GALAHAD LPA package#

purpose#

The lpa package uses the simplex method to solve a given linear program (LP). The aim is to minimize the linear objective function

\[q(x) = f + g^T x\]

subject to the general linear constraints and simple bounds

\[c_l \leq A x \leq c_u \;\;\mbox{and} \;\; x_l \leq x \leq x_u,\]

where $A$ is a given $m$ by $n$ matrix, $g$ is a vector, $f$ is a scalar, and any of the components of the vectors $c_l$, $c_u$, $x_l$ or $x_u$ may be infinite. The method offers the choice of direct and iterative solution of the key regularization subproblems, and is most suitable for problems involving a large number of unknowns $x$.

See Section 4 of $GALAHAD/doc/lpa.pdf for a brief description of the method employed and other details.

N.B. The package is simply a sophisticated interface to the HSL package LA04, and requires that a user has obtained the latter. LA04 is not included in GALAHAD but is available without charge to recognised academics, see http://www.hsl.rl.ac.uk/catalogue/la04.html. If LA04 is unavailable, the interior- point linear programming package lpb is an alternative.

terminology#

Any required solution $x$ necessarily satisfies the primal optimality conditions

\[A x = c\]

and

\[c_l \leq c \leq c_u, \;\; x_l \leq x \leq x_u,\]

the dual optimality conditions

\[g = A^{T} y + z,\;\; y = y_l + y_u \;\;\mbox{and}\;\; z = z_l + z_u,\]

and

\[y_l \geq 0, \;\; y_u \leq 0, \;\; z_l \geq 0 \;\;\mbox{and}\;\; z_u \leq 0,\]

and the complementary slackness conditions

\[( A x - c_l )^{T} y_l = 0,\;\; ( A x - c_u )^{T} y_u = 0,\;\; (x -x_l )^{T} z_l = 0 \;\;\mbox{and}\;\;(x -x_u )^{T} z_u = 0,\]

where the vectors $y$ and $z$ are known as the Lagrange multipliers for the general linear constraints, and the dual variables for the bounds, respectively, and where the vector inequalities hold component-wise.

The so-called dual to this problem is another linear program

\[- \mbox{minimize} \;\; c^{lT} y^l + c^{uT} y^u + x^{lT} z^l + x^{uT} z^u + f \;\; \mbox{subject to the dual optimality conditions}\]

that uses the same data. The solution to the two problems, it is exists, is the same, but if one is infeasible, the other is unbounded. It can be more efficient to solve the dual, particularly if $m$ is much larger than $n$.

method#

The bulk of the work is peformed by the HSL package LA04. The main subbroutine from this package requires that the input problem be transformed into the ``standard form’’

\[\begin{split}\begin{array}{rl} \mbox{minimize} & g'^T x' \\ \mbox{subject to} & A' x' = b \\ & l_i \leq x'_i \leq u_i, \;\; (i\leq k) \\ \mbox{and} & x'_l \geq 0, \;\; (i \geq l) \end{array}\end{split}\]

by introducing slack an surpulus variables, reordering and removing fixed variables and free constraints. The resulting problem involves $n'$ unknowns and $m'$ general constraints. In order to deal with the possibility that the general constraints are inconsistent or not of full rank, LA04 introduces additional ``artifical’’ variables $v$, replaces the constraints of the standard-form LP by the enlaarged set

\[A' x' + v = b,\]

and gradually encourages $v$ to zero as a first solution phase.

Once a selection of $m'$ independent (non-basic) variables is made, the enlarged constraints determine the remaining $m'$ dependent ({basic) variables. The simplex method is a scheme for systematically adjusting the choice of basic and non-basic variables until a set which defines an optimal solution of the standard-form LP is obtained. Each iteration of the simplex method requires the solution of a number of sets of linear equations whose coefficient matrix is the basis matrix $B$, made up of the columns of $[A' \;\; I]$ corresponding to the basic variables, or its transpose $B^T$. As the basis matrices for consecutive iterations are closely related, it is normally advantageous to update (rather than recompute) their factorizations as the computation proceeds. If an initial basis is not provided by the user, a set of basic variables which provide a (permuted) triangular basis matrix is found by the simple crash algorithm of Gould and Reid (1989), and initial steepest-edge weights are calculated.

