NAG CL Interface
e04mtc (handle_solve_lp_ipm)
Note: this function uses optional parameters to define choices in the problem specification and in the details of the algorithm. If you wish to use default
settings for all of the optional parameters, you need only read Sections 1 to 10 of this document. If, however, you wish to reset some or all of the settings please refer to Section 11 for a detailed description of the algorithm and to Section 12 for a detailed description of the specification of the optional parameters.
1
Purpose
e04mtc is a solver from the NAG optimization modelling suite for largescale Linear Programming (LP) problems based on an interior point method (IPM).
2
Specification
The function may be called by the names: e04mtc or nag_opt_handle_solve_lp_ipm.
3
Description
e04mtc solves a largescale linear optimization problem in the following form:
where
$n$ is the number of decision variables and
$m$ is the number of linear constraints. Here
$c$,
$x$,
${l}_{x}$,
${u}_{x}$ are
$n$dimensional vectors,
$A$ is an
$m\times n$ sparse matrix and
${l}_{A}$,
${u}_{A}$ are
$m$dimensional vectors.
e04mtc implements two algorithmic variants of the interior point method for solving linear optimization problems: the infeasible PrimalDual interior point method and homogeneous SelfDual interior point method. In general, the SelfDual algorithm has a slightly higher price per iteration, however, it is able to declare infeasibility or unboundness of the problem, whereas the PrimalDual relies, in this case, on heuristics. For a detailed description of both algorithms see
Section 11. The algorithm is chosen by the
${\mathbf{LPIPM\; Algorithm}}$, the default is PrimalDual.
e04mtc solves linear programming problems stored as a handle. The handle points to an internal data structure which defines the problem and serves as a means of communication for functions in the NAG optimization modelling suite. First, the problem handle is initialized by calling
e04rac. Then some of the functions
e04rec,
e04rfc,
e04rhc or
e04rjc may be called to formulate the objective function, bounds of the variables, and the block of linear constraints, respectively. Once the problem is fully set, the handle may be passed to the solver. When the handle is not needed anymore,
e04rzc should be called to destroy it and deallocate the memory held within. See
Section 4.1 in the
E04 Chapter Introduction for more details about the NAG optimization modelling suite.
The solver method can be modified by various optional parameters (see
Section 12) which can be set by
e04zmc and
e04zpc anytime between the initialization of the handle by
e04rac and a call to the solver. Once the solver has finished, options may be modified for the next solve. The solver may be called repeatedly with various optional parameters.
The optional parameter ${\mathbf{Task}}$ may be used to switch the problem to maximization or to ignore the objective function and find only a feasible point.
Several options may have significant impact on the performance of the solver. Even if the defaults were chosen to suit the majority of problems, it is recommended to experiment in order to find the most suitable set of options for a particular problem, see
Sections 11 and
12 for further details.
3.1
Structure of the Lagrangian Multipliers
The algorithm works internally with estimates of both the decision variables, denoted by
$x$, and the Lagrangian multipliers (dual variables), denoted by
$u$. The multipliers
$u$ are stored in the array
u and conform to the structure of the constraints.
If the simple bounds have been defined (
e04rhc was successfully called), the first
$2n$ elements of
u belong to the corresponding Lagrangian multipliers, interleaving a multiplier for the lower and the upper bound for each
${x}_{i}$. If any of the bounds were set to infinity, the corresponding Lagrangian multipliers are set to
$0$ and may be ignored.
Similarly, the following
$2m$ elements of
u belong to multipliers for the linear constraints (if
e04rjc has been successfully called). The organization is the same, i.e., the multipliers for each constraint for the lower and upper bounds are alternated and zeros are used for any missing (infinite bound) constraints.
Some solvers merge multipliers for both lower and upper inequality into one element whose sign determines the inequality. Negative multipliers are associated with the upper bounds and positive with the lower bounds. An equivalent result can be achieved with this storage scheme by subtracting the upper bound multiplier from the lower one. This is also consistent with equality constraints.
4
References
Andersen E D, Gondzio J, Mészáros C and Xu X (1996) Implementation of interior point methods for large scale linear programming HEC/Université de Genève
Colombo M and Gondzio J (2008) Further development of multiple centrality correctors for interior point methods Computational Optimization and Algorithms 41(3) 277–305
Goldfard D and Scheinberg K (2004) A productform Cholesky factorization method for handling dense columns in interior point methods for linear programming Mathematical Programming 99(1) 1–34
Gondzio J (1996) Multiple centrality corrections in a primaldual method for linear programming Computational Optimization and Algorithms 6(2) 137–156
Gondzio J (2012) Interior point methods 25 years later European Journal of Operations Research 218(3) 587–601
Hogg J D and Scott J A (2011) HSL MA97: a bitcompatible multifrontal code for sparse symmetric systems RAL Technical Report. RALTR2011024
HSL (2011) A collection of Fortran codes for largescale scientific computation
http://www.hsl.rl.ac.uk/
Karypis G and Kumar V (1998) A fast and high quality multilevel scheme for partitioning irregular graphs SIAM J. Sci. Comput. 20(1) 359–392
Mészáros C (1996) The efficient implementation of interior point methods for
linear programming and their applications PhD Thesis Eötvös Loránd University of Science, Budapest
Nocedal J and Wright S J (2006) Numerical Optimization (2nd Edition) Springer Series in Operations Research, Springer, New York
Wright S W (1997) Primaldual interior point methods SIAM, Philadelphia
Xu X, Hung PF and Ye Y (1996) A simplified homogeneous and selfdual linear programming algorithm and its implementation Annals of Operations Research 62(1) 151–171
5
Arguments

1:
$\mathbf{handle}$ – void *
Input

On entry: the handle to the problem. It needs to be initialized (e.g., by
e04rac) and to hold a problem formulation compatible with
e04mtc. It
must not be changed between calls to the NAG optimization modelling suite.

2:
$\mathbf{nvar}$ – Integer
Input

On entry: $n$, the current number of decision variables $x$ in the model.

3:
$\mathbf{x}\left[{\mathbf{nvar}}\right]$ – double
Input/Output

On entry: the input of
x is reserved for future releases of the NAG Library and it is ignored at the moment.
On exit: the final values of the variables $x$.

