GALAHAD EQP package#

purpose#

The eqp package uses an iterative method to solve a given equality-constrained quadratic program. The aim is to minimize the quadratic objective function

\[q(x) = f + g^T x + \frac{1}{2} x^T H x,\]

or the shifted-least-distance objective function

\[s(x) = f + g^T x + \frac{1}{2} \sum_{j=1}^n w_j^2 (x_j - x_j^0)^2,\]

subject to the general linear equality constraints

\[A x + c = 0,\]

where $H$ and $A$ are, respectively, given $n$ by $n$ symmetric and $m$ by $n$ general matrices, $g$, $w$, $x^0$ and $c$ are vectors, and $f$ is a scalar. The method is most suitable for problems involving a large number of unknowns $x$.

See Section 4 of $GALAHAD/doc/eqp.pdf for additional details.

terminology#

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

\[A x + c = 0 \;\;\mbox{(1)}\]

and the dual optimality conditions

\[H x + g = A^T y,\]

where the vector $y$ is known as the Lagrange multipliers for the general linear constraints.

In the shifted-least-distance case, $g$ is shifted by $-W^2 x^0$, and $H = W^2$, where $W$ is the diagonal matrix whose entries are the $w_j$.

method#

A solution to the problem is found in two phases. In the first, a point $x_F$ satisfying (1) is found. In the second, the required solution $x = x_F + s$ is determined by finding $s$ to minimize $q(s) = \frac{1}{2} s^T H s + g_F^T s + f_F^{}$ subject to the homogeneous constraints $A s = 0$, where $g_F^{} = H x_F^{} + g$ and $f_F^{} = \frac{1}{2} x_F^T H x_F^{} + g^T x_F^{} + f$. The required constrained minimizer of $q(s)$ is obtained by implictly applying the preconditioned conjugate-gradient method in the null space of $A$. Any preconditioner of the form

\[\begin{split}K_{G} = \left(\begin{array}{cc} G & A^T \\ A & 0 \end{array}\right)\end{split}\]

is suitable, and the package SBLS provides a number of possibilities. In order to ensure that the minimizer obtained is finite, an additional, precautionary trust-region constraint $\|s\| \leq \Delta$ for some suitable positive radius $\Delta$ is imposed, and the package GLTR is used to solve this additionally-constrained problem.

references#

The preconditioning aspcets are described in detail in

H. S. Dollar, N. I. M. Gould and A. J. Wathen. ``On implicit-factorization constraint preconditioners’’. In Large Scale Nonlinear Optimization (G. Di Pillo and M. Roma, eds.) Springer Series on Nonconvex Optimization and Its Applications, Vol. 83, Springer Verlag (2006) 61–82

and

H. S. Dollar, N. I. M. Gould, W. H. A. Schilders and A. J. Wathen ``On iterative methods and implicit-factorization preconditioners for regularized saddle-point systems’’. SIAM Journal on Matrix Analysis and Applications 28(1) (2006) 170–189,

while the constrained conjugate-gradient method is discussed in

N. I. M. Gould, S. Lucidi, M. Roma and Ph. L. Toint, ``Solving the trust-region subproblem using the Lanczos method’’. SIAM Journal on Optimization 9(2) (1999), 504–525.

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 columnsare 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.

The symmetric $n$ by $n$ matrix $H$ may also be presented and stored in a variety of formats. But crucially symmetry is exploited by only storing values from the lower triangular part (i.e, those entries that lie on or below the leading diagonal).

Dense storage format: The matrix $H$ 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. Since $H$ is symmetric, only the lower triangular part (that is the part $H_{ij}$ for $0 \leq j \leq i \leq n-1$) need be held. In this case the lower triangle should be stored by rows, that is component $i * i / 2 + j$ of the storage array H_val will hold the value $H_{ij}$ (and, by symmetry, $H_{ji}$) for $0 \leq j \leq i \leq n-1$. The string H_type = ‘dense’ 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 $H$, its row index i, column index j and value $H_{ij}$, $0 \leq j \leq i \leq n-1$, are stored as the $l$-th components of the integer arrays H_row and H_col and real array H_val, respectively, while the number of nonzeros is recorded as H_ne = $ne$. Note that only the entries in the lower triangle should be stored. The string H_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 $H$ the i-th component of the integer array H_ptr holds the position of the first entry in this row, while H_ptr(n) holds the total number of entries. The column indices j, $0 \leq j \leq i$, and values $H_{ij}$ of the entries in the i-th row are stored in components l = H_ptr(i), …, H_ptr(i+1)-1 of the integer array H_col, and real array H_val, respectively. Note that as before only the entries in the lower triangle should be stored. For sparse matrices, this scheme almost always requires less storage than its predecessor. The string H_type = ‘sparse_by_rows’ should be specified.

