Integer, Intent (In)	::	nclin, nnzb, irowb(nnzb), icolb(nnzb)
Integer, Intent (Inout)	::	idlc, ifail
Real (Kind=nag_wp), Intent (In)	::	bl(nclin), bu(nclin), b(nnzb)
Type (c_ptr), Intent (In)	::	handle

C Header Interface

#include nagmk26.h

void	e04rjf_ ( void *handle, const Integer nclin, const double bl[], const double bu[], const Integer nnzb, const Integer irowb[], const Integer icolb[], const double b[], Integer idlc, Integer *ifail)

3

Description

After the initialization routine e04raf has been called, e04rjf may be used to define the linear constraints

l_{B} \leq B x \leq u_{B}

of the problem unless the linear constraints have already been defined. This will typically be used for problems, such as linear programming (LP)

\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) \end{array}

(1)

quadratic programming (QP)

\begin{array}{l} \underset{x \in ℝ^{n}}{minimize} & \frac{1}{2} x^{T} H x + c^{T} x & (a) \\ subject to & l_{B} \leq B x \leq u_{B} & (b) \\ l_{x} \leq x \leq u_{x}, & (c) \end{array}

(2)

nonlinear programming (NLP)

\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}

(3)

or linear semidefinite programming (SDP)

\begin{array}{l} \underset{x \in ℝ^{n}}{minimize} & c^{T} x & (a) \\ subject to & \sum_{i = 1}^{n} x_{i} A_{i}^{k} - A_{0}^{k} ⪰ 0, k = 1, \dots, m_{A} & (b) \\ l_{B} \leq B x \leq u_{B} & (c) \\ l_{x} \leq x \leq u_{x} & (d) \end{array}

(4)

where

n

is the number of decision variables,

B

is a general

m_{B} \times n

rectangular matrix and

l_{B}

and

u_{B}

are

m_{B}

-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 of

l_{B}

u_{B}

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 routine and any later alterations to Infinite Bound Size will not affect these constraints.

See e04raf for more details.

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.

2: $nclin$ – IntegerInput

On entry:

m_{B}

, the number of linear constraints (number of rows of the matrix

B

nclin = 0

, no linear constraints will be defined and bl, bu, nnzb, irowb, icolb and b will not be referenced.

Constraint:

nclin \geq 0

3: $bl (nclin)$ – Real (Kind=nag_wp) arrayInput

4: $bu (nclin)$ – Real (Kind=nag_wp) arrayInput

On entry: bl and bu define lower and upper bounds of the linear constraints,

l_{B}

and

u_{B}

, respectively. To define the

j

th constraint as equality, set

bl (j) = bu (j) = β

, where

|β| < bigbnd

. To specify a nonexistent lower bound (i.e.,

l_{j} = - \infty

), set

bl (j) \leq - bigbnd

; to specify a nonexistent upper bound, set

bu (j) \geq bigbnd

Constraints:

$bl (j) \leq bu (j)$ , for $j = 1, 2, \dots, nclin$ ;
$bl (j) < bigbnd$ , for $j = 1, 2, \dots, nclin$ ;
$bu (j) > - bigbnd$ , for $j = 1, 2, \dots, nclin$ ;
if $bl (j) = bu (j)$ , $|bl (j)| < bigbnd$ , for $j = 1, 2, \dots, nclin$ .

5: $nnzb$ – IntegerInput

On entry: nnzb gives the number of nonzeros in matrix

B

Constraint: if

nclin > 0

nnzb > 0

6: $irowb (nnzb)$ – Integer arrayInput

7: $icolb (nnzb)$ – Integer arrayInput

8: $b (nnzb)$ – Real (Kind=nag_wp) arrayInput

On entry: arrays irowb, icolb and b store nnzb nonzeros of the sparse matrix

B

in coordinate storage (CS) format (see Section 2.1.1 in the F11 Chapter Introduction). The matrix

B

has dimensions

m_{B} \times n

, where

n

, the number of variables in the problem, was set in nvar during the initialization of the handle by e04raf. irowb specifies one-based row indices, icolb specifies one-based column indices and b specifies the values of the nonzero elements in such a way that

b_{i j} = b (l)

where

i = irowb (l)

and

j = icolb (l)

, for

l = 1, 2, \dots, nnzb

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

Constraint:

1 \leq irowb (l) \leq nclin

1 \leq icolb (l) \leq n

, for

l = 1, 2, \dots, nnzb

9: $idlc$ – IntegerInput/Output

Note: idlc is reserved for future releases of the NAG Library.

