void	e04rfc (void handle, Integer nnzc, const Integer idxc[], const double c[], Integer nnzh, const Integer irowh[], const Integer icolh[], const double h[], NagError fail)

The function may be called by the names: e04rfc or nag_opt_handle_set_quadobj.

3 Description

After the initialization function e04rac has been called, e04rfc may be used to define the objective function of the problem as a quadratic function

c^{T} x + \frac{1}{2} x^{T} H x

or a sparse linear function

c^{T} x

unless the objective function has already been defined by another function in the suite. This objective function will typically be used for 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 problems (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)

or for semidefinite programming problems 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}

(3)

The matrix

H

is a sparse symmetric

n

n

matrix. It does not need to be positive definite. 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 by e04rac and must not be changed between calls to the NAG optimization modelling suite.

2: $nnzc$ – Integer Input

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

c

nnzc = 0

c

is considered to be zero and the arrays idxc and c will not be referenced and may be NULL.

Constraint:

nnzc \geq 0

3: $idxc [nnzc]$ – const Integer Input

4: $c [nnzc]$ – const double Input

On entry: the nonzero elements of the sparse vector

c

idxc [i - 1]

must contain the index of

c [i - 1]

in the vector, for

i = 1, 2, \dots, nnzc

. The elements are stored in ascending order. Note that

n

, the number of variables in the problem, was set in nvar during the initialization of the handle by e04rac.

Constraints:

$1 \leq idxc [i - 1] \leq n$ , for $i = 1, 2, \dots, nnzc$ ;
$idxc [i - 1] < idxc [i]$ , for $i = 1, 2, \dots, nnzc - 1$ .

5: $nnzh$ – Integer Input

On entry: the number of nonzero elements in the upper triangle of the matrix

H

nnzh = 0

, the matrix

H

is considered to be zero, the objective function is linear and irowh, icolh and h will not be referenced and may be NULL.

Constraint:

nnzh \geq 0

6: $irowh [nnzh]$ – const Integer Input

7: $icolh [nnzh]$ – const Integer Input

8: $h [nnzh]$ – const double Input

On entry: arrays irowh, icolh and h store the nonzeros of the upper triangle of the matrix

H

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

h_{i j} = h [l - 1]

where

i = irowh [l - 1]

j = icolh [l - 1]

, for

l = 1, 2, \dots, nnzh

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

Constraint:

1 \leq irowh [l - 1] \leq icolh [l - 1] \leq n

, for

l = 1, 2, \dots, nnzh

9: $fail$ – 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_ALREADY_DEFINED: The objective function has already been defined.
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 initialized by e04rac or it has been corrupted.
NE_INT: On entry, $nnzc = 〈value〉$ .
Constraint: $nnzc \geq 0$ .

On entry, $nnzh = 〈value〉$ .
Constraint: $nnzh \geq 0$ .
NE_INTARR: On entry, $i = 〈value〉$ , $idxc [i - 1] = 〈value〉$ and $n = 〈value〉$ .
Constraint: $1 \leq idxc [i - 1] \leq n$ .
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, $i = 〈value〉$ , $icolh [i - 1] = 〈value〉$ and $n = 〈value〉$ .
Constraint: $1 \leq icolh [i - 1] \leq n$ .

On entry, $i = 〈value〉$ , $irowh [i - 1] = 〈value〉$ and $icolh [i - 1] = 〈value〉$ .
Constraint: $irowh [i - 1] \leq icolh [i - 1]$ (elements within the upper triangle).

On entry, $i = 〈value〉$ , $irowh [i - 1] = 〈value〉$ and $n = 〈value〉$ .
Constraint: $1 \leq irowh [i - 1] \leq n$ .

On entry, more than one element of h has row index $〈value〉$ and column index $〈value〉$ .
Constraint: each element of h 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, $i = 〈value〉$ , $idxc [i - 1] = 〈value〉$ and $idxc [i] = 〈value〉$ .
Constraint: $idxc [i - 1] < idxc [i]$ (ascending order).
NE_PHASE: The problem cannot be modified in this phase any more, the solver has already been called.

7 Accuracy

Not applicable.

8 Parallelism and Performance

e04rfc is not threaded in any implementation.

9 Further Comments

None.