Phases one (finding a feasible solution) and two (solving the standard-form LP) of the simplex method are applied, as appropriate, with the choice of entering variable as described by Goldfarb and Reid (1977) and the choice of leaving variable as proposed by Harris (1973). Refactorizations of the basis matrix are performed whenever doing so will reduce the average iteration time or there is insufficient memory for its factors. The reduced cost for the entering variable is computed afresh. If it is found to be of a different sign from the recurred value or more than 10% different in magnitude, a fresh computation of all the reduced costs is performed. Details of the factorization and updating procedures are given by Reid (1982). Iterative refinement is encouraged for the basic solution and for the reduced costs after each factorization of the basis matrix and when they are recomputed at the end of phase 1.

references#

D. Goldfarb and J. K. Reid, ``A practicable steepest-edge simplex algorithm’’. Mathematical Programming 12 (1977) 361-371.

N. I. M. Gould and J. K. Reid, ``New crash procedures for large systems of linear constraints’’. Mathematical Programming 45 (1989) 475-501.

P. M. J. Harris, ``Pivot selection methods of the Devex LP code’’. Mathematical Programming 5 (1973) 1-28.

J. K. Reid, ``A sparsity-exploiting variant of the Bartels-Golub decomposition for linear-programming bases’’. Mathematical Programming 24 (1982) 55-69.

matrix storage#

The unsymmetric $m$ by $n$ matrix $A$ may be presented and stored in a variety of convenient input formats.

Dense storage format: The matrix $A$ is stored as a compact dense matrix by rows, that is, the values of the entries of each row in turn are stored in order within an appropriate real one-dimensional array. In this case, component $n \ast i + j$ of the storage array A_val will hold the value $A_{ij}$ for $1 \leq i \leq m$, $1 \leq j \leq n$. The string A_type = ‘dense’ should be specified.

Dense by columns storage format: The matrix $A$ is stored as a compact dense matrix by columns, that is, the values of the entries of each column in turn are stored in order within an appropriate real one-dimensional array. In this case, component $m \ast j + i$ of the storage array A_val will hold the value $A_{ij}$ for $1 \leq i \leq m$, $1 \leq j \leq n$. The string A_type = ‘dense_by_columns’ should be specified.

Sparse co-ordinate storage format: Only the nonzero entries of the matrices are stored. For the $l$-th entry, $1 \leq l \leq ne$, of $A$, its row index i, column index j and value $A_{ij}$, $1 \leq i \leq m$, $1 \leq j \leq n$, are stored as the $l$-th components of the integer arrays A_row and A_col and real array A_val, respectively, while the number of nonzeros is recorded as A_ne = $ne$. The string A_type = ‘coordinate’should be specified.

Sparse row-wise storage format: Again only the nonzero entries are stored, but this time they are ordered so that those in row i appear directly before those in row i+1. For the i-th row of $A$ the i-th component of the integer array A_ptr holds the position of the first entry in this row, while A_ptr(m+1) holds the total number of entries plus one. The column indices j, $1 \leq j \leq n$, and values $A_{ij}$ of the nonzero entries in the i-th row are stored in components l = A_ptr(i), $\ldots$, A_ptr(i+1)-1, $1 \leq i \leq m$, of the integer array A_col, and real array A_val, respectively. For sparse matrices, this scheme almost always requires less storage than its predecessor. The string A_type = ‘sparse_by_rows’ should be specified.

