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 \(0 \leq i \leq m-1\), \(0 \leq j \leq n-1\). 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 \(0 \leq i \leq m-1\), \(0 \leq j \leq n-1\). 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, \(0 \leq l \leq ne-1\), of \(A\), its row index i, column index j and value \(A_{ij}\), \(0 \leq i \leq m-1\), \(0 \leq j \leq n-1\), 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) holds the total number of entries. The column indices j, \(0 \leq j \leq n-1\), 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, \(0 \leq i \leq m-1\), 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) holds the total number of entries. The row indices i, \(0 \leq i \leq m-1\), and values \(A_{ij}\) of the nonzero entries in the j-th columns are stored in components l = A_ptr(j), \(\ldots\), A_ptr(j+1)-1, \(0 \leq j \leq n-1\), 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:

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.

callable functions

overview of functions provided

// typedefs

typedef float spc_;
typedef double rpc_;
typedef int ipc_;

// structs

struct lpa_control_type;
struct lpa_inform_type;
struct lpa_time_type;

// function calls

void lpa_initialize(void **data, struct lpa_control_type* control, ipc_ *status);
void lpa_read_specfile(struct lpa_control_type* control, const char specfile[]);

void lpa_import(
    struct lpa_control_type* control,
    void **data,
    ipc_ *status,
    ipc_ n,
    ipc_ m,
    const char A_type[],
    ipc_ A_ne,
    const ipc_ A_row[],
    const ipc_ A_col[],
    const ipc_ A_ptr[]
);

void lpa_reset_control(
    struct lpa_control_type* control,
    void **data,
    ipc_ *status
);

void lpa_solve_lp(
    void **data,
    ipc_ *status,
    ipc_ n,
    ipc_ m,
    const rpc_ g[],
    const rpc_ f,
    ipc_ a_ne,
    const rpc_ A_val[],
    const rpc_ c_l[],
    const rpc_ c_u[],
    const rpc_ x_l[],
    const rpc_ x_u[],
    rpc_ x[],
    rpc_ c[],
    rpc_ y[],
    rpc_ z[],
    ipc_ x_stat[],
    ipc_ c_stat[]
);

void lpa_information(void **data, struct lpa_inform_type* inform, ipc_ *status);

void lpa_terminate(
    void **data,
    struct lpa_control_type* control,
    struct lpa_inform_type* inform
);

typedefs

typedef float spc_

spc_ is real single precision

typedef double rpc_

rpc_ is the real working precision used, but may be changed to float by defining the preprocessor variable REAL_32 or (if supported) to __real128 using the variable REAL_128.

typedef int ipc_

ipc_ is the default integer word length used, but may be changed to int64_t by defining the preprocessor variable INTEGER_64.

function calls

void lpa_initialize(void **data, struct lpa_control_type* control, ipc_ *status)

Set default control values and initialize private data

Parameters:

data	holds private internal data
control	is a struct containing control information (see lpa_control_type)
status	is a scalar variable of type ipc_, that gives the exit status from the package. Possible values are (currently): 0 The initialization was successful.

void lpa_read_specfile(struct lpa_control_type* control, const char 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 struct containing control information (see lpa_control_type)
specfile	is a character string containing the name of the specification file

void lpa_import(
    struct lpa_control_type* control,
    void **data,
    ipc_ *status,
    ipc_ n,
    ipc_ m,
    const char A_type[],
    ipc_ A_ne,
    const ipc_ A_row[],
    const ipc_ A_col[],
    const ipc_ A_ptr[]
)

Import problem data into internal storage prior to solution.

Parameters:

control	is a struct whose members provide control paramters for the remaining prcedures (see lpa_control_type)
data	holds private internal data
status	is a scalar variable of type ipc_, 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 ipc_, that holds the number of variables.
m	is a scalar variable of type ipc_, that holds the number of general linear constraints.
A_type	is a one-dimensional array of type char 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 ipc_, 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 ipc_, 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 NULL.
A_col	is a one-dimensional array of size A_ne and type ipc_, 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 NULL.
A_ptr	is a one-dimensional array of size n+1 and type ipc_, 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 NULL.

void lpa_reset_control(
    struct lpa_control_type* control,
    void **data,
    ipc_ *status
)

Reset control parameters after import if required.

