nagcpp::opt::handle_set_nlnhess (e04rl) : NAG Library, Mark 28

2 Specification

#include "e04/nagcpp_e04rl.hpp" #include "e04/nagcpp_class_CommE04RA.hpp"

template <typename COMM, typename IROWH, typename ICOLH> void function handle_set_nlnhess(COMM &comm, const types::f77_integer idf, const IROWH &irowh, const ICOLH &icolh, OptionalE04RL opt)

template <typename COMM, typename IROWH, typename ICOLH> void function handle_set_nlnhess(COMM &comm, const types::f77_integer idf, const IROWH &irowh, const ICOLH &icolh)

3 Description

After the handle has been initialized (e.g., handle_init has been called), and a nonlinear objective function

f (x)

and/or the

i

th (

1 \leq i \leq m_{g}

) nonlinear constraint function

g_{i} (x)

has been registered with handle_set_nlnconstr and handle_set_nlnobj, then handle_set_nlnhess may be used to define the sparsity structure (pattern) of the Hessians of those functions or of their Lagrangian function. Define:

$\nabla^{2} f (x) \equiv (\begin{matrix} \frac{\partial^{2} f (x)}{\partial^{2} x_{1}} & \frac{\partial^{2} f (x)}{\partial x_{2} \partial x_{1}} & \dots & \frac{\partial^{2} f (x)}{\partial x_{n} \partial x_{1}} \\ \frac{\partial^{2} f (x)}{\partial x_{1} \partial x_{2}} & \frac{\partial^{2} f (x)}{\partial^{2} x_{2}} & \dots & \frac{\partial^{2} f (x)}{\partial x_{n} \partial x_{2}} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ \frac{\partial^{2} f (x)}{\partial x_{1} \partial x_{n}} & \frac{\partial^{2} f (x)}{\partial x_{2} \partial x_{n}} & \dots & \frac{\partial^{2} f (x)}{\partial^{2} x_{n}} \end{matrix})$ , and $\nabla^{2} g_{i} (x) \equiv (\begin{matrix} \frac{\partial^{2} g_{i} (x)}{\partial^{2} x_{1}} & \frac{\partial^{2} g_{i} (x)}{\partial x_{2} \partial x_{1}} & \dots & \frac{\partial^{2} g_{i} (x)}{\partial x_{n} \partial x_{1}} \\ \frac{\partial^{2} g_{i} (x)}{\partial x_{1} \partial x_{2}} & \frac{\partial^{2} g_{i} (x)}{\partial^{2} x_{2}} & \dots & \frac{\partial^{2} g_{i} (x)}{\partial x_{n} \partial x_{2}} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ \frac{\partial^{2} g_{i} (x)}{\partial x_{1} \partial x_{n}} & \frac{\partial^{2} g_{i} (x)}{\partial x_{2} \partial x_{n}} & \dots & \frac{\partial^{2} g_{i} (x)}{\partial^{2} x_{n}} \end{matrix})$ for $1 \leq i \leq m_{g}$
handle_set_nlnhess can be used to define the following sparsity structures:
the Hessian of the Lagrangian function $σ \nabla^{2} f (x) + \sum_{i = 1}^{m_{g}} λ_{i} \nabla^{2} g_{i} (x)$ ,
the Hessian of the objective function $\nabla^{2} f (x)$ , or
the Hessian of the $i$ th constraint function $\nabla^{2} g_{i} (x)$ with $1 \leq i \leq m_{g}$ .

In general, each of the symmetric

n \times n

Hessian matrices will have its own sparsity structure. These structures can be given in separate handle_set_nlnhess calls, or merged together in the Lagrangian and given in one call. The nonzero values of the Hessians at particular points will be communicated to the NLP solver by user-supplied functions (e.g., hess for handle_solve_ipopt). The values will need to be provided in the order matching the sparsity pattern.