Sparse column-wise storage format: Once again only the nonzero entries are stored, but this time they are ordered so that those in column j appear directly before those in column j+1. For the j-th column of $A$ the j-th component of the integer array A_ptr holds the position of the first entry in this column, while A_ptr(n+1) holds the total number of entries plus one. The row indices i, $1 \leq i \leq m$, and values $A_{ij}$ of the nonzero entries in the j-th columnsare stored in components l = A_ptr(j), $\ldots$, A_ptr(j+1)-1, $1 \leq j \leq n$, of the integer array A_row, and real array A_val, respectively. As before, for sparse matrices, this scheme almost always requires less storage than the co-ordinate format. The string A_type = ‘sparse_by_columns’ should be specified.

introduction to function calls#

To solve a given problem, functions from the lpa package must be called in the following order:

lpa_initialize - provide default control parameters and set up initial data structures
lpa_read_specfile (optional) - override control values by reading replacement values from a file
lpa_import - set up problem data structures and fixed values
lpa_reset_control (optional) - possibly change control parameters if a sequence of problems are being solved
lpa_solve_lp - solve the linear program
lpa_information (optional) - recover information about the solution and solution process
lpa_terminate - deallocate data structures

See the examples section for illustrations of use.

parametric real type T#

Below, the symbol T refers to a parametric real type that may be Float32 (single precision), Float64 (double precision) or, if supported, Float128 (quadruple precision).

callable functions#

    function lpa_initialize(T, data, control, status)

Set default control values and initialize private data

Parameters:

data

holds private internal data

control

is a structure containing control information (see lpa_control_type)

status

is a scalar variable of type Int32 that gives the exit status from the package. Possible values are (currently):

0
The initialization was successful.

    function lpa_read_specfile(T, control, specfile)

Read the content of a specification file, and assign values associated with given keywords to the corresponding control parameters. An in-depth discussion of specification files is available, and a detailed list of keywords with associated default values is provided in $GALAHAD/src/lpa/LPA.template. See also Table 2.1 in the Fortran documentation provided in $GALAHAD/doc/lpa.pdf for a list of how these keywords relate to the components of the control structure.

Parameters:

control	is a structure containing control information (see lpa_control_type)
specfile	is a one-dimensional array of type Vararg{Cchar} that must give the name of the specification file

    function lpa_import(T, control, data, status, n, m,
                        A_type, A_ne, A_row, A_col, A_ptr)

Import problem data into internal storage prior to solution.

Parameters:

control	is a structure whose members provide control parameters for the remaining procedures (see lpa_control_type)
data	holds private internal data
status	is a scalar variable of type Int32 that gives the exit status from the package. Possible values are: 0 The import was successful -1 An allocation error occurred. A message indicating the offending array is written on unit control.error, and the returned allocation status and a string containing the name of the offending array are held in inform.alloc_status and inform.bad_alloc respectively. -2 A deallocation error occurred. A message indicating the offending array is written on unit control.error and the returned allocation status and a string containing the name of the offending array are held in inform.alloc_status and inform.bad_alloc respectively. -3 The restrictions n > 0 or m > 0 or requirement that A_type contains its relevant string ‘dense’, ‘coordinate’ or ‘sparse_by_rows’ has been violated.
n	is a scalar variable of type Int32 that holds the number of variables.
m	is a scalar variable of type Int32 that holds the number of general linear constraints.
A_type	is a one-dimensional array of type Vararg{Cchar} that specifies the unsymmetric storage scheme used for the constraint Jacobian, $A$. It should be one of ‘coordinate’, ‘sparse_by_rows’ or ‘dense; lower or upper case variants are allowed.
A_ne	is a scalar variable of type Int32 that holds the number of entries in $A$ in the sparse co-ordinate storage scheme. It need not be set for any of the other schemes.
A_row	is a one-dimensional array of size A_ne and type Int32 that holds the row indices of $A$ in the sparse co-ordinate storage scheme. It need not be set for any of the other schemes, and in this case can be C_NULL.
A_col	is a one-dimensional array of size A_ne and type Int32 that holds the column indices of $A$ in either the sparse co-ordinate, or the sparse row-wise storage scheme. It need not be set when the dense or diagonal storage schemes are used, and in this case can be C_NULL.
A_ptr	is a one-dimensional array of size n+1 and type Int32 that holds the starting position of each row of $A$, as well as the total number of entries, in the sparse row-wise storage scheme. It need not be set when the other schemes are used, and in this case can be C_NULL.

    function lpa_reset_control(T, control, data, status)

Reset control parameters after import if required.

