigbnd}$ is ${10}^{20}$, but this may be changed by the optional parameter $\mathbf{Infinite\; Bound\; Size}$. To specify the $j$th constraint as an equality, set $\mathbf{bl}\left[j-1\right]=\mathbf{bu}\left[j-1\right]=\beta$, say, where $\left|\beta \right|<\mathit{bigbnd}$.

Constraints:

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

11: $\mathbf{iter}$ – Integer * Input/Output

On intermediate re-entry: must remain unchanged from a previous call to e04ufc.

On final exit: the number of major iterations performed.

12: $\mathbf{istate}\left[\mathbf{n}+\mathbf{nclin}+\mathbf{ncnln}\right]$ – Integer Input/Output

On initial entry: need not be set if the (default) optional parameter $\mathbf{Cold\; Start}$ is used.

If the optional parameter $\mathbf{Warm\; Start}$ has been chosen, the elements of istate corresponding to the bounds and linear constraints define the initial working set for the procedure that finds a feasible point for the linear constraints and bounds. The active set at the conclusion of this procedure and the elements of istate corresponding to nonlinear constraints then define the initial working set for the first QP subproblem. More precisely, the first $n$ elements of istate refer to the upper and lower bounds on the variables, the next $n_L$ elements refer to the upper and lower bounds on $A_{L}x$, and the next $n_N$ elements refer to the upper and lower bounds on $c\left(x\right)$. Possible values for $\mathbf{istate}\left[j-1\right]$ are as follows:

$\mathbf{istate}\left[\mathit{j}-1\right]$	Meaning
0	The corresponding constraint is not in the initial QP working set.
1	This inequality constraint should be in the working set at its lower bound.
2	This inequality constraint should be in the working set at its upper bound.
3	This equality constraint should be in the initial working set. This value must not be specified unless $\mathbf{bl}\left[j-1\right]=\mathbf{bu}\left[j-1\right]$.

The values $\mathrm{-2}$, $\mathrm{-1}$ and $4$ are also acceptable but will be modified by the function. If e04ufc has been called previously with the same values of n, nclin and ncnln, istate already contains satisfactory information. (See also the description of the optional parameter $\mathbf{Warm\; Start}$.) The function also adjusts (if necessary) the values supplied in x to be consistent with istate.

Constraint: $\mathrm{-2}\le \mathbf{istate}\left[\mathit{j}-1\right]\le 4$, for $\mathit{j}=1,2,\dots ,\mathbf{n}+\mathbf{nclin}+\mathbf{ncnln}$.

On final exit: the status of the constraints in the QP working set at the point returned in x. The significance of each possible value of $\mathbf{istate}\left[j-1\right]$ is as follows:

$\mathbf{istate}\left[\mathit{j}-1\right]$	Meaning
$\mathrm{-2}$	This constraint violates its lower bound by more than the appropriate feasibility tolerance (see the optional parameters $\mathbf{Linear\; Feasibility\; Tolerance}$ and $\mathbf{Nonlinear\; Feasibility\; Tolerance}$). This value can occur only when no feasible point can be found for a QP subproblem.
$\mathrm{-1}$	This constraint violates its upper bound by more than the appropriate feasibility tolerance (see the optional parameters $\mathbf{Linear\; Feasibility\; Tolerance}$ and $\mathbf{Nonlinear\; Feasibility\; Tolerance}$). This value can occur only when no feasible point can be found for a QP subproblem.
$\phantom{-}0$	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 $\mathbf{bl}\left[j-1\right]=\mathbf{bu}\left[j-1\right]$.

13: $\mathbf{c}\left[\mathit{dim}\right]$ – double Input/Output

Note: the dimension, dim, of the array c must be at least $\max (1,\mathbf{ncnln})$.

On initial entry: need not be set.

On intermediate re-entry: if $\mathbf{irevcm}=4$ or $6$ and $\mathbf{needc}\left[i-1\right]>0$, $\mathbf{c}\left[i-1\right]$ must contain the value of the $i$th constraint at $x$. The remaining elements of c, corresponding to the non-positive elements of needc, are ignored.

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

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

14: $\mathbf{cjac}\left[\mathit{dim}\right]$ – double Input/Output

Note: the dimension, dim, of the array cjac must be at least $\max (1,\mathbf{ncnln}\times \mathbf{tdcj})$.

where $\mathbf{CJAC}(i,j)$ appears in this document, it refers to the array element $\mathbf{cjac}\left[(i-1)\times \mathbf{tdcj}+j-1\right]$.