Diagonal storage format: If $H$ is diagonal (i.e., $H_{ij} = 0$ for all $0 \leq i \neq j \leq n-1$) only the diagonals entries $H_{ii}$, $0 \leq i \leq n-1$ need be stored, and the first n components of the array H_val may be used for the purpose. The string H_type = ‘diagonal’ should be specified.

Multiples of the identity storage format: If $H$ is a multiple of the identity matrix, (i.e., $H = \alpha I$ where $I$ is the n by n identity matrix and $\alpha$ is a scalar), it suffices to store $\alpha$ as the first component of H_val. The string H_type = ‘scaled_identity’ should be specified.

The identity matrix format: If $H$ is the identity matrix, no values need be stored. The string H_type = ‘identity’ should be specified.

The zero matrix format: The same is true if $H$ is the zero matrix, but now the string H_type = ‘zero’ or ‘none’ should be specified.

introduction to function calls#

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

eqp_initialize - provide default control parameters and set up initial data structures
eqp_read_specfile (optional) - override control values by reading replacement values from a file
eqp_import - set up problem data structures and fixed values
eqp_reset_control (optional) - possibly change control parameters if a sequence of problems are being solved
solve the problem by calling one of
- eqp_solve_qp - solve the quadratic program
- eqp_solve_sldqp - solve the shifted least-distance problem
eqp_resolve_qp (optional) - resolve the problem with the same Hessian and Jacobian, but different $g$, $f$ and/or $c$
eqp_information (optional) - recover information about the solution and solution process
eqp_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 eqp_control_type;
struct eqp_inform_type;
struct eqp_time_type;

// function calls

void eqp_initialize(void **data, struct eqp_control_type* control, ipc_ *status);
void eqp_read_specfile(struct eqp_control_type* control, const char specfile[]);

void eqp_import(
    struct eqp_control_type* control,
    void **data,
    ipc_ *status,
    ipc_ n,
    ipc_ m,
    const char H_type[],
    ipc_ H_ne,
    const ipc_ H_row[],
    const ipc_ H_col[],
    const ipc_ H_ptr[],
    const char A_type[],
    ipc_ A_ne,
    const ipc_ A_row[],
    const ipc_ A_col[],
    const ipc_ A_ptr[]
);

void eqp_reset_control(
    struct eqp_control_type* control,
    void **data,
    ipc_ *status
);

void eqp_solve_qp(
    void **data,
    ipc_ *status,
    ipc_ n,
    ipc_ m,
    ipc_ h_ne,
    const rpc_ H_val[],
    const rpc_ g[],
    const rpc_ f,
    ipc_ a_ne,
    const rpc_ A_val[],
    rpc_ c[],
    rpc_ x[],
    rpc_ y[]
);

void eqp_solve_sldqp(
    void **data,
    ipc_ *status,
    ipc_ n,
    ipc_ m,
    const rpc_ w[],
    const rpc_ x0[],
    const rpc_ g[],
    const rpc_ f,
    ipc_ a_ne,
    const rpc_ A_val[],
    rpc_ c[],
    rpc_ x[],
    rpc_ y[]
);

void eqp_resolve_qp(
    void **data,
    ipc_ *status,
    ipc_ n,
    ipc_ m,
    const rpc_ g[],
    const rpc_ f,
    rpc_ c[],
    rpc_ x[],
    rpc_ y[]
);

void eqp_information(void **data, struct eqp_inform_type* inform, ipc_ *status);

