NAG CPP Interface
nagcpp::opt::handle_solve_lp_ipm (e04mt)

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.
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1 Purpose

handle_solve_lp_ipm is a solver from the NAG optimization modelling suite for large-scale Linear Programming (LP) problems based on an interior point method (IPM).

2 Specification

#include "e04/nagcpp_e04mt.hpp"
#include "e04/nagcpp_class_CommE04RA.hpp"
template <typename COMM, typename X, typename U, typename RINFO, typename STATS, typename MONIT>

void function handle_solve_lp_ipm(COMM &comm, X &&x, U &&u, RINFO &&rinfo, STATS &&stats, MONIT &&monit, OptionalE04MT opt)
template <typename COMM, typename X, typename U, typename RINFO, typename STATS, typename MONIT>

void function handle_solve_lp_ipm(COMM &comm, X &&x, U &&u, RINFO &&rinfo, STATS &&stats, MONIT &&monit)

3 Description

handle_solve_lp_ipm solves a large-scale linear optimization problem in the following form:
minimize xn cTx   (a) subject to   lAAxuA   (b) lxxux ,   (c) (1)
where n is the number of decision variables and m is the number of linear constraints. Here c, x, lx, ux are n-dimensional vectors, A is an m×n sparse matrix and lA, uA are m-dimensional vectors.
handle_solve_lp_ipm implements two algorithmic variants of the interior point method for solving linear optimization problems: the infeasible Primal-Dual interior point method and homogeneous Self-Dual interior point method. In general, the Self-Dual algorithm has a slightly higher price per iteration, however, it is able to declare infeasibility or unboundness of the problem, whereas the Primal-Dual relies, in this case, on heuristics. For a detailed description of both algorithms see Section 11. The algorithm is chosen by the LPIPM Algorithm, the default is Primal-Dual.
handle_solve_lp_ipm 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 handle_​init. Then some of the functions handle_​set_​linobj, handle_​set_​quadobj, handle_​set_​simplebounds and handle_​set_​linconstr 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, handle_​free should be called to destroy it and deallocate the memory held within. See Section 3.1 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 handle_​opt_​set and e04zpf (no CPP interface) anytime between the initialization of the handle by handle_​init 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 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 (handle_​set_​simplebounds 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 xi. 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 handle_​set_​linconstr 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 product-form 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 primal-dual 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 bit-compatible multifrontal code for sparse symmetric systems RAL Technical Report. RAL-TR-2011-024
HSL (2011) A collection of Fortran codes for large-scale 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) Primal-dual interior point methods SIAM, Philadelphia
Xu X, Hung P-F and Ye Y (1996) A simplified homogeneous and self-dual linear programming algorithm and its implementation Annals of Operations Research 62(1) 151–171

5 Arguments

1: comm CommE04RA Input/Output
Communication structure. An object of either the derived class CommE04RA or its base class NoneCopyableComm can be supplied. It is recommended that the derived class is used. If the base class is supplied it must first be initialized via a call to opt::handle_init (e04ra).
2: x(nvar) double array 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.
3: u(nnzu) double array Input/Output
Note: if nnzu>0, u holds Lagrange multipliers (dual variables) for the bound constraints and linear constraints. If nnzu=0, u will not be referenced.
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.
4: rinfo(100) double array 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 Primal-Dual specific information (will be 0 if the Self-Dual 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 μ (see Section 11.1).
8 Centring parameter σ (see Section 11.1).
9 Primal step length.
10 Dual step length.
11-13 Reserved for future use.
14-23 Self-Dual specific information (will be 0 if the Primal-Dual 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 τ, see (8).
19 κ, see (8).
20 Step length.
21-23 Reserved for future use.
24-99 Reserved for future use.
5: stats(100) double array Output
On exit: solver statistics as given in the table below. Note that time statistics are provided only if Stats Time is set (the default is 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.
6: monit void function Function
monit is provided to enable you to monitor the progress of the optimization. It is invoked at the end of every ith iteration where i is given by the optional parameter LPIPM Monitor Frequency (the default is 0, monit is not called).
void function monit(CommE04RA &comm, const utility::array1D<double,data_handling::ArgIntent::IN> &rinfo, const utility::array1D<double,data_handling::ArgIntent::IN> &stats)
1: comm CommE04RA Input/Output
Communication structure.
Container for:
handlevoid *
This optional parameter may be set using the method CommE04RA::handle and accessed via CommE04RA::get_handle.
On entry: the handle to the problem as provided on entry to handle_solve_lp_ipm. It may be used to query the model during the solve, and extract the current approximation of the solution by e04rxf (no CPP interface).
2: rinfo(100) double array Input
On entry: error measures and various indicators at the end of the current iteration as described in rinfo.
3: stats(100) double array 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.
7: opt OptionalE04MT Input/Output
Optional parameter container, derived from Optional.