Parameters:

control	is a struct whose members provide control paramters for the remaining prcedures (see lpa_control_type)
data	holds private internal data
status	is a scalar variable of type ipc_, that gives the exit status from the package. Possible values are: 0 The import was successful.

void lpa_solve_lp(
    void **data,
    ipc_ *status,
    ipc_ n,
    ipc_ m,
    const rpc_ g[],
    const rpc_ f,
    ipc_ a_ne,
    const rpc_ A_val[],
    const rpc_ c_l[],
    const rpc_ c_u[],
    const rpc_ x_l[],
    const rpc_ x_u[],
    rpc_ x[],
    rpc_ c[],
    rpc_ y[],
    rpc_ z[],
    ipc_ x_stat[],
    ipc_ c_stat[]
)

Solve the linear program.

Parameters:

data	holds private internal data
status	is a scalar variable of type ipc_, 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 ipc_, that holds the number of variables
m	is a scalar variable of type ipc_, that holds the number of general linear constraints.
g	is a one-dimensional array of size n and type rpc_, that holds the linear term \(g\) of the objective function. The j-th component of g, j = 0, … , n-1, contains \(g_j\).
f	is a scalar of type rpc_, that holds the constant term \(f\) of the objective function.
a_ne	is a scalar variable of type ipc_, 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 rpc_, 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 rpc_, that holds the lower bounds \(c^l\) on the constraints \(A x\). The i-th component of c_l, i = 0, … , m-1, contains \(c^l_i\).
c_u	is a one-dimensional array of size m and type rpc_, that holds the upper bounds \(c^l\) on the constraints \(A x\). The i-th component of c_u, i = 0, … , m-1, contains \(c^u_i\).
x_l	is a one-dimensional array of size n and type rpc_, that holds the lower bounds \(x^l\) on the variables \(x\). The j-th component of x_l, j = 0, … , n-1, contains \(x^l_j\).
x_u	is a one-dimensional array of size n and type rpc_, that holds the upper bounds \(x^l\) on the variables \(x\). The j-th component of x_u, j = 0, … , n-1, contains \(x^l_j\).
x	is a one-dimensional array of size n and type rpc_, that holds the values \(x\) of the optimization variables. The j-th component of x, j = 0, … , n-1, contains \(x_j\).
c	is a one-dimensional array of size m and type rpc_, that holds the residual \(c(x)\). The i-th component of c, i = 0, … , m-1, contains \(c_i(x)\).
y	is a one-dimensional array of size n and type rpc_, that holds the values \(y\) of the Lagrange multipliers for the general linear constraints. The j-th component of y, i = 0, … , m-1, contains \(y_i\).
z	is a one-dimensional array of size n and type rpc_, that holds the values \(z\) of the dual variables. The j-th component of z, j = 0, … , n-1, contains \(z_j\).
x_stat	is a one-dimensional array of size n and type ipc_, 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 ipc_, 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.

void lpa_information(void **data, struct lpa_inform_type* inform, ipc_ *status)

Provides output information

Parameters:

data	holds private internal data
inform	is a struct containing output information (see lpa_inform_type)
status	is a scalar variable of type ipc_, that gives the exit status from the package. Possible values are (currently): 0 The values were recorded successfully

void lpa_terminate(
    void **data,
    struct lpa_control_type* control,
    struct lpa_inform_type* inform
)

Deallocate all internal private storage

Parameters:

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

available structures

lpa_control_type structure

#include <galahad_lpa.h>

struct lpa_control_type {
    // components

    bool f_indexing;
    ipc_ error;
    ipc_ out;
    ipc_ print_level;
    ipc_ start_print;
    ipc_ stop_print;
    ipc_ maxit;
    ipc_ max_iterative_refinements;
    ipc_ min_real_factor_size;
    ipc_ min_integer_factor_size;
    ipc_ random_number_seed;
    ipc_ sif_file_device;
    ipc_ qplib_file_device;
    rpc_ infinity;
    rpc_ tol_data;
    rpc_ feas_tol;
    rpc_ relative_pivot_tolerance;
    rpc_ growth_limit;
    rpc_ zero_tolerance;
    rpc_ change_tolerance;
    rpc_ identical_bounds_tol;
    rpc_ cpu_time_limit;
    rpc_ clock_time_limit;
    bool scale;
    bool dual;
    bool warm_start;
    bool steepest_edge;
    bool space_critical;
    bool deallocate_error_fatal;
    bool generate_sif_file;
    bool generate_qplib_file;
    char sif_file_name[31];
    char qplib_file_name[31];
    char prefix[31];
};

detailed documentation

control derived type as a C struct

components

bool f_indexing

use C or Fortran sparse matrix indexing

ipc_ error

error and warning diagnostics occur on stream error

ipc_ out

general output occurs on stream out

ipc_ print_level

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

ipc_ start_print

any printing will start on this iteration

ipc_ stop_print

any printing will stop on this iteration

ipc_ maxit

at most maxit inner iterations are allowed

ipc_ max_iterative_refinements

maximum number of iterative refinements allowed

ipc_ min_real_factor_size

initial size for real array for the factors and other data

ipc_ min_integer_factor_size

initial size for integer array for the factors and other data

ipc_ random_number_seed

the initial seed used when generating random numbers

ipc_ sif_file_device

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

ipc_ qplib_file_device

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

rpc_ infinity

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

rpc_ tol_data

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

rpc_ feas_tol

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

rpc_ 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

rpc_ growth_limit

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

rpc_ zero_tolerance

any entry in the basis smaller than this is considered zero

rpc_ change_tolerance

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

rpc_ 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

rpc_ cpu_time_limit

the maximum CPU time allowed (-ve means infinite)

rpc_ 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

char sif_file_name[31]

name of generated SIF file containing input problem

char qplib_file_name[31]

name of generated QPLIB file containing input problem

char prefix[31]

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

#include <galahad_lpa.h>

struct lpa_time_type {
    // components

    rpc_ total;
    rpc_ preprocess;
    rpc_ clock_total;
    rpc_ clock_preprocess;
};

detailed documentation

time derived type as a C struct

components

rpc_ total

the total CPU time spent in the package

rpc_ preprocess

the CPU time spent preprocessing the problem

rpc_ clock_total

the total clock time spent in the package

rpc_ clock_preprocess

the clock time spent preprocessing the problem

lpa_inform_type structure

#include <galahad_lpa.h>

struct lpa_inform_type {
    // components

    ipc_ status;
    ipc_ alloc_status;
    char bad_alloc[81];
    ipc_ iter;
    ipc_ la04_job;
    ipc_ la04_job_info;
    rpc_ obj;
    rpc_ primal_infeasibility;
    bool feasible;
    rpc_ RINFO[40];
    struct lpa_time_type time;
    struct rpd_inform_type rpd_inform;
};

detailed documentation

inform derived type as a C struct

components

ipc_ status

return status. See LPA_solve for details

ipc_ alloc_status

the status of the last attempted allocation/deallocation

char bad_alloc[81]

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

ipc_ iter

the total number of iterations required

ipc_ la04_job

the final value of la04’s job argument

ipc_ la04_job_info

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

rpc_ obj

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

rpc_ primal_infeasibility

the value of the primal infeasibility

bool feasible

is the returned “solution” feasible?

rpc_ 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/C/lpat.c . A variety of supported constraint matrix storage formats are shown.

Notice that C-style indexing is used, and that this is flagged by setting control.f_indexing to false. The floating-point type rpc_ is set in galahad_precision.h to double by default, but to float if the preprocessor variable SINGLE is defined. Similarly, the integer type ipc_ from galahad_precision.h is set to int by default, but to int64_t if the preprocessor variable INTEGER_64 is defined.