On entry: if

idlc = 0

, new linear constraints are added to the problem definition. This is the only value allowed at the moment.

Constraint:

idlc = 0

On exit: the number of the last linear constraint added, thus nclin.

10: $ifail$ – IntegerInput/Output

On entry: ifail must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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.

$ifail = 3$: A set of linear constraints has already been defined.

$ifail = 4$: On entry, $idlc = 〈value〉$ .
Constraint: $idlc = 0$ .

$ifail = 6$: On entry, $nclin = 〈value〉$ .
Constraint: $nclin \geq 0$ .

On entry, $nnzb = 〈value〉$ .
Constraint: $nnzb > 0$ .

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

On entry, $i = 〈value〉$ , $irowb (i) = 〈value〉$ and $nclin = 〈value〉$ .
Constraint: $1 \leq irowb (i) \leq nclin$ .

On entry, more than one element of b has row index $〈value〉$ and column index $〈value〉$ .
Constraint: each element of b must have a unique row and column index.

$ifail = 10$: On entry, $j = 〈value〉$ , $bl (j) = 〈value〉$ , $bigbnd = 〈value〉$ .
Constraint: $bl (j) < bigbnd$ .

On entry, $j = 〈value〉$ , $bl (j) = 〈value〉$ and $bu (j) = 〈value〉$ .
Constraint: $bl (j) \leq bu (j)$ .

On entry, $j = 〈value〉$ , $bu (j) = 〈value〉$ , $bigbnd = 〈value〉$ .
Constraint: $bu (j) > - bigbnd$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

Not applicable.

8

Parallelism and Performance

e04rjf is not threaded in any implementation.

9

Further Comments

None.

10

Example

This example demonstrates how to use the MPS file reader e04mxf and this suite of routines to define and solve a QP problem. e04mxf uses a different output format to the one required by e04rjf, in particular, it uses the compressed column storage (CCS) (see Section 2.1.3 in the F11 Chapter Introduction) instead of the coordinate storage and the linear objective vector is included in the system matrix. Therefore a simple transformation is needed before calling e04rjf as demonstrated in the example program.

The data file stores the following problem:

minimize c^{T} x + \frac{1}{2} x^{T} H x subject to \begin{matrix} l_{B} & \leq B x & \leq u_{B}, \\ - 2 & \leq x & \leq 2, \end{matrix}

where

c = (\begin{matrix} - 4.0 \\ - 1.0 \\ - 1.0 \\ - 1.0 \\ - 1.0 \\ - 1.0 \\ - 1.0 \\ - 0.1 \\ - 0.3 \end{matrix}), H = (\begin{matrix} 2 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 2 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 2 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 2 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 2 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}),

B = (\begin{matrix} 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 4.0 \\ 1.0 & 2.0 & 3.0 & 4.0 & - 2.0 & 1.0 & 1.0 & 1.0 & 1.0 \\ 1.0 & - 1.0 & 1.0 & - 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 \end{matrix}),

l_{B} = (\begin{matrix} - 2.0 \\ - 2.0 \\ - 2.0 \end{matrix}) and u_{B} = (\begin{matrix} 1.5 \\ 1.5 \\ 4.0 \end{matrix}) .

The optimal solution (to five figures) is

x^{*} = {(2.0, - 0.23333, - 0.26667, - 0.3, - 0.1, 2.0, 2.0, - 1.7777, - 0.45555)}^{T} .

See also Section 10 in e04raf for links to further examples in this suite.

NAG Library Routine Document

e04rjf (handle_set_linconstr)

▸▿ 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

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