On initial entry: in general, cjac need not be initialized before the call to e04ufc. However, if the optional parameter $\mathbf{Derivative\; Level}=2$ or $3$, you may optionally set the constant elements of cjac. Such constant elements need not be re-assigned on subsequent intermediate exits.

If all elements of the constraint Jacobian are known (i.e., $\mathbf{Derivative\; Level}=2$ or $3$), any constant elements may be assigned to cjac one time only at the start of the optimization. An element of cjac that is not subsequently assigned during an intermediate exit will retain its initial value throughout. Constant elements may be loaded into cjac either before the call to e04ufc or during the first intermediate exit. The ability to preload constants is useful when many Jacobian elements are identically zero, in which case cjac may be initialized to zero and nonzero elements may be reset during intermediate exits.

On intermediate re-entry: if $\mathbf{irevcm}=5$ or $6$ and $\mathbf{needc}\left[i-1\right]>0$, the $i$th row of CJAC must contain the available elements of the vector $\nabla c_i$ given by

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

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 $x$. The remaining rows of CJAC, corresponding to non-positive elements of needc, are ignored. The $\mathit{i}$th row of the Jacobian should be stored in elements $\mathbf{CJAC}(\mathit{i},\mathit{j})$, for $\mathit{i}=1,2,\dots ,\mathbf{ncnln}$ and $\mathit{j}=1,2,\dots ,\mathbf{n}$.

Note that constant nonzero elements do affect the values of the constraints. Thus, if $\mathbf{CJAC}(i,j)$ is set to a constant value, it need not be reset during subsequent intermediate exits, but the value $\mathbf{CJAC}(i,j)\times \mathbf{x}\left[j-1\right]$ must nonetheless be added to $\mathbf{c}\left[i-1\right]$. For example, if $\mathbf{CJAC}(1,1)=2$ and $\mathbf{CJAC}(1,2)=\mathrm{-5}$, the term $2\times \mathbf{x}\left[0\right]-5\times \mathbf{x}\left[1\right]$ must be included in the definition of $\mathbf{c}\left[0\right]$.

It must be emphasized that, if $\mathbf{Derivative\; Level}=0$ or $1$, unassigned elements of cjac 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 $\mathbf{Difference\; Interval}$, an interval for each element of $x$ is computed automatically at the start of the optimization. The automatic procedure can usually identify constant elements of cjac, which are then computed once only by finite differences.

See also the description of the optional parameter $\mathbf{Verify}$.

On final exit: if $\mathbf{ncnln}>0$, cjac contains the Jacobian matrix of the nonlinear constraint functions at the final iterate, i.e., $\mathbf{CJAC}(\mathit{i},\mathit{j})$ contains the partial derivative of the $\mathit{i}$th constraint function with respect to the $\mathit{j}$th variable, for $\mathit{i}=1,2,\dots ,\mathbf{ncnln}$ and $\mathit{j}=1,2,\dots ,\mathbf{n}$.

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

15: $\mathbf{clamda}\left[\mathbf{n}+\mathbf{nclin}+\mathbf{ncnln}\right]$ – double Input/Output

On initial entry: need not be set if the (default) optional parameter $\mathbf{Cold\; Start}$ is used.

If the optional parameter $\mathbf{Warm\; Start}$ has been chosen, $\mathbf{clamda}\left[\mathit{j}-1\right]$ must contain a multiplier estimate for each nonlinear constraint with a sign that matches the status of the constraint specified by the istate array, for $\mathit{j}=\mathbf{n}+\mathbf{nclin}+1,\dots ,\mathbf{n}+\mathbf{nclin}+\mathbf{ncnln}$. The remaining elements need not be set. Note that if the $j$th constraint is defined as ‘inactive’ by the initial value of the istate array (i.e., $\mathbf{istate}\left[j-1\right]=0$), $\mathbf{clamda}\left[j-1\right]$ should be zero; if the $j$th constraint is an inequality active at its lower bound (i.e., $\mathbf{istate}\left[j-1\right]=1$), $\mathbf{clamda}\left[j-1\right]$ should be non-negative; if the $j$th constraint is an inequality active at its upper bound (i.e., $\mathbf{istate}\left[j-1\right]=2$), $\mathbf{clamda}\left[j-1\right]$ should be non-positive. If necessary, the function will modify clamda to match these rules.