Parameters:

control

is a structure whose members provide control parameters for the remaining procedures (see lpa_control_type)

data

holds private internal data

status

is a scalar variable of type Int32 that gives the exit status from the package. Possible values are:

0
The import was successful.

    function lpa_solve_lp(T, data, status, n, m, g, f, a_ne, A_val,
                          c_l, c_u, x_l, x_u, x, c, y, z, x_stat, c_stat)

Solve the linear program.

Parameters:

data	holds private internal data
status	is a scalar variable of type Int32 that gives the entry and exit status from the package. Possible exit values are: 0 The run was successful. -1 An allocation error occurred. A message indicating the offending array is written on unit control.error, and the returned allocation status and a string containing the name of the offending array are held in inform.alloc_status and inform.bad_alloc respectively. -2 A deallocation error occurred. A message indicating the offending array is written on unit control.error and the returned allocation status and a string containing the name of the offending array are held in inform.alloc_status and inform.bad_alloc respectively. -3 The restrictions n > 0 and m > 0 or requirement that A_type contains its relevant string ‘dense’, ‘coordinate’ or ‘sparse_by_rows’ has been violated. -5 The simple-bound constraints are inconsistent. -7 The constraints appear to have no feasible point. -9 The analysis phase of the factorization failed; the return status from the factorization package is given in the component inform.factor_status -10 The factorization failed; the return status from the factorization package is given in the component inform.factor_status. -11 The solution of a set of linear equations using factors from the factorization package failed; the return status from the factorization package is given in the component inform.factor_status. -16 The problem is so ill-conditioned that further progress is impossible. -17 The step is too small to make further impact. -18 Too many iterations have been performed. This may happen if control.maxit is too small, but may also be symptomatic of a badly scaled problem. -19 The CPU time limit has been reached. This may happen if control.cpu_time_limit is too small, but may also be symptomatic of a badly scaled problem.
n	is a scalar variable of type Int32 that holds the number of variables
m	is a scalar variable of type Int32 that holds the number of general linear constraints.
g	is a one-dimensional array of size n and type T that holds the linear term $g$ of the objective function. The j-th component of `g`, j = 1, … , n, contains $g_j$.
f	is a scalar of type T that holds the constant term $f$ of the objective function.
a_ne	is a scalar variable of type Int32 that holds the number of entries in the constraint Jacobian matrix $A$.
A_val	is a one-dimensional array of size a_ne and type T that holds the values of the entries of the constraint Jacobian matrix $A$ in any of the available storage schemes.
c_l	is a one-dimensional array of size m and type T that holds the lower bounds $c^l$ on the constraints $A x$. The i-th component of `c_l`, i = 1, … , m, contains $c^l_i$.
c_u	is a one-dimensional array of size m and type T that holds the upper bounds $c^l$ on the constraints $A x$. The i-th component of `c_u`, i = 1, … , m, contains $c^u_i$.
x_l	is a one-dimensional array of size n and type T that holds the lower bounds $x^l$ on the variables $x$. The j-th component of `x_l`, j = 1, … , n, contains $x^l_j$.
x_u	is a one-dimensional array of size n and type T that holds the upper bounds $x^l$ on the variables $x$. The j-th component of `x_u`, j = 1, … , n, contains $x^l_j$.
x	is a one-dimensional array of size n and type T that holds the values $x$ of the optimization variables. The j-th component of `x`, j = 1, … , n, contains $x_j$.
c	is a one-dimensional array of size m and type T that holds the residual $c(x)$. The i-th component of `c`, i = 1, … , m, contains $c_i(x)$.
y	is a one-dimensional array of size n and type T that holds the values $y$ of the Lagrange multipliers for the general linear constraints. The j-th component of `y`, i = 1, … , m, contains $y_i$.
z	is a one-dimensional array of size n and type T that holds the values $z$ of the dual variables. The j-th component of `z`, j = 1, … , n, contains $z_j$.
x_stat	is a one-dimensional array of size n and type Int32 that gives the optimal status of the problem variables. If x_stat(j) is negative, the variable $x_j$ most likely lies on its lower bound, if it is positive, it lies on its upper bound, and if it is zero, it lies between its bounds.
c_stat	is a one-dimensional array of size m and type Int32 that gives the optimal status of the general linear constraints. If c_stat(i) is negative, the constraint value $a_i^Tx$ most likely lies on its lower bound, if it is positive, it lies on its upper bound, and if it is zero, it lies between its bounds.