10 Example

This example demonstrates how to use nonlinear semidefinite programming to find a nearest correlation matrix satisfying additional requirements. This is a viable alternative to functions g02aac, g02abc, g02ajc or g02anc as it easily allows you to add further constraints on the correlation matrix. In this case a problem with a linear matrix inequality and a quadratic objective function is formulated to find the nearest correlation matrix in the Frobenius norm preserving the nonzero pattern of the original input matrix. However, additional box bounds (e04rhc) or linear constraints (e04rjc) can be readily added to further bind individual elements of the new correlation matrix or new matrix inequalities (e04rnc) to restrict its eigenvalues.

The problem is as follows (to simplify the notation only the upper triangular parts are shown). To a given

m

m

symmetric input matrix

G

G = (\begin{matrix} g_{11} & \dots & g_{1 m} \\ ⋱ & ⋮ \\ g_{m m} \end{matrix})

find correction terms

x_{1}, \dots, x_{n}

which form symmetric matrix

\bar{G}

\bar{G} = (\begin{matrix} {\bar{g}}_{11} & {\bar{g}}_{12} & \dots & {\bar{g}}_{1 m} \\ {\bar{g}}_{22} & \dots & {\bar{g}}_{2 m} \\ ⋱ & ⋮ \\ {\bar{g}}_{m m} \end{matrix}) = (\begin{matrix} 1 & g_{12} + x_{1} & g_{13} + x_{2} & \dots & g_{1 m} + x_{i} \\ 1 & g_{23} + x_{3} \\ 1 & ⋮ \\ ⋱ \\ 1 & g_{m - 1 m} + x_{n} \\ 1 \end{matrix})

so that the following requirements are met:

(a)It is a correlation matrix, i.e., symmetric positive semidefinite matrix with a unit diagonal. This is achieved by the way

\bar{G}

is assembled and by a linear matrix inequality

\begin{array}{l} \bar{G} = & x_{1} (\begin{matrix} 0 & 1 & 0 & \dots & 0 \\ 0 & 0 & \dots & 0 \\ 0 & \dots & 0 \\ ⋱ & ⋮ \\ 0 \end{matrix}) + x_{2} (\begin{matrix} 0 & 0 & 1 & \dots & 0 \\ 0 & 0 & \dots & 0 \\ 0 & \dots & 0 \\ ⋱ & ⋮ \\ 0 \end{matrix}) + x_{3} (\begin{matrix} 0 & 0 & 0 & \dots & 0 \\ 0 & 1 & \dots & 0 \\ 0 & \dots & 0 \\ ⋱ & ⋮ \\ 0 \end{matrix}) + \dots \\ + x_{n} (\begin{matrix} 0 & \dots & 0 & 0 & 0 \\ ⋱ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 \\ 0 & 1 \\ 0 \end{matrix}) - (\begin{matrix} - 1 & - g_{12} & - g_{13} & \dots & - g_{1 m} \\ - 1 & - g_{23} & \dots & - g_{2 m} \\ - 1 & \dots & - g_{3 m} \\ ⋱ & ⋮ \\ - 1 \end{matrix}) ⪰ 0 . \end{array}

(b) $\bar{G}$ is nearest to $G$ in the Frobenius norm, i.e., it minimizes the Frobenius norm of the difference which is equivalent to:
$minimize \frac{1}{2} \sum_{i \neq j} {({\bar{g}}_{i j} - g_{i j})}^{2} = \sum_{i = 1}^{n} x_{i}^{2} .$
(c) $\bar{G}$ preserves the nonzero structure of $G$ . This is met by defining $x_{i}$ only for nonzero elements $g_{i j}$ .

For the input matrix

G = (\begin{matrix} 2 & - 1 & 0 & 0 \\ - 1 & 2 & - 1 & 0 \\ 0 & - 1 & 2 & - 1 \\ 0 & 0 & - 1 & 2 \end{matrix})

the result is

\bar{G} = (\begin{array}{r} 1.0000 & - 0.6823 & 0.0000 & 0.0000 \\ - 0.6823 & 1.0000 & - 0.5344 & 0.0000 \\ 0.0000 & - 0.5344 & 1.0000 & - 0.6823 \\ 0.0000 & 0.0000 & - 0.6823 & 1.0000 \end{array}) .

See also Section 10 in e04rac for links to further examples in the suite.

Interfaces: FL CL AD

NAG CL Interface Introduction

E04 (Opt) Chapter Contents

E04 (Opt) Chapter Introduction

e04rf: FL CL AD

NAG CL Interfacee04rfc (handle_​set_​quadobj)

▸▿ 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
e04rfc (handle_set_quadobj)