e04rjf is a part of the NAG optimization modelling suite and adds a new block of linear constraints to the problem or modifies an individual linear constraint.

2 Specification

Fortran Interface

Subroutine e04rjf (

handle, nclin, bl, bu, nnzb, irowb, icolb, b, idlc, ifail)

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 <nag.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)

The routine may be called by the names e04rjf or nagf_opt_handle_set_linconstr.

3 Description

After the handle has been initialized (e.g., e04raf has been called), e04rjf may be used to add to the problem a new block of

m_{B}

linear constraints

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

where

B

is a general

m_{B} \times n

rectangular matrix,

n

is the current number of decision variables in the model and

l_{B}

and

u_{B}

are

m_{B}

-dimensional vectors defining the lower and upper bounds, respectively. The call can be repeated to add multiple blocks to the model.

Note that the bounds are specified for all the constraints of this block. 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.

The linear constraints can be edited. To identify the individual constraints, they are numbered starting with

1

, see idlc. A single constraint (i.e., a single row of the matrix

B

) can be modified (replaced) by e04rjf by referring to its idlc. An individual coefficient

b_{ij}

of the matrix

B

can be set or modified by e04tjf and bounds of a single constraint can be set or modified by e04tdf. Note that it is also possible to temporarily disable and enable individual constraints in the model by e04tcf and e04tbf, respectively.

Linear constraints may be present in many different types of problems, for simplicity of the notation, only one block of linear constraints is presented. For example,

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)

Quadratically Constrained Quadratic Programming (QCQP)

\begin{array}{l} \underset{x \in ℝ^{n}}{minimize} & \frac{1}{2} x^{T} H x + c^{T} x & (a) \\ subject to & \frac{1}{2} x^{T} Q_{k} x + r_{k}^{T} x + s_{k} \leq 0, k = 1, \dots, m_{Q}, & (b) \\ l_{B} \leq B x \leq u_{B}, & (c) \\ l_{x} \leq x \leq u_{x}, & (d) \end{array}

(3)

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) \\ \frac{1}{2} x^{T} Q_{k} x + r_{k}^{T} x + s_{k} \leq 0, k = 1, \dots, m_{Q}, & (c) \\ l_{B} \leq B x \leq u_{B}, & (d) \\ l_{x} \leq x \leq u_{x}, & (e) \end{array}

(4)

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}

(5)

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 (e.g., by e04raf) and must not be changed between calls to the NAG optimization modelling suite.

2: $nclin$ – Integer Input

On entry:

m_{B}

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

B

) in this block.

nclin = 0

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

Constraints:

$nclin \geq 0$ ;
if $idlc > 0$ , $nclin = 1$ .

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

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

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: $nnzb$ – Integer Input

On entry: nnzb gives the number of nonzeros in matrix

B

Constraint:

nnzb \geq 0

6: $irowb (nnzb)$ – Integer array Input

7: $icolb (nnzb)$ – Integer array Input

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

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

is the current number of decision variables in the model. 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: $idlc$ – Integer Input/Output

On entry: if

idlc = 0

, a new block of linear constraints is added to the model; otherwise,

idlc > 0

refers to the number of an existing linear constraint which will be replaced and nclin must be set to one.

Constraint:

idlc \geq 0

On exit: if

idlc = 0

, the number of the last linear constraint added. By definition, it is the number of linear constraints already defined plus nclin. Otherwise,

idlc > 0

stays unchanged.

10: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

- 1

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

- 1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

- 1

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

- 1

is recommended. 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 properly initialized or it has been corrupted.

$ifail = 2$: The problem cannot be modified right now, the solver is running.

$ifail = 4$: On entry, $idlc = ⟨ value ⟩$ .
Constraint: $idlc \geq 0$ .

On entry, $idlc = ⟨ value ⟩$ .
The given idlc does not match with any existing linear constraint.
The maximum idlc is $⟨ value ⟩$ .

$ifail = 6$: On entry, $idlc = ⟨ value ⟩$ and $nclin = ⟨ value ⟩$ .
Constraint: If $idlc > 0$ , $nclin = 1$ .

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

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

$ifail = 8$: 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 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

e04rjf is not threaded in any implementation.

9 Further Comments

9.1 Internal Changes

Internal changes have been made to this routine as follows:

At Mark 27.1: Previously, it was not possible to define more than one block of the linear constraints, modify the constraints or to edit the model once a solver had been called. These limitations have been removed and the associated error codes were removed. $ifail = 4$ and $ifail = 6$ have been extended to reflect the new use of idlc.

For details of all known issues which have been reported for the NAG Library please refer to the Known Issues.

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 e04raf for links to further examples in this suite.

e04rj: FL CL CPP AD

NAG FL Interfacee04rjf (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

9.1 Internal Changes

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
e04rjf (handle_set_linconstr)