    function lpa_information(T, data, inform, status)

Provides output information

Parameters:

data

holds private internal data

inform

is a structure containing output information (see lpa_inform_type)

status

is a scalar variable of type Int32 that gives the exit status from the package. Possible values are (currently):

0
The values were recorded successfully

    function lpa_terminate(T, data, control, inform)

Deallocate all internal private storage

Parameters:

data	holds private internal data
control	is a structure containing control information (see lpa_control_type)
inform	is a structure containing output information (see lpa_inform_type)

available structures#

lpa_control_type structure#

    struct lpa_control_type{T}
      f_indexing::Bool
      error::Int32
      out::Int32
      print_level::Int32
      start_print::Int32
      stop_print::Int32
      maxit::Int32
      max_iterative_refinements::Int32
      min_real_factor_size::Int32
      min_integer_factor_size::Int32
      random_number_seed::Int32
      sif_file_device::Int32
      qplib_file_device::Int32
      infinity::T
      tol_data::T
      feas_tol::T
      relative_pivot_tolerance::T
      growth_limit::T
      zero_tolerance::T
      change_tolerance::T
      identical_bounds_tol::T
      cpu_time_limit::T
      clock_time_limit::T
      scale::Bool
      dual::Bool
      warm_start::Bool
      steepest_edge::Bool
      space_critical::Bool
      deallocate_error_fatal::Bool
      generate_sif_file::Bool
      generate_qplib_file::Bool
      sif_file_name::NTuple{31,Cchar}
      qplib_file_name::NTuple{31,Cchar}
      prefix::NTuple{31,Cchar}

detailed documentation#

control derived type as a Julia structure

components#

Bool f_indexing

use C or Fortran sparse matrix indexing

Int32 error

error and warning diagnostics occur on stream error

Int32 out

general output occurs on stream out

Int32 print_level

the level of output required is specified by print_level (>= 2 turns on LA04 output)

Int32 start_print

any printing will start on this iteration

Int32 stop_print

any printing will stop on this iteration

Int32 maxit

at most maxit inner iterations are allowed

Int32 max_iterative_refinements

maximum number of iterative refinements allowed

Int32 min_real_factor_size

initial size for real array for the factors and other data

Int32 min_integer_factor_size

initial size for integer array for the factors and other data

Int32 random_number_seed

the initial seed used when generating random numbers

Int32 sif_file_device

specifies the unit number to write generated SIF file describing the current problem

Int32 qplib_file_device

specifies the unit number to write generated QPLIB file describing the current problem

T infinity

any bound larger than infinity in modulus will be regarded as infinite

T tol_data

the tolerable relative perturbation of the data (A,g,..) defining the problem

T feas_tol

any constraint violated by less than feas_tol will be considered to be satisfied

T relative_pivot_tolerance

pivot threshold used to control the selection of pivot elements in the matrix factorization. Any potential pivot which is less than the largest entry in its row times the threshold is excluded as a candidate