5.1Additional Quantities

1: nvar
n, the current number of decision variables x in the model.
2: nnzu
The dimension of array u

6 Exceptions and Warnings

Errors or warnings detected by the function:
Note: in some cases handle_solve_lp_ipm may return useful information.
All errors and warnings have an associated numeric error code field, errorid, stored either as a member of the thrown exception object (see errorid), or as a member of opt.ifail, depending on how errors and warnings are being handled (see Error Handling for more details).
Raises: ErrorException
errorid=1
comm::handle has not been initialized.
errorid=1
comm::handle does not belong to the NAG optimization modelling suite,
has not been initialized properly or is corrupted.
errorid=1
comm::handle has not been initialized properly or is corrupted.
errorid=2
This solver does not support the model defined in the handle.
errorid=2
The problem is already being solved.
errorid=4
On entry, nvar = value,
expected value=value.
Constraint: nvar must match the current number of variables
of the model in the comm::handle.
errorid=5
On entry, nnzu = value.
nnzu does not match the size of the Lagrangian multipliers
for constraints.
The correct value is either 0 or value.
errorid=5
On entry, nnzu = value.
nnzu does not match the size of the Lagrangian multipliers
for constraints.
The correct value is 0 for no constraints.
errorid=51
The problem was found to be primal infeasible.
errorid=52
The problem was found to be dual infeasible.
errorid=53
The problem seems to be primal or dual infeasible,
the algorithm was stopped.
errorid=10601
On entry, argument value must be a vector of size value array.
Supplied argument has value dimensions.
errorid=10601
On entry, argument value must be a vector of size value array.
Supplied argument was a vector of size value.
errorid=10601
On entry, argument value must be a vector of size value array.
The size for the supplied array could not be ascertained.
errorid=10602
On entry, the raw data component of value is null.
errorid=10603
On entry, unable to ascertain a value for value.
errorid=10605
On entry, the communication class value has not been initialized correctly.
errorid=10703
An exception was thrown during IO (writing).
errorid=-99
An unexpected error has been triggered by this routine.
errorid=-399
Your licence key may have expired or may not have been installed correctly.
errorid=-999
Dynamic memory allocation failed.
Raises: CallbackEarlyTermination
errorid=20
User requested termination during a monitoring step.
Raises: WarningException
errorid=22
Maximum number of iterations exceeded.
errorid=24
No progress, stopping early.
errorid=50
Suboptimal solution.
Raises: CallbackException
errorid=10701
An exception was thrown in a callback.
errorid=10702
The memory address for an array in a callback has changed.

7 Accuracy

The accuracy of the solution is determined by optional parameters LPIPM Stop Tolerance and LPIPM Stop Tolerance 2.
If errorid=0 on the final exit, the returned point satisfies Karush–Kuhn–Tucker (KKT) conditions to the requested accuracy (under the default settings close to ε) and thus it is a good estimate of the solution. If errorid=50, 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

Please see the description for the underlying computational routine in this section of the FL Interface documentation.

