e04rbf is a part of the NAG optimization modelling suite and modifies a model by either adding a new, or replacing or deleting an existing, quadratic or rotated quadratic cone constraint.

2 Specification

Fortran Interface

Subroutine e04rbf (

handle, gtype, lgroup, group, idgroup, ifail)

Integer, Intent (In)	::	lgroup, group(lgroup)
Integer, Intent (Inout)	::	idgroup, ifail
Character (*), Intent (In)	::	gtype
Type (c_ptr), Intent (In)	::	handle

C Header Interface

#include <nag.h>

void	e04rbf_ (void *handle, const char gtype, const Integer lgroup, const Integer group[], Integer idgroup, Integer *ifail, const Charlen length_gtype)

The routine may be called by the names e04rbf or nagf_opt_handle_set_group.

3 Description

After the initialization routine e04raf has been called, e04rbf may be used to edit a model by adding, replacing, or deleting a cone constraint

i

of dimension

m_{i}

. The supported cones are quadratic cone and rotated quadratic cone, also known as second-order cones, which are defined as follows:

•Quadratic cone:
$K_{q}^{m_{i}} ≔ \{z = (z_{1}, z_{2}, \dots, z_{m_{i}}) \in ℝ^{m_{i}} : z_{1}^{2} \geq \sum_{j = 2}^{m_{i}} z_{j}^{2}, z_{1} \geq 0\} .$ (1)

•Rotated quadratic cone:

K_{r}^{m_{i}} ≔ \{z = (z_{1}, z_{2}, \dots, z_{m_{i}}) \in ℝ^{m_{i}} : 2 z_{1} z_{2} \geq \sum_{j = 3}^{m_{i}} z_{j}^{2}, z_{1} \geq 0, z_{2} \geq 0\} .

(2)

The cone constraint is defined by its type and a subset (group) of variables. Let index set

G^{i} \subseteq \{1, 2, \dots, n\}

denote variable indices, then

x_{G^{i}}

will denote the subvector of variables

x \in ℝ^{n}

For example, if

m_{i} = 3

and

G^{i} = \{4, 1, 2\}

, then a quadratic cone constraint

x_{G^{i}} = (x_{4}, x_{1}, x_{2}) \in K_{q}^{3}

implies the inequality constraints

x_{4}^{2} \geq x_{1}^{2} + x_{2}^{2}, x_{4} \geq 0 .

Typically, this routine will be used to build second-order cone programming (SOCP) problems which might be formulated in the following way:

\begin{array}{l} \underset{x \in ℝ^{n}}{minimize} & c^{T} x & (a) \\ subject to & l_{B} \leq B x \leq u_{B}, & (b) \\ l_{x} \leq x \leq u_{x}, & (c) \\ x_{G^{i}} \in K^{m_{i}}, i = 1, \dots, r, & (d) \end{array}

(3)

where

K^{m_{i}}

is either quadratic cone or rotated quadratic cone of dimension

m_{i}

e04rbf can be called repeatedly to add, replace or delete one cone constraint at a time. See Section 3.1 in the E04 Chapter Introduction for more details about the NAG optimization modelling suite.

4 References

None.

5 Arguments

1: $handle$ – Type (c_ptr) Input

On entry: the handle to the problem. It needs to be initialized by e04raf and must not be changed between calls to the NAG optimization modelling suite.

2: $gtype$ – Character(*) Input

On entry: the type of the cone constraint, case insensitive.

$gtype ='QUAD'$ or $'Q'$: The group defines a quadratic cone.
$gtype ='RQUAD'$ or $'R'$: The group defines a rotated quadratic cone.

Constraint:

gtype ='QUAD'

'Q'

'RQUAD'

'R'

3: $lgroup$ – Integer Input

On entry:

m_{i}

, the number of the variables in the group.

lgroup = 0

, gtype and group will not be referenced, and the constraint with ID number idgroup will be deleted from the model.

Constraints:

if $gtype ='QUAD'$ or $'Q'$ , $lgroup = 0$ or $lgroup \geq 2$ ;
if $gtype ='RQUAD'$ or $'R'$ , $lgroup = 0$ or $lgroup \geq 3$ .

4: $group (lgroup)$ – Integer array Input

On entry:

G^{i}

, the indices of the variables in the constraint. If

lgroup = 0

, group is not referenced.