T growth_limit

limit to control growth in the upated basis factors. A refactorization occurs if the growth exceeds this limit

T zero_tolerance

any entry in the basis smaller than this is considered zero

T change_tolerance

any solution component whose change is smaller than a tolerence times the largest change may be considered to be zero

T identical_bounds_tol

any pair of constraint bounds (c_l,c_u) or (x_l,x_u) that are closer than identical_bounds_tol will be reset to the average of their values

T cpu_time_limit

the maximum CPU time allowed (-ve means infinite)

T clock_time_limit

the maximum elapsed clock time allowed (-ve means infinite)

Bool scale

if .scale is true, the problem will be automatically scaled prior to solution. This may improve computation time and accuracy

Bool dual

should the dual problem be solved rather than the primal?

Bool warm_start

should a warm start using the data in C_stat and X_stat be attempted?

Bool steepest_edge

should steepest-edge weights be used to detetrmine the variable leaving the basis?

Bool space_critical

if .space_critical is true, every effort will be made to use as little space as possible. This may result in longer computation time

Bool deallocate_error_fatal

if .deallocate_error_fatal is true, any array/pointer deallocation error will terminate execution. Otherwise, computation will continue

Bool generate_sif_file

if .generate_sif_file is .true. if a SIF file describing the current problem is to be generated

Bool generate_qplib_file

if .generate_qplib_file is .true. if a QPLIB file describing the current problem is to be generated

NTuple{31,Cchar} sif_file_name

name of generated SIF file containing input problem

NTuple{31,Cchar} qplib_file_name

name of generated QPLIB file containing input problem

NTuple{31,Cchar} prefix

all output lines will be prefixed by .prefix(2:LEN(TRIM(.prefix))-1) where .prefix contains the required string enclosed in quotes, e.g. “string” or ‘string’

lpa_time_type structure#

    struct lpa_time_type{T}
      total::T
      preprocess::T
      clock_total::T
      clock_preprocess::T

detailed documentation#

time derived type as a Julia structure

components#

T total

the total CPU time spent in the package

T preprocess

the CPU time spent preprocessing the problem

T clock_total

the total clock time spent in the package

T clock_preprocess

the clock time spent preprocessing the problem

lpa_inform_type structure#

    struct lpa_inform_type{T}
      status::Int32
      alloc_status::Int32
      bad_alloc::NTuple{81,Cchar}
      iter::Int32
      la04_job::Int32
      la04_job_info::Int32
      obj::T
      primal_infeasibility::T
      feasible::Bool
      RINFO::NTuple{40,T}
      time::lpa_time_type{T}
      rpd_inform::rpd_inform_type

detailed documentation#

inform derived type as a Julia structure

components#

Int32 status

return status. See LPA_solve for details

Int32 alloc_status

the status of the last attempted allocation/deallocation

NTuple{81,Cchar} bad_alloc

the name of the array for which an allocation/deallocation error occurred

Int32 iter

the total number of iterations required

Int32 la04_job

the final value of la04’s job argument

Int32 la04_job_info

any extra information from an unsuccessfull call to LA04 (LA04’s RINFO(35)

T obj

the value of the objective function at the best estimate of the solution determined by LPA_solve

T primal_infeasibility

the value of the primal infeasibility

Bool feasible

is the returned “solution” feasible?

T RINFO[40]

the information array from LA04

struct lpa_time_type time

timings (see above)

struct rpd_inform_type rpd_inform

inform parameters for RPD

Index

example calls#

This is an example of how to use the package to solve a linear program; the code is available in $GALAHAD/src/lpa/Julia/test_lpa.jl . A variety of supported constraint matrix storage formats are shown.