4:
$\mathbf{nnzu}$ – Integer
Input

On entry: the dimension of array
u.
If
${\mathbf{nnzu}}=0$,
u will not be referenced; otherwise, it needs to match the dimension of constraints defined by
e04rhc and
e04rjc as explained in
Section 3.1.
Constraint:
${\mathbf{nnzu}}\ge 0$.

5:
$\mathbf{u}\left[{\mathbf{nnzu}}\right]$ – double
Input/Output

Note: if
${\mathbf{nnzu}}>0$,
u holds Lagrange multipliers (dual variables) for the bound constraints and linear constraints. If
${\mathbf{nnzu}}=0$,
u will not be referenced and may be
NULL.
On entry: the input of
u is reserved for future releases of the NAG Library and it is ignored at the moment.
On exit: the final values of the variables $u$.

6:
$\mathbf{rinfo}\left[100\right]$ – double
Output

On exit: error measures and various indicators of the algorithm (see
Section 11 for details) as given in the table below:
$0$ 
Value of the primal objective. 
$1$ 
Value of the dual objective. 
$2$ 
Flag indicating the system formulation used by the solver, $0$: augmented system, $1$: normal equation. 
$3$ 
Factorization type, $3$: Cholesky, $4$: Bunch–Parlett. 
$4$–$13$ 
PrimalDual specific information (will be $0$ if the SelfDual algorithm is chosen). 

$4$ 
Relative dual feasibility (optimality), see (9). 
$5$ 
Relative primal feasibility, see (10). 
$6$ 
Relative duality gap (complementarity), see (11). 
$7$ 
Average complementarity error $\mu $ (see Section 11.1). 
$8$ 
Centring parameter $\sigma $ (see Section 11.1). 
$9$ 
Primal step length. 
$10$ 
Dual step length. 
$11$–$13$ 
Reserved for future use. 

$14$–$23$ 
SelfDual specific information (will be $0$ if the PrimalDual algorithm is chosen). 

$14$ 
Relative primal infeasibility, see (12). 
$15$ 
Relative dual infeasibility, see (13). 
$16$ 
Relative duality gap, see (14). 
$17$ 
Accuracy, see (15). 
$18$ 
$\tau $, see (8). 
$19$ 
$\kappa $, see (8). 
$20$ 
Step length. 
$21$–$23$ 
Reserved for future use. 

$24$–$99$ 
Reserved for future use. 

7:
$\mathbf{stats}\left[100\right]$ – double
Output

On exit: solver statistics as given in the table below. Note that time statistics are provided only if
${\mathbf{Stats\; Time}}$ is set (the default is
$\mathrm{NO}$), the measured time is returned in seconds.
$0$ 
Number of iterations. 
$1$ 
Total number of centrality correction steps performed. 
$2$ 
Total number of iterative refinements performed. 
$3$ 
Value of the perturbation added to the diagonal in the normal equation formulation or on the zero block in the augmented system formulation. 
$4$ 
Total number of factorizations performed. 
$5$ 
Total time spent in the solver. 
$6$ 
Time spent in the presolve phase. 
$7$ 
Time spent in the last iteration. 
$8$ 
Total time spent factorizing the system matrix. 
$9$ 
Total time spent backsolving the system matrix. 
$10$ 
Total time spent in the multiple centrality correctors phase. 
$11$ 
Time spent in the initialization phase. 
$12$ 
Number of nonzeros in the system matrix. 
$13$ 
Number of nonzeros in the system matrix factor. 
$14$ 
Maximum error of the backsolve. 
$15$ 
Number of columns in $A$ considered dense by the solver. 
$16$ 
Maximum number of centrality corrector steps. 
$17$–$99$ 
Reserved for future use. 

8:
$\mathbf{monit}$ – function, supplied by the user
External Function

monit is provided to enable you to monitor the progress of the optimization. It is invoked at the end of every
$i$th iteration where
$i$ is given by the optional parameter
${\mathbf{LPIPM\; Monitor\; Frequency}}$ (the default is
$0$,
monit is not called).
monit may be specified as
NULLFN.
The specification of
monit is:

1:
$\mathbf{handle}$ – void *
Input

On entry: the handle to the problem as provided on entry to
e04mtc. It may be used to query the model during the solve, and extract the current approximation of the solution by
e04rxc.

2:
$\mathbf{rinfo}\left[100\right]$ – const double
Input

On entry: error measures and various indicators at the end of the current iteration as described in
rinfo.

3:
$\mathbf{stats}\left[100\right]$ – const double
Input

On entry: solver statistics at the end of the current iteration as described in
stats, however, elements
$2$,
$3$,
$5$,
$9$,
$10$,
$11$ and
$15$ refer to the quantities in the last iteration rather than accumulated over all iterations through the whole algorithm run.

4:
$\mathbf{comm}$ – Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to
monit.
 user – double *
 iuser – Integer *
 p – Pointer
The type Pointer will be
void *. Before calling
e04mtc you may allocate memory and initialize these pointers with various quantities for use by
monit when called from
e04mtc (see
Section 3.1.1 in the Introduction to the NAG Library CL Interface).

5:
$\mathbf{inform}$ – Integer *
Input/Output

On entry: a nonnegative value.
On exit: must be set to a value describing the action to be taken by the solver on return from
monit. Specifically, if the value is negative the solution of the current problem will terminate immediately with
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_USER_STOP; otherwise, computations will continue.

9:
$\mathbf{comm}$ – Nag_Comm *

The NAG communication argument (see
Section 3.1.1 in the Introduction to the NAG Library CL Interface).

