nag_opt_handle_set_simplebounds (e04rhc) : NAG Library, Mark 26

nag_opt_handle_set_simplebounds (e04rhc) is a part of the NAG optimization modelling suite and defines bounds on the variables of the problem.

After the initialization routine nag_opt_handle_init (e04rac) has been called, nag_opt_handle_set_simplebounds (e04rhc) may be used to define the variable bounds

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

of the problem unless the bounds 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

l_{x}

and

u_{x}

are

n

-dimensional vectors. Note that upper and lower bounds are specified for all the variables. This form allows full generality in specifying various types of constraint. If certain bounds are not present, the associated elements of

l_{x}

u_{x}

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.

Candes E and Recht B (2009) Exact matrix completion via convex optimization Foundations of Computation Mathematics (Volume 9) 717–772

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:

n

, the number of decision variables

x

in the problem. It must be unchanged from the value set during the initialization of the handle by nag_opt_handle_init (e04rac).

On entry:

l_{x}

, bl and

u_{x}

, bu define lower and upper bounds on the variables, respectively. To specify a nonexistent lower bound (i.e.,

l_{j} = - \infty

), set

bl [j - 1] \leq - bigbnd

; to specify a nonexistent upper bound (i.e.,

u_{j} = \infty

), set

bu [j - 1] \geq bigbnd

. Fixing of the variables is not allowed in this release, however, this limitation will be removed in a future release.

Constraints:

$bl [j - 1] < bu [j - 1]$ , for $j = 1, 2, \dots, nvar$ ;
$bl [j - 1] < bigbnd$ , for $j = 1, 2, \dots, nvar$ ;
$bu [j - 1] > - bigbnd$ , for $j = 1, 2, \dots, nvar$ .

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.

Variable bounds have 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] < 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.

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.

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,

nvar = 〈value〉

, expected

value = 〈value〉

.
Constraint: nvar must match the value given during initialization of handle.

nag_opt_handle_set_simplebounds (e04rhc) is not threaded in any implementation.

There is a vast number of problems which can be reformulated as SDP. This example follows Candes and Recht (2009) to show how a rank minimization problem can be approximated by SDP. In addition, it demonstrates how to work with the monitor mode of nag_opt_handle_solve_pennon (e04svc).

The problem can be stated as follows: Let's have

m

respondents answering

k

questions where they express their preferences as a number between

0

and

1

or the question can be left unanswered. The task is to fill in the missing entries, i.e., to guess the unexpressed preferences. This problem falls into the category of matrix completion. The idea is to choose the missing entries to minimize the rank of the matrix as it is commonly believed that only a few factors contribute to an individual's tastes or preferences.

Rank minimization is in general NP-hard but it can be approximated by a heuristic, minimizing the nuclear norm of the matrix. The nuclear norm of a matrix is the sum of its singular values. A rank deficient matrix must have (several) zero singular values. Given the fact that the singular values are always non-negative, a minimization of the nuclear norm has the same effect as

ℓ_{1}

norm in compress sensing, i.e., it encourages many singular values to be zero and thus it can be considered as a heuristic for the original rank minimization problem.

Let

\hat{Y}

denote the partially filled in

m \times k

matrix with the valid responses on

(i, j) \in Ω

positions. We are looking for

Y

of the same size so that the valid responses are unchanged and the nuclear norm (denoted here as

{‖\cdot‖}_{*}

) is minimal.

\begin{matrix} \underset{Y}{minimize} & {‖Y‖}_{*} \\ subject to & Y_{i j} = {\hat{Y}}_{i j} for all (i, j) \in Ω . \end{matrix}

This is equivalent to

\begin{matrix} \underset{W_{1}, W_{2}, Y}{minimize} & trace (W_{1}) + trace (W_{2}) \\ subject to & Y_{i j} = {\hat{Y}}_{i j} for all (i, j) \in Ω \\ (\begin{matrix} W_{1} & Y \\ Y^{T} & W_{2} \end{matrix}) ⪰ 0 \end{matrix}

which is the linear semidefinite problem solved in this example, see Candes and Recht (2009) and the references therein for details.

This example has

m = 15

respondents and

k = 6

answers. The obtained answers are

\hat{Y} = (\begin{matrix} * & * & * & * & * & 0.4 \\ 0.6 & 0.4 & 0.8 & * & * & * \\ * & * & 0.8 & * & 0.2 & * \\ 0.8 & 0.2 & * & * & * & * \\ * & 0.4 & * & 0.0 & * & 0.2 \\ 0.4 & * & * & 0.2 & * & 0.2 \\ * & 0.8 & 0.2 & 0.6 & * & * \\ * & * & 0.2 & * & * & * \\ * & 0.4 & * & 0.6 & 0.0 & * \\ * & * & 0.4 & * & * & * \\ * & * & 0.2 & 0.2 & 0.4 & 0.4 \\ * & * & * & * & 1.0 & 0.8 \\ 1.0 & * & 0.2 & * & * & 0.6 \\ * & * & * & * & * & 0.2 \\ 0.6 & * & 0.2 & 0.4 & * & * \end{matrix})

where

*

denotes missing entries (

- 1.0

is used instead in the data file). The obtained matrix has rank

4

and it is shown below printed to

1

-digit accuracy:

Y = (\begin{matrix} 0.5 & 0.3 & 0.2 & 0.2 & 0.4 & 0.4 \\ 0.6 & 0.4 & 0.8 & 0.2 & 0.3 & 0.4 \\ 0.4 & 0.3 & 0.8 & 0.0 & 0.2 & 0.2 \\ 0.8 & 0.2 & 0.3 & 0.4 & 0.3 & 0.4 \\ 0.0 & 0.4 & 0.2 & 0.0 & 0.2 & 0.2 \\ 0.4 & 0.1 & 0.2 & 0.2 & 0.1 & 0.2 \\ 0.6 & 0.8 & 0.2 & 0.6 & 0.2 & 0.4 \\ 0.1 & 0.1 & 0.2 & 0.0 & 0.0 & 0.1 \\ 0.6 & 0.4 & 0.1 & 0.6 & 0.0 & 0.3 \\ 0.2 & 0.1 & 0.4 & 0.0 & 0.1 & 0.1 \\ 0.5 & 0.3 & 0.2 & 0.2 & 0.4 & 0.4 \\ 0.7 & 0.4 & 0.3 & 0.0 & 1.0 & 0.8 \\ 1.0 & 0.3 & 0.2 & 0.5 & 0.5 & 0.6 \\ 0.2 & 0.1 & 0.1 & 0.1 & 0.2 & 0.2 \\ 0.6 & 0.3 & 0.2 & 0.4 & 0.2 & 0.3 \end{matrix}) .

The example also turns on monitor mode of nag_opt_handle_solve_pennon (e04svc), there is a time limit introduced for the solver which is being checked at the end of every outer iteration. If the time limit is reached, the routine is stopped by setting

inform = 0

within the monitor step.

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

NAG Library Function Document

nag_opt_handle_set_simplebounds (e04rhc)

▸▿ 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_simplebounds (e04rhc)

▸▿ 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_simplebounds (e04rhc)