e04rbc 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

#include <nag.h>

void	e04rbc (void handle, const char gtype, Integer lgroup, const Integer group[], Integer idgroup, NagError fail)

The function may be called by the names: e04rbc or nag_opt_handle_set_group.

3 Description

After the initialization function e04rac has been called, e04rbc 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 function 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}

e04rbc can be called repeatedly to add, replace or delete one cone constraint at a time. 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 by e04rac and must not be changed before the call to e04rbc.

2: $gtype$ – const char * 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 may be NULL, 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: $group [lgroup]$ – const Integer Input

On entry:

G^{i}

, the indices of the variables in the constraint. If

lgroup = 0

, group is not referenced and may be NULL.

Constraint:

1 \leq group [k] \leq n

, for

k = 0, 1, \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: $fail$ – 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_HANDLE: 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 e04rac or it has been corrupted.
NE_INDICES: On entry, $k = 〈value〉$ , $group [k - 1] = 〈value〉$ and $n = 〈value〉$ .
Constraint: $1 \leq group [k - 1] \leq n$ .
NE_INT: On entry, $idgroup = 〈value〉$ .
Constraint: $idgroup \geq 0$ .

On entry, $lgroup = 〈value〉$ .
Constraint: $lgroup \geq 0$ .
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_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 in this phase any more, the solver has already been called.
NE_REF_MATCH: On entry, $idgroup = 〈value〉$ .
The given idgroup does not match with any cone constraint already defined.
NE_REPEAT_CONEVAR_DF: On entry, variable with index $i = 〈value〉$ has been defined in a cone in a previous call to this function.
Constraint: each variable may be defined in one cone constraint at most.
NE_REPEAT_CONEVAR_SM: On entry, $group [i - 1] = group [j - 1] = 〈value〉$ for $i = 〈value〉$ and $j = 〈value〉$ .
Constraint: elements in group cannot repeat.
NE_STANDARD_ERRORS: 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 $lgroup \geq 3$ .
NE_STR_UNKNOWN: On entry, $gtype = 〈value〉$ .
Constraint: $gtype = QUAD$ , $Q$ , $RQUAD$ or $R$ .

7 Accuracy

Not applicable.

8 Parallelism and Performance

e04rbc 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

e04rb: FL CL CPP AD

NAG CL Interfacee04rbc (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 CL Interface
e04rbc (handle_set_group)