9 Further Comments

9.1 Description of the Printed Output

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 Print File and Monitoring File. Optional parameters Print Level, Print Solution and 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 (Print File=6, Print Level=2), six sections are printed to the standard output: The iteration log varies depending on which algorithm has been selected to solve the problem (Primal-Dual or Self-Dual).
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 e04zpf (no CPP interface). The output might look as follows:
     Lp Presolve                   =                 Yes     * d
     Lpipm Algorithm               =         Primal-dual     * d
     Lpipm Centrality Correctors   =                  -6     * U
     Lpipm Iteration Limit         =                 100     * d
Problem statistics
If Print Level2, 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 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 Print Level2, the solver prints the status of each iteration.
Primal-Dual algorithm
If 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 μ, 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  5.82E-02  2.09E-01  2.12E+01  4.7E+02   2
 2 -1.09237E+02 -2.81254E+03  3.12E-03  4.84E-15  4.84E-01  5.3E+01   0
 3 -2.10923E+02 -6.07429E+02  4.61E-04  4.99E-14  3.70E-02  7.8E+00   0
If 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  5.82E-02  2.09E-01  2.12E+01  4.7E+02  9.57E-01  9.92E-01  1.54E-12   2
 2 -1.09237E+02 -2.81254E+03  3.12E-03  4.84E-15  4.84E-01  5.3E+01  1.00E+00  9.46E-01  3.02E-12   0
 3 -2.10923E+02 -6.07429E+02  4.61E-04  4.99E-14  3.70E-02  7.8E+00  1.00E+00  8.53E-01  7.66E-11   0
Self-Dual algorithm
If 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 τ 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  4.98E-04  3.11E-02  2.58E-02  3.5E-01   0
 2  7.09323E+00  3.62789E+00  1.89E-04  1.18E-02  9.79E-03  1.4E-01   0
 3 -1.33628E+01  5.87563E+00  1.94E-05  1.21E-03  1.01E-03  3.3E-02   0
If Print Level=3, the solver also prints for each iteration ρA (15), the value of the variable κ, 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  4.98E-04  3.11E-02  2.58E-02  2.4E+00  3.5E-01  1.0E+00  6.52E-01  3.43E-13   0
 2  7.09323E+00  3.62789E+00  1.89E-04  1.18E-02  9.79E-03  7.5E-01  1.4E-01  9.8E-01  6.21E-01  2.85E-13   0
 3 -1.33628E+01  5.87563E+00  1.94E-05  1.21E-03  1.01E-03  2.8E+00  3.3E-02  7.9E-01  8.97E-01  9.31E-13   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 Print Level>3, for both the Primal-Dual and the Self-Dual algorithms, each iteration produces more information that expands over several lines. This additional information contains: The output might look as follows:
----------- Details of Iteration   1 ------------
method                            Normal Equation
sigma                                    6.72E-02
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        1.605940E-15
Relative primal infeasibility        1.210247E-16
Absolute dual infeasibility          4.272004E-12
Relative dual infeasibility          6.068111E-15
Absolute complementarity gap         9.387105E-07
Relative complementarity gap         2.507021E-10
Iterations                                      7
It starts with the status line of the overall result which matches the ifail value and is followed by the final primal and dual objective values as well as the error measures and iteration count.
Optionally, if 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 Print Solution=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  -1.00000E-02   -1.00000E-02    1.00000E-02
    2  -1.00000E-01   -1.00000E-01    1.50000E-01
    3  -1.00000E-02    3.00000E-02    3.00000E-02
    4  -4.00000E-02    2.00000E-02    2.00000E-02
    5  -1.00000E-01   -6.74853E-02    5.00000E-02
    6  -1.00000E-02   -2.28013E-03         inf
    7  -1.00000E-02   -2.34528E-04         inf
If Print Solution=YES or 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  -1.00000E-02    3.30098E-01    1.00000E-02    0.00000E+00
    2  -1.00000E-01    1.43844E-02    1.50000E-01    0.00000E+00
    3  -1.00000E-02    0.00000E+00    3.00000E-02    9.09967E-02
    4  -4.00000E-02    0.00000E+00    2.00000E-02    7.66124E-02
    5  -1.00000E-01    4.92258E-12    5.00000E-02    0.00000E+00
    6  -1.00000E-02    2.42274E-11         inf       0.00000E+00
    7  -1.00000E-02    4.83752E-12         inf       0.00000E+00