Note that the Hessians are automatically deleted whenever the underlying functions change. For example, if handle_set_nlnconstr is called to redefine the nonlinear constraints, all individual constraints Hessians or Hessian of the Lagrangian would be deleted. If a nonlinear objective function was changed to linear, the Hessian of the objective function or of the Lagrangian would be deleted. handle_set_nlnhess can work either with individual Hessians or with the Hessian of the Lagrangian but not both. Therefore, if the Hessian of the Lagrangian was defined and handle_set_nlnhess was called to define an individual Hessian of the constraint, the Hessian of the Lagrangian would be removed, and vice versa. Hessians can be redefined by multiple calls of handle_set_nlnhess.

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

5 Arguments

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).

idf

– types::f77_integer Input

On entry: specifies the functions for which a Hessian sparsity structure is provided in nnzh, irowh and icolh.

$idf = −1$: The sparsity structure of the Hessian of the Lagrangian is provided.
$idf = 0$: The sparsity structure of the Hessian of the objective function is provided.
$idf > 0$: The sparsity structure of the Hessian of the idfth constraint function is provided.

The value of idf will also determine how an NLP solver will call the user-supplied functions that evaluate these nonzeros at particular points of the decision variable space, i.e., whether the solver will expect the nonzero values of the objective and constraint Hessians in separate calls or merged in the Lagrangian Hessian, in one call. See, for example, hess of handle_solve_ipopt.

Constraint:

−1 \leq idf \leq m_{g}

Note:

m_{g}

, the number of nonlinear constraints registered with the handle.

irowh (nnzh)

– types::f77_integer array Input

On entry: arrays irowh and icolh store the nonzeros of the upper triangle of the matrix

H

in coordinate storage (CS) format (see Section 2.1.1 in the F11 Chapter Introduction). irowh specifies one-based row indices, icolh specifies one-based column indices and specifies the values of the nonzero elements in such a way that

h_{i j} = H (l - 1)

where

i = irowh (l - 1)

and

j = icolh (l - 1)

, for

l = 1, 2, \dots, nnzh

. No particular order is expected, but elements should not repeat.

Constraint:

1 \leq irowh (l - 1) \leq icolh (l - 1) \leq n

, for

l = 1, 2, \dots, nnzh

icolh (nnzh)

– types::f77_integer array Input

On entry: arrays irowh and icolh store the nonzeros of the upper triangle of the matrix

H

h_{i j} = H (l - 1)

where

i = irowh (l - 1)

and

j = icolh (l - 1)

, for

l = 1, 2, \dots, nnzh

. No particular order is expected, but elements should not repeat.

Constraint:

1 \leq irowh (l - 1) \leq icolh (l - 1) \leq n

, for

l = 1, 2, \dots, nnzh

opt

– OptionalE04RL Input/Output

Optional parameter container, derived from Optional.

5.1Additional Quantities

1: $nnzh$: The number of nonzero elements in the upper triangle of the matrix $H$

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 right now, the solver is running.

$errorid = 2$: Neither nonlinear objective nor nonlinear constraints are present.
The structure of the Hessian cannot be defined.

$errorid = 2$: No nonlinear objective has been defined, its Hessian cannot be set.

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

$errorid = 7$: On entry, $idf = ⟨ value ⟩$ .
Constraint: $−1 \leq idf \leq ⟨ value ⟩$ .

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

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

$errorid = 8$: On entry, $i = ⟨ value ⟩$ , $irowh [i - 1] = ⟨ value ⟩$ and
$icolh [i - 1] = ⟨ value ⟩$ .
Constraint: $irowh [i - 1] \leq icolh [i - 1]$ (elements within the upper triangle).

$errorid = 8$: On entry, more than one element of structural matrix $H$ has row index
$⟨ value ⟩$ and column index $⟨ value ⟩$ .
Constraint: each element of structural matrix $H$ must have a unique row
and column index.

$errorid = 10601$: 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.

NAG CPP Interface
nagcpp::opt::handle_set_nlnhess (e04rl)

▸▿ 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 Interfacenagcpp::opt::handle_set_nlnhess (e04rl)

▸▿ 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_nlnhess (e04rl)