e04rkc is a part of the NAG optimization modelling suite and defines or overwrites nonlinear constraints in the model, their number as well as the sparsity structure of their first derivatives.

2 Specification

#include <nag.h>

void	e04rkc (void handle, Integer ncnln, const double bl[], const double bu[], Integer nnzgd, const Integer irowgd[], const Integer icolgd[], NagError fail)

The function may be called by the names: e04rkc or nag_opt_handle_set_nlnconstr.

3 Description

After the handle has been initialized (e.g., e04rac has been called), e04rkc may be used to define the nonlinear constraints

l_{g} \leq g (x) \leq u_{g}

of the problem. If the nonlinear constraints have already been defined, they will be overwritten and their Hessians (or the Hessian of the Lagrangian) will be removed. If e04rkc is called with

m_{g} = 0

, existing nonlinear constraints are removed and no new ones are added. This will typically be used for nonlinear programming problems (NLP) of the kind:

\begin{array}{l} \underset{x \in ℝ^{n}}{minimize} & f (x) & (a) \\ subject to & l_{g} \leq g (x) \leq u_{g}, & (b) \\ l_{B} \leq B x \leq u_{B}, & (c) \\ l_{x} \leq x \leq u_{x}, & (d) \end{array}

(1)

where

n

is the number of the decision variables

x

m_{g}

is the number of the nonlinear constraints (in (1)(b)) and

g (x)

l_{g}

and

u_{g}

are

m_{g}

-dimensional vectors.

Note that upper and lower bounds are specified for all the constraints. This form allows full generality in specifying various types of constraint. In particular, the

j

th constraint may be defined as an equality by setting

l_{j} = u_{j}

. If certain bounds are not present, the associated elements

l_{j}

u_{j}

may be set to special values that are treated as

- \infty

+ \infty

. See the description of the optional parameter

Infinite Bound Size

which is common among all solvers in the suite. Its value is denoted as

bigbnd

further in this text. Note that the bounds are interpreted based on its value at the time of calling this function and any later alterations to

Infinite Bound Size

will not affect these constraints.

Since each nonlinear constraint is most likely to involve a small subset of the decision variables, the partial derivatives of the constraint functions with respect to those variables are best expressed as a sparse Jacobian matrix of

m_{g}

rows and

n

columns. The row and column positions of all the nonzero derivatives must be registered with the handle through e04rkc. The values of the nonlinear constraint functions and their nonzero gradients at particular points in the decision variable space will be communicated to the NLP solver by user-supplied functions (e.g., confun and congrd for e04stc).

An individual set of bounds can be also modified by e04tdc. Note that it is also possible to temporarily disable and enable individual constraints in the model by calling e04tcc and e04tbc, respectively.

See Section 4.1 in the E04 Chapter Introduction for more details about the NAG optimization modelling suite.

4 References

None.

5 Arguments

1: $handle$ – void * Input

On entry: the handle to the problem. It needs to be initialized (e.g., by e04rac) and must not be changed between calls to the NAG optimization modelling suite.

2: $ncnln$ – Integer Input

On entry:

m_{g}

, the number of nonlinear constraints (number of rows of the Jacobian matrix).

ncnln = 0

, no nonlinear constraints will be defined and any existing nonlinear constraint will be removed from the model. In this case, bl, bu, nnzgd, irowgd and icolgd will not be referenced and may be NULL.

Constraint:

ncnln \geq 0

3: $bl [ncnln]$ – const double Input

4: $bu [ncnln]$ – const double Input

On entry: bl and bu define lower and upper bounds of the nonlinear constraints,

l_{g}

and

u_{g}

, respectively. To define the

j

th constraint as equality, set

bl [j - 1] = bu [j - 1] = β

, where

| β | < bigbnd

. To specify a nonexistent lower bound (i.e.,

l_{j} = - \infty

), set

bl [j - 1] \leq - bigbnd

; to specify a nonexistent upper bound, set

bu [j - 1] \geq bigbnd

Constraints:

$bl [j - 1] \leq bu [j - 1]$ , for $j = 1, 2, \dots, ncnln$ ;
$bl [j - 1] < bigbnd$ , for $j = 1, 2, \dots, ncnln$ ;
$bu [j - 1] > - bigbnd$ , for $j = 1, 2, \dots, ncnln$ .