On final exit: the values of the QP multipliers from the last QP subproblem. $\mathbf{clamda}\left[j-1\right]$ should be non-negative if $\mathbf{istate}\left[j-1\right]=1$ and non-positive if $\mathbf{istate}\left[j-1\right]=2$.

16: $\mathbf{objf}$ – double * Input/Output

On initial entry: need not be set.

On intermediate re-entry: if $\mathbf{irevcm}=1$ or $3$, objf must be set to the value of the objective function at $x$.

On final exit: the value of the objective function at the final iterate.

17: $\mathbf{objgrd}\left[\mathbf{n}\right]$ – double Input/Output

On initial entry: need not be set.

On intermediate re-entry: if $\mathbf{irevcm}=2$ or $3$, objgrd must contain the available elements of the gradient evaluated at $x$.

See also the description of the optional parameter $\mathbf{Verify}$.

On final exit: the gradient of the objective function at the final iterate (or its finite difference approximation).

18: $\mathbf{r}\left[\mathbf{n}\times \mathbf{tdr}\right]$ – double Input/Output

Note: the $(i,j)$th element of the matrix $R$ is stored in $\mathbf{r}\left[(i-1)\times \mathbf{tdr}+j-1\right]$.

On initial entry: need not be initialized if the (default) optional parameter $\mathbf{Cold\; Start}$ is used.

If the optional parameter $\mathbf{Warm\; Start}$ has been chosen, r must contain the upper triangular Cholesky factor $R$ of the initial approximation of the Hessian of the Lagrangian function, with the variables in the natural order. Elements not in the upper triangular part of r are assumed to be zero and need not be assigned.

On final exit: if $\mathbf{Hessian}=\mathrm{NO}$, r contains the upper triangular Cholesky factor $R$ of $Q^{\mathrm{T}}\tilde{H}Q$, an estimate of the transformed and reordered Hessian of the Lagrangian at $x$ (see (6) in Section 11.1).

If $\mathbf{Hessian}=\mathrm{YES}$, r contains the upper triangular Cholesky factor $R$ of $H$, the approximate (untransformed) Hessian of the Lagrangian, with the variables in the natural order.

19: $\mathbf{x}\left[\mathbf{n}\right]$ – double Input/Output

On initial entry: an initial estimate of the solution.

On intermediate exit: the point $x$ at which the objective function, constraint functions or their derivatives are to be evaluated.

On final exit: the final estimate of the solution.

20: $\mathbf{needc}\left[\max (1,\mathbf{ncnln})\right]$ – Integer Output

On intermediate exit: if $\mathbf{irevcm}\ge 4$, needc specifies the indices of the elements of c and/or cjac that must be assigned. If $\mathbf{needc}\left[i-1\right]>0$, the $i$th element of c and/or the available elements of the $i$th row of cjac must be evaluated at $x$.

21: $\mathbf{iwork}\left[\mathbf{liwork}\right]$ – Integer Communication Array

22: $\mathbf{liwork}$ – Integer Input

On initial entry: the dimension of the array iwork.

Constraint: $\mathbf{liwork}\ge 3\times \mathbf{n}+\mathbf{nclin}+2\times \mathbf{ncnln}$.

23: $\mathbf{work}\left[\mathbf{lwork}\right]$ – double Communication Array

24: $\mathbf{lwork}$ – Integer Input

On initial entry: the dimension of the array work.

Constraints:

• if $\mathbf{ncnln}=0$ and $\mathbf{nclin}=0$, $\mathbf{lwork}\ge 21\times \mathbf{n}+2$;
• if $\mathbf{ncnln}=0$ and $\mathbf{nclin}>0$, $\mathbf{lwork}\ge 2\times {\mathbf{n}}^2+21\times \mathbf{n}+11\times \mathbf{nclin}+2$;
• if $\mathbf{ncnln}>0$ and $\mathbf{nclin}\ge 0$, $\mathbf{lwork}\ge 2\times {\mathbf{n}}^2+\mathbf{n}\times \mathbf{nclin}+2\times \mathbf{n}\times \mathbf{ncnln}+21\times \mathbf{n}+11\times \mathbf{nclin}+22\times \mathbf{ncnln}+1$.

The contents of the communication arrays iwork and work must not be altered between calls to e04ufc.