Linear constraints dual variables:
  idx   Lower bound        Value      Upper bound        Value
    1  -1.30000E-01    0.00000E+00   -1.30000E-01    1.43111E+00
    2       -inf       0.00000E+00   -4.90000E-03    4.07810E-10
    3       -inf       0.00000E+00   -6.40000E-03    5.64870E-10
    4       -inf       0.00000E+00   -3.70000E-03    1.25984E-10
    5       -inf       0.00000E+00   -1.20000E-03    1.87338E-11
    6  -9.92000E-02    1.50098E+00         inf       0.00000E+00
    7  -3.00000E-03    1.51661E+00    2.00000E-03    0.00000E+00

10 Example

This example demonstrates how to use handle_solve_lp_ipm to solve a small LP problem with the two algorithms implemented (Primal-Dual and Self-Dual). The solver is called twice on the same handle with different values of optional parameters.
We solve the following linear programming problem:
minimize
-0.02x1 -0.2x2 -0.2x3 -0.2x4 -0.2x5 +0.04x6 +0.04x7  
subject to the bounds
-0.01x10.01 -0.10x20.15 -0.01x30.03 -0.04x40.02 -0.10x50.05 -0.01x60.00 -0.01x70.00  
and the general constraints
x1 + x2 + x3 + x4 + x5 + x6 + x7 = -0.13 0.15x1 + 0.04x2 + 0.02x3 + 0.04x4 + 0.02x5 + 0.01x6 + 0.03x7 -0.0049 0.03x1 + 0.05x2 + 0.08x3 + 0.02x4 + 0.06x5 + 0.01x6 -0.0064 00.02x1 + 0.04x2 + 0.01x3 + 0.02x4 + 0.02x5 -0.0037 0.02x1 + 0.03x2 + 0.01x5 -0.0012 -0.0992 0.70x1 + 0.75x2 + 0.80x3 + 0.75x4 + 0.80x5 + 0.97x6 -0.003 0.02x1 + 0.06x2 + 0.08x3 + 0.12x4 + 0.02x5 + 0.01x6 + 0.97x7 -0.002.  

10.1 Example Program

Source FileDataResults
ex_e04mt.cppNoneex_e04mt.r

11 Algorithmic Details

This section contains the description of the underlying algorithms used in handle_solve_lp_ipm, which implements the infeasible Primal-Dual and homogeneous Self-Dual 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
minimize xn cTx   (a) subject to   Ax=b   (b) x0   (c) (2)
where c, xn, bm and Am×n with full row rank. The dual formulation for (2) is given by
maximize ym , zn bTy   (a) subject to   ATy+z=c   (b) z0   (c) y​ (free)   (d) (3)
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 so-called Karush–Kuhn–Tucker (KKT) conditions, which are stated as follows:
ATy+z = c    (dual feasibility, optimality) Ax = b    (primal feasibility) XZe = 0    (complementarity) (x,z) 0 (4)
where we define X and Z as the diagonal matrices with the elements xi and zi, respectively, and e=(1,1,,1)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 non-negative constraints are preserved at every iteration.