/* lpat.c */
/* Full test for the LPA C interface using C sparse matrix indexing */

#include <stdio.h>
#include <math.h>
#include "galahad_precision.h"
#include "galahad_cfunctions.h"
#include "galahad_lpa.h"
#ifdef REAL_128
#include <quadmath.h>
#endif

int main(void) {

    // Derived types
    void *data;
    struct lpa_control_type control;
    struct lpa_inform_type inform;

    // Set problem data
    ipc_ n = 3; // dimension
    ipc_ m = 2; // number of general constraints
    rpc_ g[] = {0.0, 2.0, 0.0};   // linear term in the objective
    rpc_ f = 1.0;  // constant term in the objective
    ipc_ A_ne = 4; // Jacobian elements
    ipc_ A_row[] = {0, 0, 1, 1}; // row indices
    ipc_ A_col[] = {0, 1, 1, 2}; // column indices
    ipc_ A_ptr[] = {0, 2, 4}; // row pointers
    rpc_ A_val[] = {2.0, 1.0, 1.0, 1.0 }; // values
    rpc_ c_l[] = {1.0, 2.0};   // constraint lower bound
    rpc_ c_u[] = {2.0, 2.0};   // constraint upper bound
    rpc_ x_l[] = {-1.0, - INFINITY, - INFINITY}; // variable lower bound
    rpc_ x_u[] = {1.0, INFINITY, 2.0}; // variable upper bound

    // Set output storage
    rpc_ c[m]; // constraint values
    ipc_ x_stat[n]; // variable status
    ipc_ c_stat[m]; // constraint status
    char st = ' ';
    ipc_ status;

    printf(" C sparse matrix indexing\n\n");

    printf(" basic tests of lp storage formats\n\n");

    for( ipc_ d=1; d <= 3; d++){

        // Initialize LPA
        lpa_initialize( &data, &control, &status );

        // Set user-defined control options
        control.f_indexing = false; // C sparse matrix indexing

        // Start from 0
        rpc_ x[] = {0.0,0.0,0.0};
        rpc_ y[] = {0.0,0.0};
        rpc_ z[] = {0.0,0.0,0.0};

        switch(d){
            case 1: // sparse co-ordinate storage
                st = 'C';
                lpa_import( &control, &data, &status, n, m,
                           "coordinate", A_ne, A_row, A_col, NULL );
                lpa_solve_lp( &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 );
                break;
            printf(" case %1" i_ipc_ " break\n",d);
            case 2: // sparse by rows
                st = 'R';
                lpa_import( &control, &data, &status, n, m,
                            "sparse_by_rows", A_ne, NULL, A_col, A_ptr );
                lpa_solve_lp( &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 );
                break;
            case 3: // dense
                st = 'D';
                ipc_ A_dense_ne = 6; // number of elements of A
                rpc_ A_dense[] = {2.0, 1.0, 0.0, 0.0, 1.0, 1.0};
                lpa_import( &control, &data, &status, n, m,
                            "dense", A_ne, NULL, NULL, NULL );
                lpa_solve_lp( &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 );
                break;
            }
        lpa_information( &data, &inform, &status );

        if(inform.status == 0){
#ifdef REAL_128
// interim replacement for quad output: $GALAHAD/include/galahad_pquad_f.h
#include "galahad_pquad_f.h"
#else
            printf("%c:%6" i_ipc_ " iterations. Optimal objective "
                   "value = %.2f status = %1" i_ipc_ "\n",
                   st, inform.iter, inform.obj, inform.status);
#endif
        }else{
            printf("%c: LPA_solve exit status = %1" i_ipc_ "\n",
                   st, inform.status);
        }
        //printf("x: ");
        //for( ipc_ i = 0; i < n; i++) printf("%f ", x[i]);
        //printf("\n");
        //printf("gradient: ");
        //for( ipc_ i = 0; i < n; i++) printf("%f ", g[i]);
        //printf("\n");

        // Delete internal workspace
        lpa_terminate( &data, &control, &inform );
    }
}

This is the same example, but now fortran-style indexing is used; the code is available in $GALAHAD/src/lpa/C/lpatf.c .