25: $\mathbf{cwsav}\left[400\right]$ – char Communication Array

26: $\mathbf{lwsav}\left[120\right]$ – Nag_Boolean Communication Array

27: $\mathbf{iwsav}\left[610\right]$ – Integer Communication Array

28: $\mathbf{rwsav}\left[475\right]$ – double Communication Array

The arrays lwsav, iwsav, rwsav and cwsav MUST NOT be altered between calls to any of the functions e04wbc, e04udc, e04uec or e04ufc.

29: $\mathbf{fail}$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

If the printed output before the overflow error contains a warning about serious ill-conditioning in the working set when adding the $j$th constraint, it may be possible to avoid the difficulty by increasing the magnitude of the $\mathbf{Linear\; Feasibility\; Tolerance}$ and/or the optional parameter $\mathbf{Nonlinear\; Feasibility\; Tolerance}$ and rerunning the program. If the message recurs even after this change, the offending linearly dependent constraint (with index ‘$j$’) must be removed from the problem. If overflow occurs in one of the user-supplied functions (e.g., if the nonlinear functions involve exponentials or singularities), it may help to specify tighter bounds for some of the variables (i.e., reduce the gap between the appropriate $l_j$ and $u_j$).

NE_ALLOC_FAIL

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.

NE_BAD_PARAM

On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.

NE_BOUND

On entry, the bounds on linear constraint $⟨\mathit{\text{value}}⟩$ are inconsistent: $\mathbf{bl}\left[⟨\mathit{\text{value}}⟩\right]=⟨\mathit{\text{value}}⟩$ and $\mathbf{bu}\left[⟨\mathit{\text{value}}⟩\right]=⟨\mathit{\text{value}}⟩$.

On entry, the bounds on nonlinear constraint $⟨\mathit{\text{value}}⟩$ are inconsistent: $\mathbf{bl}\left[⟨\mathit{\text{value}}⟩\right]=⟨\mathit{\text{value}}⟩$ and $\mathbf{bu}\left[⟨\mathit{\text{value}}⟩\right]=⟨\mathit{\text{value}}⟩$.

On entry, the bounds on variable $⟨\mathit{\text{value}}⟩$ are inconsistent: $\mathbf{bl}\left[⟨\mathit{\text{value}}⟩\right]=⟨\mathit{\text{value}}⟩$ and $\mathbf{bu}\left[⟨\mathit{\text{value}}⟩\right]=⟨\mathit{\text{value}}⟩$.

NE_BOUND_EQ

On entry, the equal bounds on linear constraint $⟨\mathit{\text{value}}⟩$ are infinite, because $\mathbf{bl}\left[⟨\mathit{\text{value}}⟩\right]=\mathrm{beta}$ and $\mathbf{bu}\left[⟨\mathit{\text{value}}⟩\right]=\mathrm{beta}$, but $\left|\mathrm{beta}\right|\ge \mathrm{bigbnd}$: $\mathrm{beta}=⟨\mathit{\text{value}}⟩$ and $\mathrm{bigbnd}=⟨\mathit{\text{value}}⟩$.

On entry, the equal bounds on nonlinear constraint $⟨\mathit{\text{value}}⟩$ are infinite, because $\mathbf{bl}\left[⟨\mathit{\text{value}}⟩\right]=\mathrm{beta}$ and $\mathbf{bu}\left[⟨\mathit{\text{value}}⟩\right]=\mathrm{beta}$, but $\left|\mathrm{beta}\right|\ge \mathrm{bigbnd}$: $\mathrm{beta}=⟨\mathit{\text{value}}⟩$ and $\mathrm{bigbnd}=⟨\mathit{\text{value}}⟩$.

On entry, the equal bounds on variable $⟨\mathit{\text{value}}⟩$ are infinite, because $\mathbf{bl}\left[⟨\mathit{\text{value}}⟩\right]=\mathrm{beta}$ and $\mathbf{bu}\left[⟨\mathit{\text{value}}⟩\right]=\mathrm{beta}$, but $\left|\mathrm{beta}\right|\ge \mathrm{bigbnd}$: $\mathrm{beta}=⟨\mathit{\text{value}}⟩$ and $\mathrm{bigbnd}=⟨\mathit{\text{value}}⟩$.

NE_DERIV_ERRORS

Large errors found in the derivatives.

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 fail will occur if the verification process indicated that at least one gradient or Jacobian element had no correct figures. You should refer to 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 in such a test, 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.

NE_INT

On entry, $\mathbf{irevcm}=⟨\mathit{\text{value}}⟩$.
Constraint: $6\ge \mathbf{irevcm}\ge 0$.