11.1 The Infeasible-interior-point Primal-Dual 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
Ax = b ATy+z = c XZe = σμe (x,z) > 0. (5)
Here μ is the average complementarity error at the current iteration μ=xTz/n, called duality measure, and σ(0,1), called centring parameter, is the reduction factor that we wish to achieve in the duality measure.
Each iteration of the Primal-Dual algorithm makes one step of Newton's method applied to the perturbed first order optimality conditions (5) with a given σ and μ. In particular, a Newton search direction is computed by solving a system of linear equations and a length of the step α is determined so that the bounds (x,z)>0 are not violated. The residual of (4) and μ define stopping criteria and the algorithm terminates once they are reduced to the requested accuracy, see Section 11.3.
Given an x, z+n and ym, Newton's direction is obtained by solving the following system:
( A 0 0 0 AT I Z 0 X ) ( Δx Δs Δz ) = ( rb rc μe-XZe ) , (6)
where
rb=b-Ax, (Primal infeasibilities) rc=c-ATy-z (Dual infeasibilities)  
denote the violations of the primal and the dual constraints, respectively. Primal and dual infeasibilities, rb and rc, are reduced at the same rate (1-α), given a step size α(0,1). The Primal-Dual 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, Δx and Δz are used to compute the maximum step sizes in primal space (αP) and dual space (αD) such that the non-negativity of variables is preserved
αP= min{1, ρ· max{α0: xk + αΔxk0 } } αD= min{1, ρ· max{α0: zk+ αΔzk0 } } ,  
where 0<ρ<1 is a reduction parameter close to 1, typically ρ(0.9,1). The next iterate is updated as follows:
xk+1 xk + αPΔxk (yk+1,zk+1) (yk,zk) + αD (Δyk,Δzk)  
Finally, the barrier parameter μ 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 non-negativity constraints x0 by a logarithmic barrier term j=1nlogxj with a barrier parameter τ>0. This approach leads to the primal logarithmic barrier problem defined as
minimize xn cTx - τ j=1n logxj subject to   Ax=b . (7)
The Lagrangian formulation for (7) is
L(x,y,τ) = cTx-τ j=1 n logxj - yT (Ax-b) ,  
and the first order optimality conditions for problem (7) are
xL = c-τX-1e-ATy=0 yL = b-Ax=0, x > 0(Barrier avoids x=0).  
Finally, by introducing Z=τX-1 and z=Ze the same system of equations (5) is formulated.

11.2 Homogeneous Self-Dual Algorithm

A homogeneous Self-Dual (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. Self-Dual 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 Self-Dual linear feasibility (HLF) model as follows:
minimize x,zn , ym , τ,κ 0 subject to   Ax-bτ=0 ATy+z-cτ=0 -cTx + bTy-κ=0 x,τ,z,κ0,y​ (free) , (8)
where τ and κ are two additional variables. The model (8) is a Self-Dual linear programming problem with zero right-hand side and a zero objective vector. If (x^,τ^,y^,z^,κ^) is a strictly complementarity solution for (8), then if τ^>0, the linear programming problem has an optimal solution given by
(x*,y*,z*) = (x^,y^,z^) / τ^,  
and the duality gap is given by cTx*-bTy*=κ^/τ^=0. The homogeneous algorithm is an application of the Primal-Dual method for the computation of a strictly complementarity solution to (8).
Homogeneous and Self-Dual interior point methods have several advantages besides an inherent ability to detect infeasibility (which improves the detection of divergence in Primal-Dual 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 Primal-Dual 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 Primal-Dual algorithm
The Primal-Dual 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:
Furthermore, final absolute primal and dual infeasibilities, and duality gap are also returned. Here ε1 may be set using LPIPM Stop Tolerance and the norm denotes the 2-norm.
11.3.1.2 Self-Dual algorithm
Similar to the Primal-Dual algorithm, the homogeneous Self-Dual algorithm is stopped when the following measures are satisfied to the requested accuracy. 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
ρA = |cTx-bTy| τ+ |bTy| ε2 . (15)
Here ε1 and ε2 may be set using LPIPM Stop Tolerance and 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 Self-Dual algorithm is stopped if the above termination conditions are met within a small factor and τ>1000κ. This measure is tracked after the first 10 iterations.

