NAG CL Interface
e04rhc (handle_​set_​simplebounds)

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1 Purpose

e04rhc is a part of the NAG optimization modelling suite and sets bounds on the variables of the problem.

2 Specification

#include <nag.h>
void  e04rhc (void *handle, Integer nvar, const double bl[], const double bu[], NagError *fail)
The function may be called by the names: e04rhc or nag_opt_handle_set_simplebounds.

3 Description

After the handle has been initialized (e.g., e04rac has been called), e04rhc may be used to define the variable bounds lxxux of the problem. If the bounds have already been defined, they will be overwritten. Individual bounds may also be set by e04tdc.
This will typically be used for problems, such as:
Linear Programming (LP)
minimize xn cTx   (a) subject to   lBBxuB,   (b) lxxux ,   (c) (1)
Quadratic Programming (QP)
minimize xn 12 xTHx + cTx   (a) subject to lBBxuB,   (b) lxxux,   (c) (2)
Nonlinear Programming (NLP)
minimize xn f(x)   (a) subject to lgg(x)ug,   (b) lBBxuB,   (c) lxxux,   (d) (3)
or linear Semidefinite Programming (SDP)
minimize xn cTx   (a) subject to   i=1 n xi Aik - A0k 0 ,  k=1,,mA ,   (b) lBBxuB,   (c) lxxux,   (d) (4)
where lx and ux 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. In particular, the jth variable may be fixed by setting lj=uj. If certain bounds are not present, the associated elements of lx or ux may be set to special values that are treated as - or +. 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.
See Section 4.1 in the E04 Chapter Introduction for more details about the NAG optimization modelling suite.

4 References

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

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: nvar Integer Input
On entry: n, the current number of decision variables x in the model.
3: bl[nvar] const double Input
4: bu[nvar] const double Input
On entry: lx, bl and ux, bu define lower and upper bounds on the variables, respectively. To fix the jth variable, set bl[j-1]=bu[j-1]=β, where |β|<bigbnd. To specify a nonexistent lower bound (i.e., lj=-), set bl[j-1]-bigbnd; to specify a nonexistent upper bound (i.e., uj=), set bu[j-1]bigbnd.
  • bl[j-1]bu[j-1], for j=1,2,,nvar;
  • bl[j-1]<bigbnd, for j=1,2,,nvar;
  • bu[j-1]>-bigbnd, for j=1,2,,nvar.
5: 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

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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 properly initialized 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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
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.
The problem cannot be modified right now, the solver is running.
On entry, nvar=value, expected value=value.
Constraint: nvar must match the current number of variables of the model in the handle.

7 Accuracy

Not applicable.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
e04rhc is not threaded in any implementation.

9 Further Comments

9.1 Internal Changes

Internal changes have been made to this function as follows:
For details of all known issues which have been reported for the NAG Library please refer to the Known Issues.

10 Example

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 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 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 Y^ denote the partially filled in m×k matrix with the valid responses on (i,j)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 ·*) is minimal.
minimizeY Y* subject to   Yij = Y^ij  for all  (i,j)Ω.  
This is equivalent to
minimize W1, W2, Y trace (W1)+ trace (W2) subject to   Yij = Y^ij  for all  (i,j)Ω , ( W1 Y YT W2 ) 0  
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
Y^ = ( * * * * * 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 * * )  
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 = ( 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 ) .  
The example also turns on monitor mode of 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 function is stopped by setting inform=0 within the monitor step.
See also e04rac for links to further examples in the suite.

10.1 Program Text

Program Text (e04rhce.c)

10.2 Program Data

Program Data (e04rhce.d)

10.3 Program Results

Program Results (e04rhce.r)