On entry, $\mathbf{n}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{n}>0$.

On entry, $\mathbf{nclin}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{nclin}\ge 0$.

On entry, $\mathbf{ncnln}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{ncnln}\ge 0$.

NE_INT_2

On entry, $\mathbf{tdr}=⟨\mathit{\text{value}}⟩$ and $\mathbf{n}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{tdr}\ge \mathbf{n}$.

NE_INT_3

On entry, $\mathbf{tda}=⟨\mathit{\text{value}}⟩$, $\mathbf{nclin}=⟨\mathit{\text{value}}⟩$ and $\mathbf{n}=⟨\mathit{\text{value}}⟩$.
Constraint: if $\mathbf{nclin}>0$, $\mathbf{tda}\ge \mathbf{n}$;
otherwise $\mathbf{tda}\ge 1$.

On entry, $\mathbf{tdcj}=⟨\mathit{\text{value}}⟩$, $\mathbf{ncnln}=⟨\mathit{\text{value}}⟩$ and $\mathbf{n}=⟨\mathit{\text{value}}⟩$.
Constraint: if $\mathbf{ncnln}>0$, $\mathbf{tdcj}\ge \mathbf{n}$;
otherwise $\mathbf{tdcj}\ge 1$.

NE_INTERNAL_ERROR

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.

NE_LIN_NOT_FEASIBLE

No feasible point for the linear constraints.

e04ufc has terminated without finding 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 $\mathbf{Linear\; Feasibility\; Tolerance}$, or no feasible point could be found in the number of iterations specified by the optional parameter $\mathbf{Minor\; Iteration\; Limit}$. 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 $\mathbf{Linear\; Feasibility\; Tolerance}$ is greater than $\sigma$. For example, if all elements of $A_L$ are of order unity and are accurate to only three decimal places, $\mathbf{Linear\; Feasibility\; Tolerance}$ should be at least ${10}^{\mathrm{-3}}$.

NE_NO_IMPROVEMENT

Current point cannot be improved upon.

$x$ does not satisfy the first-order Kuhn–Tucker conditions (see Section 11.1), and no improved point for the merit function (see Section 9.1) could be found during the final linesearch.

This sometimes occurs because an overly stringent accuracy has been requested, i.e., the value of the optional parameter $\mathbf{Optimality\; Tolerance}$ ($\text{default value}={\epsilon }_r^{0.8}$, where ${\epsilon }_r$ is the value of the optional parameter $\mathbf{Function\; Precision}$) is too small. In this case you should apply the four tests described under $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_NOERROR to determine whether or not the final solution is acceptable (see Gill et al. (1981), for a discussion of the attainable accuracy).

If many iterations have occurred in which essentially no progress has been made and e04ufc has failed completely to move from the initial point, values set by the calling program for the objective or constraint functions or their derivatives during intermediate exits may be incorrect. You should refer to comments under $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_DERIV_ERRORS and check the gradients using the optional parameter $\mathbf{Verify}$. Unfortunately, there may be small errors in the objective and constraint gradients that cannot be detected by the verification process. Finite difference approximations to first derivatives are catastrophically affected by even small inaccuracies. An indication of this situation is a dramatic alteration in the iterates if the finite difference interval is altered. One might also suspect this type of error if a switch is made to central differences even when Norm Gz and Violtn (see Section 9.1) are large.

Another possibility is that the search direction has become inaccurate because of ill-conditioning in the Hessian approximation or the matrix of constraints in the working set; either form of ill-conditioning tends to be reflected in large values of Mnr (the number of iterations required to solve each QP subproblem; see Section 9.1).

If the condition estimate of the projected Hessian (Cond Hz; see Section 9.1) is extremely large, it may be worthwhile rerunning e04ufc from the final point with the optional parameter $\mathbf{Warm\; Start}$. In this situation, istate and clamda should be left unaltered and $\mathbf{r}$ should be reset to the identity matrix.

If the matrix of constraints in the working set is ill-conditioned (i.e., Cond T is extremely large; see Section 13), it may be helpful to run e04ufc with a relaxed value of the optional parameter $\mathbf{Feasibility\; Tolerance}$. (Constraint dependencies are often indicated by wide variations in size in the diagonal elements of the matrix $T$, whose diagonals will be printed if $\mathbf{Major\; Print\; Level}\ge 30$.)

NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

NE_NONLIN_NOT_FEASIBLE

No feasible point for the nonlinear constraints.