11.3.2 Infeasibility/Unboundedness Detection

11.3.2.1 Primal-Dual algorithm
The Primal-Dual 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 Self-Dual algorithm is recommended, see Section 11.3.2.2.
11.3.2.2 Self-Dual algorithm
The problem is concluded to be primal or dual infeasible if one of the following conditions hold:
  1. 1.Both the relative primal (12) and dual (13) feasibility of the HLF model (8) are satisfied and the value of τ satisfies
    τ ε2 max(1,κ)  
  2. 2.or if the following inequalities hold
    μ ε2 μ0  
    τ ε2 min(1,κ) .  
Then the problem is declared dual infeasible if cTx<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 Δz from the last block of equations as follows:
( -D2 AT A 0 ) ( Δx Δy ) = ( r h ) , (16)
where
D2 = Z-1X r = rc-X-1(μe-XZe) h = rb  
and Δz=-X-1(μe-XZe)-X-1ZΔx. 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 Δx, to a positive definite system usually called normal equations defined as
(AD2AT)Δy=AD2r+h (17)
and
Δz = -rc-ATΔy Δx = -Z-1(μe-XZe)-XZ-1Δz.  
Typically, formulation (17) is preferred for many problems as the system matrix can be factorized by a sparse Cholesky. However, this brings some well-known disadvantages: Ill-conditioning 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 AD2AT 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 fill-in caused by dense columns and can handle free variables directly in the formulation.
handle_solve_lp_ipm 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 product-form Cholesky factorization as described in Goldfard and Scheinberg (2004). It is also possible to manually override the automatic choice via the optional parameter 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 mixed-precision is employed for reducing the roundoff errors produced during the solution of the system. When the condition number of the system AD2AT prevents the satisfactory use of iterative refinement, handle_solve_lp_ipm switches automatically to an augmented system formulation, reporting RS (Restart) in the last column of the iteration log (I). Furthermore, handle_solve_lp_ipm provides several scaling techniques to adjust the numerical characteristics of the problem data, see LPIPM Scaling.
Finally, factorization of the system matrix can degrade sparsity, so the resulting fill-in can be large, therefore, several ordering techniques are included to minimize it. handle_solve_lp_ipm 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 affine-scaling direction and its correction step, called Mehrotra's predictor-corrector. However, Mehrotra's predictor-corrector technique aims to correct the affine-direction in a full step, which is considered an aggressive approach.
handle_solve_lp_ipm implements a high-order method, called weighted multiple centrality correctors (WMCC), see Gondzio (1996) and Colombo and Gondzio (2008). This technique attempts to correct the affine-direction 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 non-repeatable results. To avoid an undesired behaviour, you can fix the maximum number of WMCC with optional parameter LPIPM Centrality Correctors.