/* lpatf.c */
/* Full test for the LPA C interface using Fortran sparse matrix indexing */

#include <stdio.h>
#include <math.h>
#include "galahad_precision.h"
#include "galahad_cfunctions.h"
#include "galahad_lpa.h"
#ifdef REAL_128
#include <quadmath.h>
#endif

int main(void) {

    // Derived types
    void *data;
    struct lpa_control_type control;
    struct lpa_inform_type inform;

    // Set problem data
    ipc_ n = 3; // dimension
    ipc_ m = 2; // number of general constraints
    rpc_ g[] = {0.0, 2.0, 0.0};   // linear term in the objective
    rpc_ f = 1.0;  // constant term in the objective
    ipc_ A_ne = 4; // Jacobian elements
    ipc_ A_row[] = {1, 1, 2, 2}; // row indices
    ipc_ A_col[] = {1, 2, 2, 3}; // column indices
    ipc_ A_ptr[] = {1, 3, 5}; // row pointers
    rpc_ A_val[] = {2.0, 1.0, 1.0, 1.0 }; // values
    rpc_ c_l[] = {1.0, 2.0};   // constraint lower bound
    rpc_ c_u[] = {2.0, 2.0};   // constraint upper bound
    rpc_ x_l[] = {-1.0, - INFINITY, - INFINITY}; // variable lower bound
    rpc_ x_u[] = {1.0, INFINITY, 2.0}; // variable upper bound

    // Set output storage
    rpc_ c[m]; // constraint values
    ipc_ x_stat[n]; // variable status
    ipc_ c_stat[m]; // constraint status
    char st = ' ';
    ipc_ status;

    printf(" Fortran sparse matrix indexing\n\n");

    printf(" basic tests of lp storage formats\n\n");

    for( ipc_ d=1; d <= 3; d++){

        // Initialize LPA
        lpa_initialize( &data, &control, &status );

        // Set user-defined control options
        control.f_indexing = true; // Fortran sparse matrix indexing

        // Start from 0
        rpc_ x[] = {0.0,0.0,0.0};
        rpc_ y[] = {0.0,0.0};
        rpc_ z[] = {0.0,0.0,0.0};

        switch(d){
            case 1: // sparse co-ordinate storage
                st = 'C';
                lpa_import( &control, &data, &status, n, m,
                           "coordinate", A_ne, A_row, A_col, NULL );
                lpa_solve_lp( &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 );
                break;
            printf(" case %1" i_ipc_ " break\n",d);
            case 2: // sparse by rows
                st = 'R';
                lpa_import( &control, &data, &status, n, m,
                            "sparse_by_rows", A_ne, NULL, A_col, A_ptr );
                lpa_solve_lp( &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 );
                break;
            case 3: // dense
                st = 'D';
                ipc_ A_dense_ne = 6; // number of elements of A
                rpc_ A_dense[] = {2.0, 1.0, 0.0, 0.0, 1.0, 1.0};
                lpa_import( &control, &data, &status, n, m,
                            "dense", A_ne, NULL, NULL, NULL );
                lpa_solve_lp( &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 );
                break;
            }
        lpa_information( &data, &inform, &status );

        if(inform.status == 0){
#ifdef REAL_128
// interim replacement for quad output: $GALAHAD/include/galahad_pquad_f.h
#include "galahad_pquad_f.h"
#else
            printf("%c:%6" i_ipc_ " iterations. Optimal objective "
                   "value = %.2f status = %1" i_ipc_ "\n",
                   st, inform.iter, inform.obj, inform.status);
#endif
        }else{
            printf("%c: LPA_solve exit status = %1" i_ipc_ "\n",
                   st, inform.status);
        }
        //printf("x: ");
        //for( ipc_ i = 0; i < n; i++) printf("%f ", x[i]);
        //printf("\n");
        //printf("gradient: ");
        //for( ipc_ i = 0; i < n; i++) printf("%f ", g[i]);
        //printf("\n");

        // Delete internal workspace
        lpa_terminate( &data, &control, &inform );
    }
}