e04rjc 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

#include <nag.h>

void	e04rjc (void handle, Integer nclin, const double bl[], const double bu[], Integer nnzb, const Integer irowb[], const Integer icolb[], const double b[], Integer idlc, NagError *fail)

The function may be called by the names: e04rjc or nag_opt_handle_set_linconstr.

3 Description

After the handle has been initialized (e.g., e04rac has been called), e04rjc 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 function 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 e04rjc by referring to its idlc. An individual coefficient

b_{ij}

of the matrix

B

can be set or modified by e04tjc and bounds of a single constraint can be set or modified by e04tdc. Note that it is also possible to temporarily disable and enable individual constraints in the model by e04tcc and e04tbc, 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 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: $nclin$ – 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 and may be NULL.

Constraints:

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

3: $bl [nclin]$ – const double Input

4: $bu [nclin]$ – const double 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 - 1] = bu [j - 1] = β

, where

| β | < bigbnd

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

l_{j} = - \infty

), set

bl [j - 1] \leq - bigbnd

; to specify a nonexistent upper bound, set

bu [j - 1] \geq bigbnd

Constraints:

$bl [j - 1] \leq bu [j - 1]$ , for $j = 1, 2, \dots, nclin$ ;
$bl [j - 1] < bigbnd$ , for $j = 1, 2, \dots, nclin$ ;
$bu [j - 1] > - bigbnd$ , for $j = 1, 2, \dots, nclin$ ;
if $bl [j - 1] = bu [j - 1]$ , $| bl [j - 1] | < bigbnd$ , 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: $irowb [nnzb]$ – const Integer Input

7: $icolb [nnzb]$ – const Integer Input

8: $b [nnzb]$ – const double 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 - 1]

where

i = irowb [l - 1]

and

j = icolb [l - 1]

, for

l = 1, 2, \dots, nnzb

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

Constraint:

1 \leq irowb [l - 1] \leq nclin

1 \leq icolb [l - 1] \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: $fail$ – 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_BOUND: On entry, $j = ⟨ value ⟩$ , $bl [j - 1] = ⟨ value ⟩$ , $bigbnd = ⟨ value ⟩$ .
Constraint: $bl [j - 1] < bigbnd$ .

On entry, $j = ⟨ value ⟩$ , $bl [j - 1] = ⟨ value ⟩$ and $bu [j - 1] = ⟨ value ⟩$ .
Constraint: $bl [j - 1] \leq bu [j - 1]$ .

On entry, $j = ⟨ value ⟩$ , $bu [j - 1] = ⟨ value ⟩$ , $bigbnd = ⟨ value ⟩$ .
Constraint: $bu [j - 1] > - bigbnd$ .
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: 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: $nnzb \geq 0$ .
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, $i = ⟨ value ⟩$ , $icolb [i - 1] = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $1 \leq icolb [i - 1] \leq n$ .

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

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.
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 right now, the solver is running.
NE_REF_MATCH: 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 $⟨ value ⟩$ .

7 Accuracy

Not applicable.

8 Parallelism and Performance

e04rjc is not threaded in any implementation.

9 Further Comments

9.1 Internal Changes

Internal changes have been made to this function 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. $fail . code =$ NE_REF_MATCH and $fail . code =$ NE_INT 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 e04mxc and this suite of functions to define and solve a QP problem. e04mxc uses a different output format to the one required by e04rjc, 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 e04rjc 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 e04rac for links to further examples in this suite.

e04rj: FL CL CPP AD

NAG CL Interfacee04rjc (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 CL Interface
e04rjc (handle_set_linconstr)