10:
$\mathbf{fail}$ – 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 $\u27e8\mathit{\text{value}}\u27e9$ had an illegal value.
 NE_DIM_MATCH

On entry,
${\mathbf{nnzu}}=\u27e8\mathit{\text{value}}\u27e9$.
nnzu does not match the size of the Lagrangian multipliers for constraints.
The correct value is
$0$ for no constraints.
On entry,
${\mathbf{nnzu}}=\u27e8\mathit{\text{value}}\u27e9$.
nnzu does not match the size of the Lagrangian multipliers for constraints.
The correct value is either
$0$ or
$\u27e8\mathit{\text{value}}\u27e9$.
 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_INFEASIBLE

The problem was found to be primal infeasible.
The primal infeasibility was detected either during the presolve phase or by the SelfDual algorithm.
 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_MAYBE_INFEASIBLE

The problem seems to be primal or dual infeasible, the algorithm was stopped.
This error is returned if the internal heuristics detected the problem to be primal or dual infeasible. It is only raised by the PrimalDual algorithm. It is recommended to rerun the problem with the SelfDual algorithm to confirm the infeasibility.
 NE_NO_IMPROVEMENT

No progress, stopping early.
The solver predicted that it is unable to make further progress and stopped prematurely. This might be due to the scaling of the problem, its conditioning or numerical difficulties.
 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 is already being solved.
 NE_REF_MATCH

On entry,
${\mathbf{nvar}}=\u27e8\mathit{\text{value}}\u27e9$, expected
$\mathrm{value}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint:
nvar must match the current number of variables of the model in the
handle.
 NE_SETUP_ERROR

This solver does not support the model defined in the handle.
 NE_TOO_MANY_ITER

Maximum number of iterations exceeded.
 NE_UNBOUNDED

The problem was found to be dual infeasible.
The dual infeasibility or unboundness was detected during the presolve phase or by the SelfDual algorithm.
 NE_USER_STOP

User requested termination during a monitoring step.
 NW_NOT_CONVERGED

Suboptimal solution.
The solver predicted that it is unable to reach a better estimate of the solution. However, the error measures indicate that the point is a reasonable approximation.
7
Accuracy
The accuracy of the solution is determined by optional parameters ${\mathbf{LPIPM\; Stop\; Tolerance}}$ and ${\mathbf{LPIPM\; Stop\; Tolerance\; 2}}$.
If
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR on the final exit, the returned point satisfies Karush–Kuhn–Tucker (KKT) conditions to the requested accuracy (under the default settings close to
$\sqrt{\epsilon}$) and thus it is a good estimate of the solution. If
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NW_NOT_CONVERGED, some of the convergence conditions were not fully satisfied but the point is a reasonable estimate and still usable. Please refer to
Section 11.3 and the description of the particular options.
8
Parallelism and Performance
Background information to multithreading can be found in the
Multithreading documentation.
e04mtc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
e04mtc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the
Users' Note for your implementation for any additional implementationspecific information.
The solver can print information to give an overview of the problem and of the progress of the computation. The output may be sent to two independent streams (files) which are set by optional parameters ${\mathbf{Print\; File}}$ and ${\mathbf{Monitoring\; File}}$. Optional parameters ${\mathbf{Print\; Level}}$, ${\mathbf{Print\; Solution}}$ and ${\mathbf{Print\; Options}}$ determine the exposed level of detail. This allows, for example, a detailed log file to be generated while the condensed information is displayed on the screen.
By default (
${\mathbf{Print\; File}}=6$,
${\mathbf{Print\; Level}}=2$), six sections are printed to the standard output:
 Header
 Optional parameters list
 Problem statistics
 Iteration log
 Summary
 Solution
The iteration log varies depending on which algorithm has been selected to solve the problem (PrimalDual or SelfDual).
Header
The header is a message indicating the start of the solver. It should look like:

E04MT, Interior point method for LP problems

Optional parameters list
The list shows all options of the solver, each displayed on one line. The output contains the option name, its current value and an indicator for how it was set. The options unchanged from the default setting are noted by ‘d’, options you set are noted by ‘U’, and options reset by the solver are noted by ‘S’. Note that the output format is compatible with the file format expected by
e04zpc. The output might look as follows:
Lp Presolve = Yes * d
Lpipm Algorithm = Primaldual * d
Lpipm Centrality Correctors = 6 * U
Lpipm Iteration Limit = 100 * d
Problem statistics
If
${\mathbf{Print\; Level}}\ge 2$, statistics on the original and the presolved problems are printed. More detailed statistics as well as a list of the presolve operations are also printed for
${\mathbf{Print\; Level}}$ $3$ or above, for example:
Problem Statistics
No of variables 7
free (unconstrained) 0
bounded 7
No of lin. constraints 7
nonzeroes 41
Objective function Linear
Presolved Problem Measures
No of variables 13
free (unconstrained) 0
No of lin. constraints 7
nonzeroes 47
Iteration log
If ${\mathbf{Print\; Level}}\ge 2$, the solver prints the status of each iteration.
PrimalDual algorithm
If
${\mathbf{Print\; Level}}=2$, the output shows the iteration number (
$0$ represents the starting point), the current primal and dual objective value, KKT measures (optimality, feasibility and complementarity), average complementarity error
$\mu $, number of centrality correction steps (MCC) performed and a column for additional information (I). Note that all these values are also available in
rinfo and
stats. The output might look as follows:

it pobj  dobj  optim  feas  compl  mu  mcc  I

0 2.02532E+03 7.37272E+02 7.71E+00 4.91E+00 2.16E+00 4.4E+03
1 2.13398E+01 1.62136E+04 5.82E02 2.09E01 2.12E+01 4.7E+02 2
2 1.09237E+02 2.81254E+03 3.12E03 4.84E15 4.84E01 5.3E+01 0
3 2.10923E+02 6.07429E+02 4.61E04 4.99E14 3.70E02 7.8E+00 0
If
${\mathbf{Print\; Level}}=3$, the solver also prints for each iteration the primal and dual steps as well as the maximum error of the backsolves performed. The output takes the following form:

it pobj  dobj  optim  feas  compl  mu  pstep  dstep  errbs  mcc  I

0 2.02532E+03 7.37272E+02 7.71E+00 4.91E+00 2.16E+00 4.4E+03
1 2.13398E+01 1.62136E+04 5.82E02 2.09E01 2.12E+01 4.7E+02 9.57E01 9.92E01 1.54E12 2
2 1.09237E+02 2.81254E+03 3.12E03 4.84E15 4.84E01 5.3E+01 1.00E+00 9.46E01 3.02E12 0
3 2.10923E+02 6.07429E+02 4.61E04 4.99E14 3.70E02 7.8E+00 1.00E+00 8.53E01 7.66E11 0
SelfDual algorithm
If
${\mathbf{Print\; Level}}=2$, the output shows the iteration number (
$0$ represents the starting point), the current primal and dual objective value, convergence measures (primal infeasibility
(12), dual infeasibility
(13) and duality gap
(14)), the value of the additional variable
$\tau $ and the number of centrality correction steps performed. The output might look as follows:

it pobj  dobj  p.inf  d.inf  d.gap  tau  mcc  I

0 1.01907E+01 0.00000E+00 6.98E+02 1.12E+01 1.35E+01 1.0E+00
1 5.80391E+00 2.39478E+00 4.98E04 3.11E02 2.58E02 3.5E01 0
2 7.09323E+00 3.62789E+00 1.89E04 1.18E02 9.79E03 1.4E01 0
3 1.33628E+01 5.87563E+00 1.94E05 1.21E03 1.01E03 3.3E02 0
If
${\mathbf{Print\; Level}}=3$, the solver also prints for each iteration
${\rho}_{A}$ (15), the value of the variable
$\kappa $, the step size as well as the maximum error of the backsolves performed. The output might look as follows:

it pobj  dobj  p.inf  d.inf  d.gap  rhoa  tau  kappa  step  errbs  mcc  I

0 1.01907E+01 0.00000E+00 6.98E+02 1.12E+01 1.35E+01 1.0E+01
1 5.80391E+00 2.39478E+00 4.98E04 3.11E02 2.58E02 2.4E+00 3.5E01 1.0E+00 6.52E01 3.43E13 0
2 7.09323E+00 3.62789E+00 1.89E04 1.18E02 9.79E03 7.5E01 1.4E01 9.8E01 6.21E01 2.85E13 0
3 1.33628E+01 5.87563E+00 1.94E05 1.21E03 1.01E03 2.8E+00 3.3E02 7.9E01 8.97E01 9.31E13 0
Occasionally, when numerical instabilities are too big, the solver will restart the iteration and switch to an augmented system formulation. In such cases the letters RS will be printed in the information column (I).
If
${\mathbf{Print\; Level}}>3$, for both the PrimalDual and the SelfDual algorithms, each iteration produces more information that expands over several lines. This additional information contains:
 The method used (normal equation, augmented system);
 The centring parameter $\sigma $;
 The total number of iterative refinements performed;
 The number of iterative refinements performed in the centrality correction steps;
 The number of factorizations performed at the current iteration;
 The type of factorization performed (Cholesky, Bunch–Parlett);
 The value of the perturbation added to the diagonal in the normal equation formulation or on the zero block in the augmented system formulation;
 The total time spent in the iteration if ${\mathbf{Stats\; Time}}$ is not set to NO.
The output might look as follows:
 Details of Iteration 1 
method Normal Equation
sigma 6.72E02
iterative refinements 0
iterative refinements wmcc 0
factorizations 1
matrix type Cholesky
diagonal perturbation 0.00E+00
time iteration 0.02 sec
Summary
Once the solver finishes, a detailed summary is produced:

Status: converged, an optimal solution found

Final primal objective value 4.647531E+02
Final dual objective value 4.647531E+02
Absolute primal infeasibility 1.605940E15
Relative primal infeasibility 1.210247E16
Absolute dual infeasibility 4.272004E12
Relative dual infeasibility 6.068111E15
Absolute complementarity gap 9.387105E07
Relative complementarity gap 2.507021E10
Iterations 7
It starts with the status line of the overall result which matches the
fail value and is followed by the final primal and dual objective values as well as the error measures and iteration count.
Optionally, if
${\mathbf{Stats\; Time}}$ is set, the timings of the different parts of the algorithm are displayed. It might look as follows:
Timing
Total time 28.78 sec
Presolver 0.07 sec ( 0.2%)
Core 28.71 sec ( 99.8%)
Initialization 1.67 sec ( 5.8%)
Factorization 16.18 sec ( 56.5%)
Compute directions 5.29 sec ( 18.5%)
Weighted MCC 5.50 sec ( 19.2%)
Iterative refinement 0.24 sec ( 0.8%)
Solution
If
${\mathbf{Print\; Solution}}=\mathrm{X}$, the values of the primal variables and their bounds on the primary and secondary outputs. It might look as follows:
Primal variables:
idx Lower bound Value Upper bound
1 1.00000E02 1.00000E02 1.00000E02
2 1.00000E01 1.00000E01 1.50000E01
3 1.00000E02 3.00000E02 3.00000E02
4 4.00000E02 2.00000E02 2.00000E02
5 1.00000E01 6.74853E02 5.00000E02
6 1.00000E02 2.28013E03 inf
7 1.00000E02 2.34528E04 inf
If
${\mathbf{Print\; Solution}}=\mathrm{YES}$ or
$\mathrm{ALL}$, the values of the dual variables are also printed. It should look as follows:
Box bounds dual variables:
idx Lower bound Value Upper bound Value
1 1.00000E02 3.30098E01 1.00000E02 0.00000E+00
2 1.00000E01 1.43844E02 1.50000E01 0.00000E+00
3 1.00000E02 0.00000E+00 3.00000E02 9.09967E02
4 4.00000E02 0.00000E+00 2.00000E02 7.66124E02
5 1.00000E01 4.92258E12 5.00000E02 0.00000E+00
6 1.00000E02 2.42274E11 inf 0.00000E+00
7 1.00000E02 4.83752E12 inf 0.00000E+00
Linear constraints dual variables:
idx Lower bound Value Upper bound Value
1 1.30000E01 0.00000E+00 1.30000E01 1.43111E+00
2 inf 0.00000E+00 4.90000E03 4.07810E10
3 inf 0.00000E+00 6.40000E03 5.64870E10
4 inf 0.00000E+00 3.70000E03 1.25984E10
5 inf 0.00000E+00 1.20000E03 1.87338E11
6 9.92000E02 1.50098E+00 inf 0.00000E+00
7 3.00000E03 1.51661E+00 2.00000E03 0.00000E+00
10
Example
This example demonstrates how to use e04mtc to solve a small LP problem with the two algorithms implemented (PrimalDual and SelfDual). The solver is called twice on the same handle with different values of optional parameters.
We solve the following linear programming problem:
minimize
subject to the bounds
and the general constraints
10.1
Program Text
10.2
Program Data
10.3
Program Results
11
Algorithmic Details
This section contains the description of the underlying algorithms used in
e04mtc, which implements the infeasible PrimalDual and homogeneous SelfDual methods. For further details, see
Andersen et al. (1996),
Gondzio (2012),
Mészáros (1996) and
Wright (1997).
For simplicity, we consider the following primal linear programming formulation
where
$c$,
$x\in {\mathbb{R}}^{n}$,
$b\in {\mathbb{R}}^{m}$ and
$A\in {\mathbb{R}}^{m\times n}$ with full row rank. The dual formulation for
(2) is given by
where
$y$ and
$z$ denote the dual variables. Solutions of the primal
(2) and dual
(3) problem are connected by the strong duality theory (see for example,
Nocedal and Wright (2006)) and are characterized by the first order optimality conditions, the socalled Karush–Kuhn–Tucker (KKT) conditions, which are stated as follows:
where we define
$X$ and
$Z$ as the diagonal matrices with the elements
${x}_{i}$ and
${z}_{i}$, respectively, and
$e={(1,1,\dots ,1)}^{\mathrm{T}}$ as the
$n$vector of ones.
The underlying algorithm applies an iterative method to find an optimal solution
$({x}^{*},{y}^{*},{z}^{*})$ of the system
(4) employing variants of Newton's method and modifying the search directions and step lengths so that the nonnegative constraints are preserved at every iteration.
11.1
The Infeasibleinteriorpoint PrimalDual Algorithm
A direct solution of the nonlinear system of equations
(4) by Newton's method is impractical as it exhibits slow progress towards the solution. Instead, a sequence of the following
perturbed KKT conditions are solved so that the complementarity is driven to zero through the iteration sequence
Here $\mu $ is the average complementarity error at the current iteration $\mu ={x}^{\mathrm{T}}z/n$, called duality measure, and $\sigma \in (0,1)$, called centring parameter, is the reduction factor that we wish to achieve in the duality measure.
Each iteration of the PrimalDual algorithm makes one step of Newton's method applied to the perturbed first order optimality conditions
(5) with a given
$\sigma $ and
$\mu $. In particular, a Newton search direction is computed by solving a system of linear equations and a length of the step
$\alpha $ is determined so that the bounds
$(x,z)>0$ are not violated. The residual of
(4) and
$\mu $ define stopping criteria and the algorithm terminates once they are reduced to the requested accuracy, see
Section 11.3.
Given an
$x$,
$z\in {\mathbb{R}}_{+}^{n}$ and
$y\in {\mathbb{R}}^{m}$, Newton's direction is obtained by solving the following system:
where
denote the violations of the primal and the dual constraints, respectively. Primal and dual infeasibilities,
${r}_{b}$ and
${r}_{c}$, are reduced at the same rate
$(1\alpha )$, given a step size
$\alpha \in (0,1)$. The PrimalDual algorithm does not require feasibility of the solutions during the optimization process. Feasibility is attained during the process as optimality is approached.
Once the system is solved,
$\Delta x$ and
$\Delta z$ are used to compute the maximum step sizes in primal space
$\left({\alpha}_{P}\right)$ and dual space (
${\alpha}_{D}$) such that the nonnegativity of variables is preserved
where
$0<\rho <1$ is a reduction parameter close to
$1$, typically
$\rho \in (0.9,1)$. The next iterate is updated as follows:
Finally, the barrier parameter
$\mu $ is decreased by a given factor and the process is repeated until the stopping criteria (see
Section 11.3) or maximum number of iterations is reached.
11.1.1
The Barrier Method
Note that there is also another way to obtain the perturbed KKT conditions
(5). They can be derived starting from the primal formulation
(2) and replacing the nonnegativity constraints
$x\ge 0$ by a logarithmic barrier term
${\sum}_{j=1}^{n}\mathrm{log}{x}_{j}$ with a
barrier parameter $\tau >0$. This approach leads to the primal logarithmic barrier problem defined as
The Lagrangian formulation for
(7) is
and the first order optimality conditions for problem
(7) are
Finally, by introducing
$Z=\tau {X}^{\mathrm{1}}$ and
$z=Ze$ the same system of equations
(5) is formulated.
11.2
Homogeneous SelfDual Algorithm
A homogeneous SelfDual (HSD) embedding of the primal linear programming and its dual was proposed in
Xu et al. (1996). As its name suggest, the HSD and its dual are equivalent. SelfDual formulations embed the original problem
(2) in a larger linear programming problem such that the latter is primal and dual feasible, with known feasible points, and from which solution we can extract optimal solutions or certificates of infeasibility of the original problem.
We define the homogeneous and SelfDual linear feasibility (HLF) model as follows:
where
$\tau $ and
$\kappa $ are two additional variables. The model
(8) is a SelfDual linear programming problem with zero righthand side and a zero objective vector. If
$(\hat{x},\hat{\tau},\hat{y},\hat{z},\hat{\kappa})$ is a strictly complementarity solution for
(8), then if
$\hat{\tau}>0$, the linear programming problem has an optimal solution given by
and the duality gap is given by
${c}^{\mathrm{T}}{x}^{*}{b}^{\mathrm{T}}{y}^{*}=\hat{\kappa}/\hat{\tau}=0$. The homogeneous algorithm is an application of the PrimalDual method for the computation of a strictly complementarity solution to
(8).
Homogeneous and SelfDual interior point methods have several advantages besides an inherent ability to detect infeasibility (which improves the detection of divergence in PrimalDual algorithms). The HSD model includes the ease of finding a suitable starting point and it is generally more robust in the presence of free variables. However, some disadvantages need to be noted: HSD is larger than the original problem. In particular, it increases the number of linear equations solved per iteration by one, requiring an extra backsolve step, which make it slightly slower than the PrimalDual algorithm. Moreover, numerical experiments indicate that the required number of iterations on feasible problems might be slightly increased.
11.3
Stopping Criteria
11.3.1
Convergence – Optimal Termination
11.3.1.1
PrimalDual algorithm
The PrimalDual algorithm is stopped when the first order optimality conditions are satisfied to the requested accuracy. These conditions are relative primal and dual feasibility and duality gap, defined as:
 relative dual feasibility
 relative primal feasibility
 relative duality gap
Furthermore, final absolute primal and dual infeasibilities, and duality gap are also returned. Here ${\epsilon}_{1}$ may be set using ${\mathbf{LPIPM\; Stop\; Tolerance}}$ and the norm denotes the 2norm.
11.3.1.2
SelfDual algorithm
Similar to the PrimalDual algorithm, the homogeneous SelfDual algorithm is stopped when the following measures are satisfied to the requested accuracy.
 relative primal feasibility
 relative dual feasibility
 relative duality gap
which measure the relative reduction in the primal, dual and gap infeasibility, respectively. In addition, an extra measure is considered to quantify the accuracy in the objective function, which is given by
Here ${\epsilon}_{1}$ and ${\epsilon}_{2}$ may be set using ${\mathbf{LPIPM\; Stop\; Tolerance}}$ and ${\mathbf{LPIPM\; Stop\; Tolerance\; 2}}$, respectively.
Premature termination is triggered if the current iteration exhibits fast convergence and the optimality measures lie within a small range. In particular, the SelfDual algorithm is stopped if the above termination conditions are met within a small factor and $\tau >1000\kappa $. This measure is tracked after the first $10$ iterations.
11.3.2
Infeasibility/Unboundedness Detection
11.3.2.1
PrimalDual algorithm
The PrimalDual algorithm detects infeasible problems fairly reliably by using a set of heuristics. When several of these heuristics classify the problem as infeasible throughout a sufficient number of iterations, the algorithm is stopped.
Note that in order to obtain a certificate of infeasibility, the use of homogeneous SelfDual algorithm is recommended, see
Section 11.3.2.2.
11.3.2.2
SelfDual algorithm
The problem is concluded to be primal or dual infeasible if one of the following conditions hold:

1.Both the relative primal (12) and dual (13) feasibility of the HLF model (8) are satisfied and the value of $\tau $ satisfies

2.or if the following inequalities hold
Then the problem is declared dual infeasible if ${c}^{\mathrm{T}}x<0$ or primal infeasible otherwise.
11.3.3
Suboptimal Solution
The solver stops prematurely and reports suboptimal solution when it predicts that the current estimate of the solution will not be improved in subsequent iterations. In most cases the returned solution should be acceptable.
11.4
Solving the KKT System
The solution of the Newton system of equations
(6) is the most computationally costly operation. In practice, system
(6) is reduced to the
augmented system by eliminating
$\Delta z$ from the last block of equations
as follows:
where
and
$\Delta z={X}^{\mathrm{1}}(\mu eXZe){X}^{\mathrm{1}}Z\Delta x$. This is a system of
$m+n$ variables and constraints, symmetric and indefinite. Submatrix
$D$ is diagonal and positive definite.
The system
(16) can be reduced further by eliminating
$\Delta x$, to a positive definite system usually called
normal equations defined as
and
Typically, formulation
(17) is preferred for many problems as the system matrix can be factorized by a sparse Cholesky. However, this brings some wellknown disadvantages: Illconditioning of the system is often observed during the final stages of the algorithm, and free (unbounded) variables require certain modifications. If matrix
$A$ contains dense columns (columns with relatively many nonzeros), then
$A{D}^{2}{A}^{\mathrm{T}}$ has many nonzeros, which in turn makes the factorization expensive. On the other hand, solving the augmented system is usually slower, but it normally avoids the fillin caused by dense columns and can handle free variables directly in the formulation.
e04mtc can detect and handle dense columns effectively: depending on the number and the density of the ‘dense’ columns, the solver may either choose to directly use an augmented system formulation or to treat these columns separately in a productform Cholesky factorization as described in
Goldfard and Scheinberg (2004). It is also possible to manually override the automatic choice via the optional parameter
${\mathbf{LPIPM\; System\; Formulation}}$ and let the solver use a normal equations or an augmented system formulation.
Badly scaled optimal solutions may present numerical challenges, therefore, iterative refinement using mixedprecision is employed for reducing the roundoff errors produced during the solution of the system. When the condition number of the system $A{D}^{2}{A}^{\mathrm{T}}$ prevents the satisfactory use of iterative refinement, e04mtc switches automatically to an augmented system formulation, reporting RS (Restart) in the last column of the iteration log (I). Furthermore, e04mtc provides several scaling techniques to adjust the numerical characteristics of the problem data, see ${\mathbf{LPIPM\; Scaling}}$.
Finally, factorization of the system matrix can degrade sparsity, so the resulting fillin can be large, therefore, several ordering techniques are included to minimize it.
e04mtc uses Harwell packages MA97 (see
Hogg and Scott (2011) and
HSL (2011)) for the underlying sparse linear algebra factorization and MC68 approximate minimum degree algorithm, and METIS (
Karypis and Kumar (1998)) nested dissection algorithm for the ordering.
11.5
Weighted Multiple Centrality Correctors
As previously stated, the factorization of the system at every iteration usually accounts for most of the computation time, therefore, it is always desirable to reuse the factors if possible and to reduce the total number of iterations. An efficient computational method is obtained by splitting the computation of the Newton direction into two steps, namely the affinescaling direction and its correction step, called Mehrotra's predictorcorrector. However, Mehrotra's predictorcorrector technique aims to correct the affinedirection in a full step, which is considered an aggressive approach.
e04mtc implements a highorder method, called weighted multiple centrality correctors (WMCC), see
Gondzio (1996) and
Colombo and Gondzio (2008). This technique attempts to correct the affinedirection recursively as long as the step sizes increase at least by a fraction of a given aspiration level, up to a maximum number of times. The heuristic to determine the maximum number of WMCC is based on the ratio between the cost of the factorization and that of backsolving and, therefore, may lead to
nonrepeatable results. To avoid an undesired behaviour, you can fix the maximum number of WMCC with optional parameter
${\mathbf{LPIPM\; Centrality\; Correctors}}$.
11.6
Further Details
e04mtc includes an advance preprocessing phase (called presolve) to reduce the dimensions of the problem before passing it to the solver. The reduction in problem size generally improves the behaviour of the solver, shortening the total computation time. In addition, infeasibility may also be detected during preprocessing. The default behaviour of the presolve can be modified by ${\mathbf{LP\; Presolve}}$.
The initial point ${x}_{0}$ is always computed using heuristics. Effective warmstarting strategies for interior point methods are still subject to intensive academic research, and it is not available in this release.
12
Optional Parameters
Several optional parameters in e04mtc define choices in the problem specification or the algorithm logic. In order to reduce the number of formal arguments of e04mtc these optional parameters have associated default values that are appropriate for most problems. Therefore, you need only specify those optional parameters whose values are to be different from their default values.
The remainder of this section can be skipped if you wish to use the default values for all optional parameters.
The optional parameters can be changed by calling
e04zmc anytime between the initialization of the handle and the call to the solver. Modification of the optional parameters during intermediate monitoring stops is not allowed. Once the solver finishes, the optional parameters can be altered again for the next solve.
If any options are set by the solver (typically those with the choice of
$\mathrm{AUTO}$), their value can be retrieved by
e04znc. If the solver is called again, any such arguments are reset to their default values and the decision is made again.
The following is a list of the optional parameters available. A full description of each optional parameter is provided in
Section 12.1.
12.1
Description of the Optional Parameters
For each option, we give a summary line, a description of the optional parameter and details of constraints.
The summary line contains:
 the keywords;
 a parameter value,
