e04rtc is a part of the NAG optimization modelling suite and defines a new, or edits an existing, quadratic objective function or constraint of the problem using a factor of the quadratic coefficient matrix.

2 Specification

#include <nag.h>

void	e04rtc (void handle, double s, Integer nnzr, const Integer idxr[], const double r[], Integer mf, Integer nnzf, const Integer irowf[], const Integer icolf[], const double f[], Integer idqc, NagError *fail)

The function may be called by the names: e04rtc or nag_opt_handle_set_qconstr_fac.

3 Description

After the handle has been initialized (e.g., e04rac has been called), e04rtc may be used to edit a model by adding or replacing a quadratic objective function or constraint of the form

\frac{1}{2} x^{T} F^{T} F x + r^{T} x

(1)

and

\frac{1}{2} x^{T} F^{T} F x + r^{T} x + s \leq 0,

(2)

respectively.

The matrix

F

is a sparse

m \times n

matrix. It can be viewed as the factor of the symmetric matrix

Q = F^{T} F

in a general quadratic function

\frac{1}{2} x^{T} Q x + r^{T} x + s .

(3)

It is also acceptable if

F

is a zero matrix, in which case the corresponding objective function or constraint becomes linear. If you have the full matrix

Q

as input data, please call function e04rsc instead. Note that it is possible to temporarily disable and enable individual constraints in the model by 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: $s$ – double Input

On entry: the constant term in quadratic constraint.

idqc = −1

, s will not be referenced.

3: $nnzr$ – Integer Input

On entry: the number of nonzero elements in the sparse vector

r

nnzr = 0

r

is considered to be zero and the arrays idxr and r will not be referenced and may be NULL.

Constraint:

nnzr \geq 0

4: $idxr [nnzr]$ – const Integer Input

5: $r [nnzr]$ – const double Input

On entry: the nonzero elements of the sparse vector

r

idxr [i - 1]

must contain the index of

r [i - 1]

in the vector, for

i = 1, 2, \dots, nnzr

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

n

is the current number of variables in the problem.

Constraint:

1 \leq idxr [i - 1] \leq n

, for

i = 1, 2, \dots, nnzr

6: $mf$ – Integer Input

On entry:

m

, row dimension of matrix

F

Constraint:

mf > 0

7: $nnzf$ – Integer Input

On entry: the number of nonzero elements in the matrix

F

nnzf = 0

, the matrix

F

is considered to be zero, the objective function or constraint is linear and mf, irowf, icolf and f will not be referenced and may be NULL.

Constraint:

nnzf \geq 0

8: $irowf [nnzf]$ – const Integer Input

9: $icolf [nnzf]$ – const Integer Input

10: $f [nnzf]$ – const double Input

On entry: arrays irowf, icolf and f store the nonzeros of the matrix

F

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

F_{i j} = f [l - 1]

where

i = irowf [l - 1]

j = icolf [l - 1]

, for

l = 1, 2, \dots, nnzf

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

Constraint:

1 \leq irowf [l - 1] \leq mf, ​ 1 \leq icolf [l - 1] \leq n

, for

l = 1, 2, \dots, nnzf

11: $idqc$ – Integer * Input/Output

On entry:

$idqc = 0$: A new quadratic constraint is created.
$idqc > 0$: Specifies the index of an existing constraint to be replaced. i.e., replaces the idqcth constraint.
$idqc = −1$: A new quadratic objective is created and will replace any previously defined objective function.

Constraint:

idqc \geq −1

On exit: if

idqc = 0

on entry, then idqc is overwritten with the index of the new quadratic constraint. By definition, this is the number of quadratic constraints already defined plus one. Otherwise, idqc stays unchanged.

12: $fail$ – 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 properly initialized or it has been corrupted.
NE_INT: nnzr and nnzf cannot be zero at the same time.

On entry, $idqc = ⟨ value ⟩$ .
Constraint: $idqc \geq −1$ .

On entry, $mf = ⟨ value ⟩$ .
Constraint: $mf > 0$ .

On entry, $nnzf = ⟨ value ⟩$ .
Constraint: $nnzf \geq 0$ .

On entry, $nnzr = ⟨ value ⟩$ .
Constraint: $nnzr \geq 0$ .
NE_INTARR: On entry, $i = ⟨ value ⟩$ , $idxr [i - 1] = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $1 \leq idxr [i - 1] \leq n$ .
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, $i = ⟨ value ⟩$ , $icolf [i - 1] = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $1 \leq icolf [i - 1] \leq n$ .

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

On entry, more than one element of f has row index $⟨ value ⟩$ and column index $⟨ value ⟩$ .
Constraint: each element of f 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_NOT_INCREASING: On entry, more than one element of idxr has index $⟨ value ⟩$ .
Constraint: each element of idxr must have a unique index.
NE_PHASE: The problem cannot be modified right now, the solver is running.
NE_REF_MATCH: On entry, $idqc = ⟨ value ⟩$ .
The given idqc does not match with any quadratic constraint already defined.

7 Accuracy

Not applicable.

8 Parallelism and Performance

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

e04rtc is not threaded in any implementation.

9 Further Comments

None.

10 Example

This example demonstrates how to define and solve a convex quadratic programming problem where the quadratic term is defined by its factors.

We solve the following norm minimization problem:

\begin{array}{l} \underset{x \in R^{3}}{minimize} & {‖ A x - b ‖}_{2}^{2} \\ subject to & e^{T} x = 1, \\ l_{x} \leq x \leq u_{x}, \end{array}

where

A = (\begin{array}{l} 0.493 & 0.382 & 0.0 \\ 0.0 & 0.270 & 0.475 \end{array}), b = (\begin{matrix} 0.2 \\ 0.4 \end{matrix}),

l_{x} = (\begin{matrix} - 1.0 \\ - 1.0 \\ - 1.0 \end{matrix}), u_{x} = (\begin{matrix} 1.0 \\ 1.0 \\ 1.0 \end{matrix}),

and

e \in R^{3}

is vector of all ones. Note that

{‖ A x - b ‖}_{2}^{2} = x^{T} A^{T} A x - 2 b^{T} A x + b^{T} b

which is a convex quadratic function.

The optimal solution (to five significant figures) is

x^{*} = {(1.0000, - 0.97221, 0.97221)}^{T},

and the objective function value without the constant term

b^{T} b

0.13130

e04rt: FL CL CPP AD PY MB

NAG CL Interfacee04rtc (handle_​set_​qconstr_​fac)

▸▿ 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
e04rtc (handle_set_qconstr_fac)