void eqp_terminate(
    void **data,
    struct eqp_control_type* control,
    struct eqp_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 eqp_initialize(void **data, struct eqp_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 eqp_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 eqp_read_specfile(struct eqp_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/eqp/EQP.template. See also Table 2.1 in the Fortran documentation provided in $GALAHAD/doc/eqp.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 eqp_control_type)
specfile	is a character string containing the name of the specification file

void eqp_import(
    struct eqp_control_type* control,
    void **data,
    ipc_ *status,
    ipc_ n,
    ipc_ m,
    const char H_type[],
    ipc_ H_ne,
    const ipc_ H_row[],
    const ipc_ H_col[],
    const ipc_ H_ptr[],
    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 eqp_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’, ‘sparse_by_rows’, ‘diagonal’, ‘scaled_identity’, ‘identity’, ‘zero’ or ‘none’ has been violated. -23 An entry from the strict upper triangle of $H$ has been specified.
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.
H_type	is a one-dimensional array of type char that specifies the symmetric storage scheme used for the Hessian, $H$. It should be one of ‘coordinate’, ‘sparse_by_rows’, ‘dense’, ‘diagonal’, ‘scaled_identity’, ‘identity’, ‘zero’ or ‘none’, the latter pair if $H=0$; lower or upper case variants are allowed.
H_ne	is a scalar variable of type ipc_, that holds the number of entries in the lower triangular part of $H$ in the sparse co-ordinate storage scheme. It need not be set for any of the other schemes.
H_row	is a one-dimensional array of size H_ne and type ipc_, that holds the row indices of the lower triangular part of $H$ in the sparse co-ordinate storage scheme. It need not be set for any of the other three schemes, and in this case can be NULL.
H_col	is a one-dimensional array of size H_ne and type ipc_, that holds the column indices of the lower triangular part of $H$ in either the sparse co-ordinate, or the sparse row-wise storage scheme. It need not be set when the dense, diagonal or (scaled) identity storage schemes are used, and in this case can be NULL.
H_ptr	is a one-dimensional array of size n+1 and type ipc_, that holds the starting position of each row of the lower triangular part of $H$, 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.
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 eqp_reset_control(
    struct eqp_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 eqp_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:

1. The import was successful.

void eqp_solve_qp(
    void **data,
    ipc_ *status,
    ipc_ n,
    ipc_ m,
    ipc_ h_ne,
    const rpc_ H_val[],
    const rpc_ g[],
    const rpc_ f,
    ipc_ a_ne,
    const rpc_ A_val[],
    rpc_ c[],
    rpc_ x[],
    rpc_ y[]
)

Solve the quadratic program when the Hessian $H$ is available.

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’, ‘sparse_by_rows’, ‘diagonal’, ‘scaled_identity’, ‘identity’, ‘zero’ or ‘none’ has been violated. -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. -23 An entry from the strict upper triangle of $H$ has been specified.
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.
h_ne	is a scalar variable of type ipc_, that holds the number of entries in the lower triangular part of the Hessian matrix $H$.
H_val	is a one-dimensional array of size h_ne and type rpc_, that holds the values of the entries of the lower triangular part of the Hessian matrix $H$ in any of the available storage schemes.
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	is a one-dimensional array of size m and type rpc_, that holds the linear term $c$ in the constraints. The i-th component of c, i = 0, … , m-1, contains $c_i$.
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 $x_j$.
y	is a one-dimensional array of size n and type rpc_, that holds the values $y$ of the Lagrange multipliers for the linear constraints. The j-th component of y, i = 0, … , m-1, contains $y_i$.

void eqp_solve_sldqp(
    void **data,
    ipc_ *status,
    ipc_ n,
    ipc_ m,
    const rpc_ w[],
    const rpc_ x0[],
    const rpc_ g[],
    const rpc_ f,
    ipc_ a_ne,
    const rpc_ A_val[],
    rpc_ c[],
    rpc_ x[],
    rpc_ y[]
)