where the letters $a$, $i$ and $r$ denote options that take character, integer and real values respectively;
 the default value, where the symbol $\epsilon $ is a generic notation for machine precision (see X02AJC).
All options accept the value $\mathrm{DEFAULT}$ to return single options to their default states.
Keywords and character values are case and white space insensitive.
This special keyword may be used to reset all optional parameters to their default values. Any value given with this keyword will be ignored.
Infinite Bound Size  $r$  Default $\text{}={10}^{20}$ 
This defines the ‘infinite’ bound $\mathit{bigbnd}$ in the definition of the problem constraints. Any upper bound greater than or equal to $\mathit{bigbnd}$ will
be regarded as $+\infty $ (and similarly any lower bound less than or equal to $\mathit{bigbnd}$ will be regarded as $\infty $). Note that a modification of this optional parameter does not influence constraints which have already been defined; only the constraints formulated after the change will be affected.
Constraint: ${\mathbf{Infinite\; Bound\; Size}}\ge 1000$.
LP Presolve  $a$  Default $=\mathrm{FULL}$ 
This parameter allows you to reduce the level of presolving of the problem or turn it off completely. If the presolver is turned off, the solver will try to handle the problem you have given. In such a case, the presence of fixed variables or linear dependencies in the constraint matrix can cause numerical instabilities to occur. In normal circumstances, it is recommended to use the full presolve which is the default.
Constraint: ${\mathbf{LP\; Presolve}}=\mathrm{FULL}$, $\mathrm{BASIC}$ or $\mathrm{NO}$.
LPIPM Algorithm  $a$  Default $=\mathrm{PRIMALDUAL}$ 
As described in
Section 11,
e04mtc implements the infeasible PrimalDual algorithm, see
Section 11.1, and the homogeneous SelfDual algorithm, see
Section 11.2. This parameter controls which one to use.
Constraint: ${\mathbf{LPIPM\; Algorithm}}=\mathrm{PRIMALDUAL}$, $\mathrm{PD}$, $\mathrm{SELFDUAL}$ or $\mathrm{SD}$.
LPIPM Centrality Correctors  $i$  Default $\text{}=6$ 
This parameter controls the number of centrality correctors (see
Section 11.5) used at each iteration. Each corrector step attempts to improve the current iterate for the price of additional solve(s) of the factorized system matrix in order to reduce the total number of iterations. Therefore, it trades the additional solves of the system with the number of factorizations. The more expensive the factorization is with respect to the solve, the more corrector steps should be allowed.
If
$i>0$, the maximum number of corrector steps will be computed by timing heuristics (the ratio between the times of the factorization and the solve in the first iteration) but will not be greater than
$i$. The number computed by the heuristic can be recovered after the solve or during a monitoring step in
stats. This might cause nonrepeatable results.
If $i<0$, the maximum number of corrector steps will be set to $\lefti\right$.
If it is set to $0$, no additional centrality correctors will be used and the algorithm reverts to Mehrotra's predictorcorrector.
LPIPM Iteration Limit  $i$  Default $\text{}=100$ 
The maximum number of iterations to be performed by
e04mtc. Setting the option too low might lead to
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_TOO_MANY_ITER.
Constraint: ${\mathbf{LPIPM\; Iteration\; Limit}}\ge 1$.
LPIPM Max Iterative Refinement  $i$  Default $\text{}=5$ 
This parameter controls the maximum number of iterative refinement iterations (see
Section 11.4) used at each main iteration when
${\mathbf{LPIPM\; System\; Formulation}}=\mathrm{Normal\; Equations}$. When solving the Normal Equations linear system for numerically challenging problems, mixedprecision iterative refinement may be used until the roundoff errors are reduced to an acceptable level or until the number of refinements reached its maximum value set by this parameter.
Constraint: ${\mathbf{LPIPM\; Max\; Iterative\; Refinement}}\ge 0$.
LPIPM Scaling  $a$  Default $=\mathrm{ARITHMETIC}$ 
This parameter controls the type of scaling to be applied on the constraint matrix $A$ before solving the problem. More precisely, the scaling procedure will try to find diagonal matrices ${D}_{1}$ and ${D}_{2}$ such that the values in ${D}_{1}A{D}_{2}$ are of a similar order of magnitude. The solver is less likely to run into numerical difficulties when the constraint matrix is well scaled.
Constraint: ${\mathbf{LPIPM\; Scaling}}=\mathrm{ARITHMETIC}$, $\mathrm{GEOMETRIC}$ or $\mathrm{NONE}$.
LPIPM Monitor Frequency  $i$  Default $\text{}=0$ 
This parameter defines the frequency of how often function
monit is called. If
$i>0$, the solver calls
monit at the end of every
$i$th iteration. If it is set to
$0$, the function is not called at all.
Constraint: ${\mathbf{LPIPM\; Monitor\; Frequency}}\ge 0$.
LPIPM Stop Tolerance  $r$  Default $=\sqrt{\epsilon}$ 
This parameter sets the value
${\epsilon}_{1}$ which is the tolerance for the convergence measures in the stopping criteria, see
Section 11.3.
Constraint: ${\mathbf{LPIPM\; Stop\; Tolerance}}>\epsilon $.
LPIPM Stop Tolerance 2  $r$  Default $={\epsilon}^{0.6}$ 
This parameter sets the additional tolerance
${\epsilon}_{2}$ used in the stopping criteria for the SelfDual algorithm, see
Section 11.3.
Constraint: ${\mathbf{LPIPM\; Stop\; Tolerance\; 2}}>\epsilon $.
LPIPM System Formulation  $a$  Default $=\mathrm{AUTO}$ 
As described in
Section 11.4,
e04mtc can internally work either with the normal equations formulation
(17) or with the augmented system
(16). A brief discussion of advantages and disadvantages is presented in
Section 11.4. Option
$\mathrm{AUTO}$ leaves the decision to the solver based on the structure of the constraints and it is the recommended option. This will typically lead to the normal equations formulation unless there are many dense columns or the system is significantly cheaper to factorize as the augmented system. Note that in some cases even if
${\mathbf{LPIPM\; System\; Formulation}}=\mathrm{NORMAL\; EQUATIONS}$ the solver might switch the formulation through the computation to the augmented system due to numerical instabilities or computational cost.
Constraint: ${\mathbf{LPIPM\; System\; Formulation}}=\mathrm{AUTO}$, $\mathrm{AUGMENTED\; SYSTEM}$, $\mathrm{AS}$, $\mathrm{NORMAL\; EQUATIONS}$ or $\mathrm{NE}$.
Monitoring File  $i$  Default $=\mathrm{1}$ 
(See
Section 3.1.1 in the Introduction to the NAG Library CL Interface for further information on NAG data types.)
If
$i\ge 0$, the
Nag_FileID number (as returned from
x04acc)
for the secondary (monitoring) output. If set to
$\mathrm{1}$, no secondary output is provided. The following information is output to the unit:

–a listing of the optional parameters if set by ${\mathbf{Print\; Options}}$;

–problem statistics, the iteration log and the final status as set by ${\mathbf{Monitoring\; Level}}$;

–the solution if set by ${\mathbf{Print\; Solution}}$.
Constraint: ${\mathbf{Monitoring\; File}}\ge \mathrm{1}$.
Monitoring Level  $i$  Default $=4$ 
This parameter sets the amount of information detail that will be printed by the solver to the secondary output. The meaning of the levels is the same as with ${\mathbf{Print\; Level}}$.
Constraint: $0\le {\mathbf{Monitoring\; Level}}\le 5$.
Print File  $i$  Default
$=\mathrm{Nag\_FileID\; number\; associated\; with\; stdout}$

(See
Section 3.1.1 in the Introduction to the NAG Library CL Interface for further information on NAG data types.)
If
$i\ge 0$, the
Nag_FileID number (as returned from
x04acc,
stdout as the default)
for the primary output of the solver. If
${\mathbf{Print\; File}}=\mathrm{1}$, the primary output is completely turned off independently of other settings. The following information is output to the unit:

–a listing of optional parameters if set by ${\mathbf{Print\; Options}}$;

–problem statistics, the iteration log and the final status from the solver as set by ${\mathbf{Print\; Level}}$;

–the solution if set by ${\mathbf{Print\; Solution}}$.
Constraint: ${\mathbf{Print\; File}}\ge \mathrm{1}$.
Print Level  $i$  Default $=2$ 
This parameter defines how detailed information should be printed by the solver to the primary output.
$\mathit{i}$ 
Output 
$0$ 
No output from the solver 
$1$ 
Only the final status and the primal and dual objective value 
$2$ 
Problem statistics, one line per iteration showing the progress of the solution with respect to the convergence measures, final status and statistics 
$3$ 
As level $2$ but each iteration line is longer, including step lengths and errors 
$4,5$ 
As level $3$ but further details of each iteration are presented 
Constraint: $0\le {\mathbf{Print\; Level}}\le 5$.
Print Options  $a$  Default $=\mathrm{YES}$ 
If ${\mathbf{Print\; Options}}=\mathrm{YES}$, a listing of optional parameters will be printed to the primary and secondary output.
Constraint: ${\mathbf{Print\; Options}}=\mathrm{YES}$ or $\mathrm{NO}$.
Print Solution  $a$  Default $=\mathrm{NO}$ 
If ${\mathbf{Print\; Solution}}=\mathrm{X}$, the final values of the primal variables are printed on the primary and secondary outputs.
If ${\mathbf{Print\; Solution}}=\mathrm{YES}$ or $\mathrm{ALL}$, in addition to the primal variables, the final values of the dual variables are printed on the primary and secondary outputs.
Constraint: ${\mathbf{Print\; Solution}}=\mathrm{YES}$, $\mathrm{NO}$, $\mathrm{X}$ or $\mathrm{ALL}$.
Stats Time  $a$  Default $=\mathrm{NO}$ 
This parameter allows you to turn on timings of various parts of the algorithm to give a better overview of where most of the time is spent. This might be helpful for a choice of different solving approaches. It is possible to choose between CPU and wall clock time. Choice $\mathrm{YES}$ is equivalent to $\mathrm{WALL\; CLOCK}$.
Constraint: ${\mathbf{Stats\; Time}}=\mathrm{YES}$, $\mathrm{NO}$, $\mathrm{CPU}$ or $\mathrm{WALL\; CLOCK}$.
Task  $a$  Default $=\mathrm{MINIMIZE}$ 
This parameter specifies the required direction of the optimization. If ${\mathbf{Task}}=\mathrm{FEASIBLEPOINT}$, the objective function (if set) is ignored and the algorithm stops as soon as a feasible point is found with respect to the given tolerance. If no objective function is set, ${\mathbf{Task}}$ reverts to $\mathrm{FEASIBLEPOINT}$ automatically.
Constraint: ${\mathbf{Task}}=\mathrm{MINIMIZE}$, $\mathrm{MAXIMIZE}$ or $\mathrm{FEASIBLE\; POINT}$.