5: $nnzgd$ – Integer Input

On entry: nnzgd gives the number of nonzeros in the Jacobian matrix.

Constraint: if

ncnln > 0

nnzgd > 0

6: $irowgd [nnzgd]$ – const Integer Input

7: $icolgd [nnzgd]$ – const Integer Input

On entry: arrays irowgd and icolgd store the sparsity structure (pattern) of the Jacobian matrix as nnzgd nonzeros in coordinate storage (CS) format (see Section 2.1.1 in the F11 Chapter Introduction). The matrix has dimensions

ncnln \times n

. irowgd specifies one-based row indices and icolgd specifies one-based column indices. No particular order of elements is expected, but elements should not repeat and the same order should be used when the Jacobian is evaluated for the solver, e.g., the value of

\frac{\partial g_{i}}{\partial x_{j}}

where

i = irowgd [l - 1]

and

j = icolgd [l - 1]

should be stored in

gdx [l - 1]

in congrd in e04stc, for

l = 1, 2, \dots, nnzgd

Constraints:

$1 \leq irowgd [l - 1] \leq ncnln$ , for $l = 1, 2, \dots, nnzgd$ ;
$1 \leq icolgd [l - 1] \leq n$ , for $l = 1, 2, \dots, nnzgd$ .

8: $fail$ – NagError * Input/Output

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

6 Error Indicators and Warnings

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 $⟨ value ⟩$ had an illegal value.
NE_BOUND: On entry, $j = ⟨ value ⟩$ , $bl [j - 1] = ⟨ value ⟩$ , $bigbnd = ⟨ value ⟩$ .
Constraint: $bl [j - 1] < bigbnd$ .

On entry, $j = ⟨ value ⟩$ , $bl [j - 1] = ⟨ value ⟩$ and $bu [j - 1] = ⟨ value ⟩$ .
Constraint: $bl [j - 1] \leq bu [j - 1]$ .

On entry, $j = ⟨ value ⟩$ , $bu [j - 1] = ⟨ value ⟩$ , $bigbnd = ⟨ value ⟩$ .
Constraint: $bu [j - 1] > - bigbnd$ .
NE_HANDLE: The supplied handle does not define a valid handle to the data structure for the NAG optimization modelling suite. It has not been properly initialized or it has been corrupted.
NE_INT: On entry, $ncnln = ⟨ value ⟩$ .
Constraint: $ncnln \geq 0$ .

On entry, $nnzgd = ⟨ value ⟩$ .
Constraint: $nnzgd > 0$ .
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_INVALID_CS: On entry, $i = ⟨ value ⟩$ , $icolgd [i - 1] = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $1 \leq icolgd [i - 1] \leq n$ .

On entry, $i = ⟨ value ⟩$ , $irowgd [i - 1] = ⟨ value ⟩$ and $ncnln = ⟨ value ⟩$ .
Constraint: $1 \leq irowgd [i - 1] \leq ncnln$ .

On entry, more than one element of structural Jacobian matrix has row index $⟨ value ⟩$ and column index $⟨ value ⟩$ .
Constraint: each element of structural Jacobian matrix must have a unique row and column index.
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_PHASE: The problem cannot be modified right now, the solver is running.

7 Accuracy

Not applicable.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

e04rkc is not threaded in any implementation.

9 Further Comments

9.1 Internal Changes

Internal changes have been made to this function as follows:

At Mark 27.1: Previously, it was not possible to modify the nonlinear constraints once they were set or to edit the model once a solver had been called. These limitations have been removed and the associated error codes were removed.

For details of all known issues which have been reported for the NAG Library please refer to the Known Issues.