No feasible point could be found for the nonlinear constraints. The problem may have no feasible solution. This means that there has been a sequence of QP subproblems for which no feasible point could be found (indicated by I at the end of each line of intermediate printout produced by the major iterations; see Section 9.1). 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.) If the infeasible subproblems occur from the very first major iteration, it is highly likely that no feasible point exists. If infeasibilities occur when earlier subproblems have been feasible, small constraint inconsistencies may be present. You should check the validity of constraints with negative values of istate. If you are convinced that a feasible point does exist, e04ufc should be restarted at a different starting point.

NE_TOO_MANY

Too many major iterations.

The limiting number of iterations (as determined by the optional parameter $\mathbf{Major\; Iteration\; Limit}$) has been reached.

If the algorithm appears to be making satisfactory progress, optional parameter $\mathbf{Major\; Iteration\; Limit}$ may be too small. If so, either increase its value and rerun e04ufc or, alternatively, rerun e04ufc using the optional parameter $\mathbf{Warm\; Start}$. If the algorithm seems to be making little or no progress however, you should check for incorrect gradients or ill-conditioning as described under $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_NO_IMPROVEMENT.

Note that ill-conditioning in the working set is sometimes resolved automatically by the algorithm, in which case performing additional iterations may be helpful. However, ill-conditioning in the Hessian approximation tends to persist once it has begun, so that allowing additional iterations without altering $\mathbf{r}$ is usually inadvisable. If the quasi-Newton update of the Hessian approximation was reset during the latter major iterations (i.e., an R occurs at the end of each line of intermediate printout; see Section 9.1), it may be worthwhile to try a $\mathbf{Warm\; Start}$ at the final point as suggested above.

NE_USER_STOP

User requested termination by setting irevcm negative during an intermediate exit.

NE_WARM_START

On entry with a Warm Start, $\mathbf{istate}\left[⟨\mathit{\text{value}}⟩\right]=⟨\mathit{\text{value}}⟩$.

NW_NOT_CONVERGED

Optimal solution found, but requested accuracy not achieved.

The final iterate $x$ satisfies the first-order Kuhn–Tucker conditions (see Section 11.1) to the accuracy requested, but the sequence of iterates has not yet converged. e04ufc was terminated because no further improvement could be made in the merit function (see Section 9.1).

The most common situation is that you ask for a solution with accuracy that is not attainable with the given precision of the problem (as specified by the optional parameter $\mathbf{Function\; Precision}$). This condition will also occur if, by chance, an iterate is an ‘exact’ Kuhn–Tucker point, but the change in the variables was significant at the previous iteration. (This situation often happens when minimizing very simple functions, such as quadratics.)

7 Accuracy

8 Parallelism and Performance

9 Further Comments

9.1 Description of the Printed Output

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

11 Algorithmic Details

11.1 Overview

11.2 Solution of the Quadratic Programming Subproblem

11.3 The Merit Function

11.4 The Quasi-Newton Update

12 Optional Parameters

• Central Difference Interval
• Cold Start
• Crash Tolerance
• Defaults
• Derivative Level
• Difference Interval
• Feasibility Tolerance
• Function Precision
• Hessian
• Infinite Bound Size
• Infinite Step Size
• Iteration Limit
• Iters
• Itns
• Linear Feasibility Tolerance
• Line Search Tolerance
• List
• Major Iteration Limit
• Major Print Level
• Minor Iteration Limit
• Minor Print Level
• Monitoring File
• Nolist
• Nonlinear Feasibility Tolerance
• Optimality Tolerance
• Print Level
• Start Constraint Check At Variable
• Start Objective Check At Variable
• Step Limit
• Stop Constraint Check At Variable
• Stop Objective Check At Variable
• Verify
• Verify Constraint Gradients
• Verify Gradients
• Verify Level
• Verify Objective Gradients
• Warm Start

Begin * Example options file 
   Print level = 5 
End

 e04uec("Print Level = 1",lwsav,iwsav,rwsav,&fail)

12.1 Description of the Optional Parameters

• the keywords, where the minimum abbreviation of each keyword is underlined (if no characters of an optional qualifier are underlined, the qualifier may be omitted);
• a parameter value, where the letters $a$, $i$ and $r$ denote options that take character, integer and real values respectively;
• the default value, where the symbol $\epsilon$ is a generic notation for machine precision (see X02AJC), and ${\epsilon }_r$ denotes the relative precision of the objective function $\mathbf{Function\; Precision}$.