# test_lpa.jl
# Simple code to test the Julia interface to LPA

using GALAHAD
using Test
using Printf
using Accessors
using Quadmath

function test_lpa(::Type{T}) where T
  # Derived types
  data = Ref{Ptr{Cvoid}}()
  control = Ref{lpa_control_type{T}}()
  inform = Ref{lpa_inform_type{T}}()

  # Set problem data
  n = 3 # dimension
  m = 2 # number of general constraints
  g = T[0.0, 2.0, 0.0]  # linear term in the objective
  f = one(T)  # constant term in the objective
  A_ne = 4 # Jacobian elements
  A_row = Cint[1, 1, 2, 2]  # row indices
  A_col = Cint[1, 2, 2, 3]  # column indices
  A_ptr = Cint[1, 3, 5]  # row pointers
  A_val = T[2.0, 1.0, 1.0, 1.0]  # values
  c_l = T[1.0, 2.0]  # constraint lower bound
  c_u = T[2.0, 2.0]  # constraint upper bound
  x_l = T[-1.0, -Inf, -Inf]  # variable lower bound
  x_u = T[1.0, Inf, 2.0]  # variable upper bound

  # Set output storage
  c = zeros(T, m) # constraint values
  x_stat = zeros(Cint, n) # variable status
  c_stat = zeros(Cint, m) # constraint status
  st = ' '
  status = Ref{Cint}()

  @printf(" Fortran sparse matrix indexing\n\n")
  @printf(" basic tests of lp storage formats\n\n")

  for d in 1:3
    # Initialize LPA
    lpa_initialize(T, data, control, status)

    # Set user-defined control options
    @reset control[].f_indexing = true # Fortran sparse matrix indexing

    # Start from 0
    x = T[0.0, 0.0, 0.0]
    y = T[0.0, 0.0]
    z = T[0.0, 0.0, 0.0]

    # sparse co-ordinate storage
    if d == 1
      st = 'C'
      lpa_import(T, control, data, status, n, m,
                 "coordinate", A_ne, A_row, A_col, C_NULL)

      lpa_solve_lp(T, data, status, n, m, g, f,
                   A_ne, A_val, c_l, c_u, x_l, x_u, x, c, y, z,
                   x_stat, c_stat)
    end

    # sparse by rows
    if d == 2
      st = 'R'
      lpa_import(T, control, data, status, n, m,
                 "sparse_by_rows", A_ne, C_NULL, A_col, A_ptr)

      lpa_solve_lp(T, data, status, n, m, g, f,
                   A_ne, A_val, c_l, c_u, x_l, x_u, x, c, y, z,
                   x_stat, c_stat)
    end

    # dense
    if d == 3
      st = 'D'
      A_dense_ne = 6 # number of elements of A
      A_dense = T[2.0, 1.0, 0.0, 0.0, 1.0, 1.0]
      lpa_import(T, control, data, status, n, m,
                 "dense", A_ne, C_NULL, C_NULL, C_NULL)

      lpa_solve_lp(T, data, status, n, m, g, f,
                   A_dense_ne, A_dense, c_l, c_u, x_l, x_u,
                   x, c, y, z, x_stat, c_stat)
    end

    lpa_information(T, data, inform, status)

    if inform[].status == 0
      @printf("%c:%6i iterations. Optimal objective value = %5.2f status = %1i\n", st,
              inform[].iter, inform[].obj, inform[].status)
    else
      @printf("%c: LPA_solve exit status = %1i\n", st, inform[].status)
    end

    # @printf("x: ")
    # for i = 1:n
    #   @printf("%f ", x[i])
    # end
    # @printf("\n")
    # @printf("gradient: ")
    # for i = 1:n
    #   @printf("%f ", g[i])
    # end
    # @printf("\n")

    # Delete internal workspace
    lpa_terminate(T, data, control, inform)
  end

  return 0
end

@testset "LPA" begin
  @test test_lpa(Float32) == 0
  @test test_lpa(Float64) == 0
  @test test_lpa(Float128) == 0
end