handle_set_nlnconstr is a part of the NAG optimization modelling suite and defines the number of nonlinear constraints of the problem as well as the sparsity structure of their first derivatives.

2 Specification

#include "e04/nagcpp_e04rk.hpp"
#include "e04/nagcpp_class_CommE04RA.hpp"

template <typename COMM, typename BL, typename BU, typename IROWGD, typename ICOLGD>

void function handle_set_nlnconstr(COMM &comm, const BL &bl, const BU &bu, const IROWGD &irowgd, const ICOLGD &icolgd, OptionalE04RK opt)

template <typename COMM, typename BL, typename BU, typename IROWGD, typename ICOLGD>

void function handle_set_nlnconstr(COMM &comm, const BL &bl, const BU &bu, const IROWGD &irowgd, const ICOLGD &icolgd)

3 Description

After the initialization function handle_init has been called, handle_set_nlnconstr may be used to define the nonlinear constraints

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

of the problem unless the nonlinear constraints have already been defined. 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 handle_set_nlnconstr.

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 handle_solve_ipopt).

See Section 3.1 in the E04 Chapter Introduction for more details about the NAG optimization modelling suite.

4 References

None.

5 Arguments

1: $comm$ – CommE04RA Input/Output

Communication structure. An object of either the derived class CommE04RA or its base class NoneCopyableComm can be supplied. It is recommended that the derived class is used. If the base class is supplied it must first be initialized via a call to opt::handle_init (e04ra).

2: $bl (ncnln)$ – double array 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 $j = 1, 2, \dots, ncnln$ .

3: $bu (ncnln)$ – double array 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 $j = 1, 2, \dots, ncnln$ .

4: $irowgd (nnzgd)$ – types::f77_integer array 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 handle_solve_ipopt, 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 $l = 1, 2, \dots, nnzgd$ .

5: $icolgd (nnzgd)$ – types::f77_integer array Input

ncnln \times n

\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 handle_solve_ipopt, 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 $l = 1, 2, \dots, nnzgd$ .

6: $opt$ – OptionalE04RK Input/Output

Optional parameter container, derived from Optional.

5.1Additional Quantities

1: $ncnln$: $m_{g}$ , the number of nonlinear constraints (number of rows of the Jacobian matrix).
2: $nnzgd$: nnzgd gives the number of nonzeros in the Jacobian matrix.

6 Exceptions and Warnings

Errors or warnings detected by the function:

All errors and warnings have an associated numeric error code field, errorid, stored either as a member of the thrown exception object (see errorid), or as a member of opt.ifail, depending on how errors and warnings are being handled (see Error Handling for more details).

Raises: ErrorException

$errorid = 1$: comm::handle has not been initialized.

$errorid = 1$: comm::handle does not belong to the NAG optimization modelling suite,
has not been initialized properly or is corrupted.

$errorid = 1$: comm::handle has not been initialized properly or is corrupted.

$errorid = 2$: The problem cannot be modified in this phase any more, the solver
has already been called.

$errorid = 2$: The Hessian of the nonlinear objective has already been defined,
nonlinear constraints cannot be added.

$errorid = 3$: A set of nonlinear constraints has already been defined.

$errorid = 6$: On entry, $nnzgd = ⟨ value ⟩$ .
Constraint: $nnzgd > 0$ .

$errorid = 6$: On entry, $ncnln = ⟨ value ⟩$ .
Constraint: $ncnln \geq 0$ .

$errorid = 8$: On entry, $i = ⟨ value ⟩$ , $irowgd [i - 1] = ⟨ value ⟩$ and
$ncnln = ⟨ value ⟩$ .
Constraint: $1 \leq irowgd [i - 1] \leq ncnln$ .

$errorid = 8$: On entry, $i = ⟨ value ⟩$ , $icolgd [i - 1] = ⟨ value ⟩$ and
$n = ⟨ value ⟩$ .
Constraint: $1 \leq icolgd [i - 1] \leq n$ .

$errorid = 8$: 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.

$errorid = 10$: On entry, $j = ⟨ value ⟩$ , $bl [j - 1] = ⟨ value ⟩$ and
$bu [j - 1] = ⟨ value ⟩$ .
Constraint: $bl [j - 1] \leq bu [j - 1]$ .

$errorid = 10$: On entry, $j = ⟨ value ⟩$ , $bl [j - 1] = ⟨ value ⟩$ ,
$bigbnd = ⟨ value ⟩$ .
Constraint: $bl [j - 1] < bigbnd$ .

$errorid = 10$: On entry, $j = ⟨ value ⟩$ , $bu [j - 1] = ⟨ value ⟩$ ,
$bigbnd = ⟨ value ⟩$ .
Constraint: $bu [j - 1] > - bigbnd$ .

$errorid = 10601$: On entry, argument $⟨ value ⟩$ must be a vector of size $⟨ value ⟩$ array.
Supplied argument has $⟨ value ⟩$ dimensions.

$errorid = 10601$: On entry, argument $⟨ value ⟩$ must be a vector of size $⟨ value ⟩$ array.
Supplied argument was a vector of size $⟨ value ⟩$ .

$errorid = 10601$: On entry, argument $⟨ value ⟩$ must be a vector of size $⟨ value ⟩$ array.
The size for the supplied array could not be ascertained.

$errorid = 10602$: On entry, the raw data component of $⟨ value ⟩$ is null.

$errorid = 10603$: On entry, unable to ascertain a value for $⟨ value ⟩$ .

$errorid = 10605$: On entry, the communication class $⟨ value ⟩$ has not been initialized correctly.

$errorid = - 99$: An unexpected error has been triggered by this routine.

$errorid = - 399$: Your licence key may have expired or may not have been installed correctly.

$errorid = - 999$: Dynamic memory allocation failed.

7 Accuracy

Not applicable.

8 Parallelism and Performance

Please see the description for the underlying computational routine in this section of the FL Interface documentation.

9 Further Comments

None.

10 Example

Examples of the use of this method may be found in the examples for: handle_solve_ipopt.

NAG Library Manual, Mark 27

Interfaces: FL CL CPP AD

NAG CPP Interface Introduction

E04 (Opt) Chapter Contents

e04rk: FL CL CPP AD

Namespace: nagcpp

Namespace: opt

Function: handle_set_nlnconstr

NAG CPP Interfacenagcpp::opt::handle_set_nlnconstr (e04rk)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

5.1Additional Quantities

6 Exceptions and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

NAG CPP Interface
nagcpp::opt::handle_set_nlnconstr (e04rk)