11.6 Further Details

handle_solve_lp_ipm 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 LP Presolve.
The initial point x0 is always computed using heuristics. Effective warm-starting 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 handle_solve_lp_ipm define choices in the problem specification or the algorithm logic. In order to reduce the number of formal arguments of handle_solve_lp_ipm 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 handle_​opt_​set 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 AUTO), their value can be retrieved by handle_​opt_​get. 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:
All options accept the value DEFAULT to return single options to their default states.
Keywords and character values are case and white space insensitive.
Defaults
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 SizerDefault =1020
This defines the ‘infinite’ bound bigbnd in the definition of the problem constraints. Any upper bound greater than or equal to bigbnd will be regarded as + (and similarly any lower bound less than or equal to -bigbnd will be regarded as -). 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: Infinite Bound Size1000.
LP PresolveaDefault =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: LP Presolve=FULL, BASIC or NO.
LPIPM AlgorithmaDefault =PRIMAL-DUAL
As described in Section 11, handle_solve_lp_ipm implements the infeasible Primal-Dual algorithm, see Section 11.1, and the homogeneous Self-Dual algorithm, see Section 11.2. This parameter controls which one to use.
Constraint: LPIPM Algorithm=PRIMAL-DUAL, PD, SELF-DUAL or SD.
LPIPM Centrality CorrectorsiDefault =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 non-repeatable results.
If i<0, the maximum number of corrector steps will be set to |i|.
If it is set to 0, no additional centrality correctors will be used and the algorithm reverts to Mehrotra's predictor-corrector.
LPIPM Iteration LimitiDefault =100
The maximum number of iterations to be performed by handle_solve_lp_ipm. Setting the option too low might lead to errorid=22.
Constraint: LPIPM Iteration Limit1.
LPIPM Max Iterative RefinementiDefault =5
This parameter controls the maximum number of iterative refinement iterations (see Section 11.4) used at each main iteration when LPIPM System Formulation=Normal Equations. When solving the Normal Equations linear system for numerically challenging problems, mixed-precision 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: LPIPM Max Iterative Refinement0.
LPIPM ScalingaDefault =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 D1 and D2 such that the values in D1AD2 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: LPIPM Scaling=ARITHMETIC, GEOMETRIC or NONE.
LPIPM Monitor FrequencyiDefault =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 ith iteration. If it is set to 0, the function is not called at all.
Constraint: LPIPM Monitor Frequency0.
LPIPM Stop TolerancerDefault =ε
This parameter sets the value ε1 which is the tolerance for the convergence measures in the stopping criteria, see Section 11.3.
Constraint: LPIPM Stop Tolerance>ε.
LPIPM Stop Tolerance 2rDefault =ε0.6
This parameter sets the additional tolerance ε2 used in the stopping criteria for the Self-Dual algorithm, see Section 11.3.
Constraint: LPIPM Stop Tolerance 2>ε.
LPIPM System FormulationaDefault =AUTO
As described in Section 11.4, handle_solve_lp_ipm 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 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 LPIPM System Formulation=NORMAL EQUATIONS the solver might switch the formulation through the computation to the augmented system due to numerical instabilities or computational cost.
Constraint: LPIPM System Formulation=AUTO, AUGMENTED SYSTEM, AS, NORMAL EQUATIONS or NE.
Monitoring FileiDefault =-1
If i0, the unit number for the secondary (monitoring) output. If set to -1, no secondary output is provided. The following information is output to the unit:
Constraint: Monitoring File-1.
Monitoring LeveliDefault =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 Print Level.
Constraint: 0Monitoring Level5.
Print FileiDefault =advisory message unit number
If i0, the unit number for the primary output of the solver. If Print File=-1, the primary output is completely turned off independently of other settings. The default value is the advisory message unit number as defined by x04abf (no CPP interface) at the time of the optional parameters initialization, e.g., at the initialization of the handle. The following information is output to the unit:
Constraint: Print File-1.
Print LeveliDefault =2
This parameter defines how detailed information should be printed by the solver to the primary output.
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: 0Print Level5.
Print OptionsaDefault =YES
If Print Options=YES, a listing of optional parameters will be printed to the primary and secondary output.
Constraint: Print Options=YES or NO.
Print SolutionaDefault =NO
If Print Solution=X, the final values of the primal variables are printed on the primary and secondary outputs.
If Print Solution=YES or ALL, in addition to the primal variables, the final values of the dual variables are printed on the primary and secondary outputs.
Constraint: Print Solution=YES, NO, X or ALL.
Stats TimeaDefault =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 YES is equivalent to WALL CLOCK.
Constraint: Stats Time=YES, NO, CPU or WALL CLOCK.
TaskaDefault =MINIMIZE
This parameter specifies the required direction of the optimization. If Task=FEASIBLE POINT, 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, Task reverts to FEASIBLE POINT automatically.
Constraint: Task=MINIMIZE, MAXIMIZE or FEASIBLE POINT.