Constraint:

1 \leq group (k) \leq n

, for

k = 1, 2, \dots, lgroup

, where

n

is the number of decision variables in the problem. The elements must not repeat and each variable can appear in one cone at most, see Section 9.

5: $idgroup$ – Integer Input/Output

On entry:

$idgroup = 0$: A new cone constraint is created.
$idgroup > 0$: $i$ , the ID number of the existing constraint to be deleted or replaced.

Constraint:

idgroup \geq 0

On exit: if

idgroup = 0

on entry, the ID number of the new cone constraint is returned. By definition, this is the number of the cone constraints already defined plus one. Otherwise, idgroup stays unchanged.

6: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

- 1 or 1

. If you are unfamiliar with this argument you should refer to Section 4 in the Introduction to the NAG Library FL Interface for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 or 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, the recommended value is

- 1

. When the value $- 1 or 1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: The supplied handle does not define a valid handle to the data structure for the NAG optimization modelling suite. It has not been initialized by e04raf or it has been corrupted.

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

$ifail = 3$: On entry, variable with index $i = 〈value〉$ has been defined in a cone in a previous call to this routine.
Constraint: each variable may be defined in one cone constraint at most.

$ifail = 4$: On entry, $idgroup = 〈value〉$ .
The given idgroup does not match with any cone constraint already defined.

$ifail = 5$: On entry, $gtype = 〈value〉$ and $lgroup = 〈value〉$ .
Constraint: if $gtype ='QUAD'$ or $'Q'$ , $lgroup = 0$ or $lgroup \geq 2$ .

On entry, $gtype = 〈value〉$ and $lgroup = 〈value〉$ .
Constraint: if $gtype ='RQUAD'$ or $'R'$ , $lgroup = 0$ or $lgroup \geq 3$ .

On entry, $lgroup = 〈value〉$ .
Constraint: $lgroup \geq 0$ .

$ifail = 6$: On entry, $gtype = 〈value〉$ .
Constraint: $gtype ='QUAD'$ , $'Q'$ , $'RQUAD'$ or $'R'$ .

$ifail = 7$: On entry, $idgroup = 〈value〉$ .
Constraint: $idgroup \geq 0$ .

$ifail = 8$: On entry, $k = 〈value〉$ , $group (k) = 〈value〉$ and $n = 〈value〉$ .
Constraint: $1 \leq group (k) \leq n$ .

$ifail = 9$: On entry, $group (i) = group (j) = 〈value〉$ for $i = 〈value〉$ and $j = 〈value〉$ .
Constraint: elements in group cannot repeat.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

Not applicable.

8 Parallelism and Performance

e04rbf is not threaded in any implementation.

9 Further Comments

Overlapping of cones is not supported, which means each variable may be defined in one cone at most. However by adding auxiliary variables, you can achieve the same effect. For example, if

x_{a} \in K^{m_{1}}

and

x_{a} \in K^{m_{2}}

, you can add one more variable

x_{b} = x_{a}

and set

x_{a} \in K^{m_{1}}

x_{b} \in K^{m_{2}}

10 Example

This example solves the following SOCP problem

minimize 10.0 x_{1} + 20.0 x_{2} + x_{3}

subject to the bounds

\begin{matrix} - 2.0 & \leq & x_{1} & \leq & 2.0 \\ - 2.0 & \leq & x_{2} & \leq & 2.0 \end{matrix}

the general linear constraints

\begin{matrix} - 0.1 x_{1} & - & 0.1 x_{2} & + & x_{3} & \leq & 1.5 \\ 1.0 & \leq & - 0.06 x_{1} & + & x_{2} & + & x_{3} \end{matrix}

and the cone constraint

(x_{3}, x_{1}, x_{2}) \in K_{q}^{3} .

The optimal solution (to five significant figures) is

x^{*} = {(- 1.2682, - 4.0843, 1.3323)}^{T},

and the objective function value is

- 19.518

Interfaces: FL CL AD

NAG FL Interface Introduction

E04 (Opt) Chapter Contents

E04 (Opt) Chapter Introduction

e04rb: FL CL AD

NAG FL Interfacee04rbf (handle_​set_​group)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
e04rbf (handle_set_group)