nag_opt_handle_set_linconstr (e04rjc) : NAG Library, Mark 26

nag_opt_handle_set_linconstr (e04rjc) is a part of the NAG optimization modelling suite and defines the block of linear constraints of the problem.

After the initialization function nag_opt_handle_init (e04rac) has been called, nag_opt_handle_set_linconstr (e04rjc) 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 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}

(1)

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}

(2)

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}

(3)

or semidefinite programming with bilinear matrix inequalities (BMI-SDP)

\begin{array}{l} \underset{x \in ℝ^{n}}{minimize} & \frac{1}{2} x^{T} H x + c^{T} x & (a) \\ subject to & \sum_{i,j = 1}^{n} x_{i} x_{j} Q_{i j}^{k} + \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

of the solvers in the suite, nag_opt_handle_solve_ipopt (e04stc) and nag_opt_handle_solve_pennon (e04svc). 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.

See nag_opt_handle_init (e04rac) for more details.

On entry: the handle to the problem. It needs to be initialized by nag_opt_handle_init (e04rac) and must not be changed.

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 and may be NULL.

Constraint:

nclin \geq 0

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 $j = 1, 2, \dots, nclin$ .

On entry: nnzb gives the number of nonzeros in matrix

B

Constraint: if

nclin > 0

nnzb > 0

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 nag_opt_handle_init (e04rac). 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

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

The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.

A set of linear constraints has already been defined.

On entry, argument

〈value〉

had an illegal value.

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

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 nag_opt_handle_init (e04rac) or it has been corrupted.

On entry,

nclin = 〈value〉

.
Constraint:

nclin \geq 0

On entry,

nnzb = 〈value〉

.
Constraint:

nnzb > 0

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.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.

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.

Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

The problem cannot be modified in this phase any more, the solver has already been called.

On entry,

idlc = 〈value〉

.
Constraint:

idlc = 0

nag_opt_handle_set_linconstr (e04rjc) is not threaded in any implementation.

This example demonstrates how to use the MPS file reader nag_opt_miqp_mps_read (e04mxc) and this suite of functions to define and solve a QP problem. nag_opt_miqp_mps_read (e04mxc) uses a different output format to the one required by nag_opt_handle_set_linconstr (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 nag_opt_handle_set_linconstr (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 Section 10 in nag_opt_handle_init (e04rac) for links to further examples in this suite.

NAG Library Function Document

nag_opt_handle_set_linconstr (e04rjc)

▸▿ 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 Library Function Documentnag_opt_handle_set_linconstr (e04rjc)

▸▿ 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 Library Function Document

nag_opt_handle_set_linconstr (e04rjc)