Solve the shifted least-distance quadratic 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’, ‘sparse_by_rows’, ‘diagonal’, ‘scaled_identity’, ‘identity’, ‘zero’ or ‘none’ has been violated. -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. -23 An entry from the strict upper triangle of $H$ has been specified.
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.
w	is a one-dimensional array of size n and type rpc_, that holds the values of the weights $w$.
x0	is a one-dimensional array of size n and type rpc_, that holds the values of the shifts $x^0$.
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	is a one-dimensional array of size m and type rpc_, that holds the linear term $c$ in the constraints. The i-th component of c, i = 0, … , m-1, contains $c_i$.
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 $x_j$.
y	is a one-dimensional array of size n and type rpc_, that holds the values $y$ of the Lagrange multipliers for the linear constraints. The j-th component of y, i = 0, … , m-1, contains $y_i$.

void eqp_resolve_qp(
    void **data,
    ipc_ *status,
    ipc_ n,
    ipc_ m,
    const rpc_ g[],
    const rpc_ f,
    rpc_ c[],
    rpc_ x[],
    rpc_ y[]
)

Resolve the quadratic program or shifted least-distance quadratic program when some or all of the data $g$, $f$ and $c$ has changed

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’, ‘sparse_by_rows’, ‘diagonal’, ‘scaled_identity’, ‘identity’, ‘zero’ or ‘none’ has been violated. -7 The constraints appear to have no feasible point. -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. -23 An entry from the strict upper triangle of $H$ has been specified.
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.
c	is a one-dimensional array of size m and type rpc_, that holds the linear term $c$ in the constraints. The i-th component of c, i = 0, … , m-1, contains $c_i$.
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 $x_j$.
y	is a one-dimensional array of size n and type rpc_, that holds the values $y$ of the Lagrange multipliers for the linear constraints. The j-th component of y, i = 0, … , m-1, contains $y_i$.

void eqp_information(void **data, struct eqp_inform_type* inform, ipc_ *status)

Provides output information

Parameters:

data

holds private internal data

inform

is a struct containing output information (see eqp_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 eqp_terminate(
    void **data,
    struct eqp_control_type* control,
    struct eqp_inform_type* inform
)

Deallocate all internal private storage

Parameters:

data	holds private internal data
control	is a struct containing control information (see eqp_control_type)
inform	is a struct containing output information (see eqp_inform_type)

available structures#

eqp_control_type structure#

#include <galahad_eqp.h>

struct eqp_control_type {
    // components

    bool f_indexing;
    ipc_ error;
    ipc_ out;
    ipc_ print_level;
    ipc_ factorization;
    ipc_ max_col;
    ipc_ indmin;
    ipc_ valmin;
    ipc_ len_ulsmin;
    ipc_ itref_max;
    ipc_ cg_maxit;
    ipc_ preconditioner;
    ipc_ semi_bandwidth;
    ipc_ new_a;
    ipc_ new_h;
    ipc_ sif_file_device;
    rpc_ pivot_tol;
    rpc_ pivot_tol_for_basis;
    rpc_ zero_pivot;
    rpc_ inner_fraction_opt;
    rpc_ radius;
    rpc_ min_diagonal;
    rpc_ max_infeasibility_relative;
    rpc_ max_infeasibility_absolute;
    rpc_ inner_stop_relative;
    rpc_ inner_stop_absolute;
    rpc_ inner_stop_inter;
    bool find_basis_by_transpose;
    bool remove_dependencies;
    bool space_critical;
    bool deallocate_error_fatal;
    bool generate_sif_file;
    char sif_file_name[31];
    char prefix[31];
    struct fdc_control_type fdc_control;
    struct sbls_control_type sbls_control;
    struct gltr_control_type gltr_control;
};

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

ipc_ factorization

the factorization to be used. Possible values are /li 0 automatic /li 1 Schur-complement factorization /li 2 augmented-system factorization (OBSOLETE)

ipc_ max_col

the maximum number of nonzeros in a column of A which is permitted with the Schur-complement factorization (OBSOLETE)

ipc_ indmin

an initial guess as to the integer workspace required by SBLS (OBSOLETE)

ipc_ valmin

an initial guess as to the real workspace required by SBLS (OBSOLETE)

ipc_ len_ulsmin

an initial guess as to the workspace required by ULS (OBSOLETE)

ipc_ itref_max

the maximum number of iterative refinements allowed (OBSOLETE)

ipc_ cg_maxit

the maximum number of CG iterations allowed. If cg_maxit < 0, this number will be reset to the dimension of the system + 1

ipc_ preconditioner

the preconditioner to be used for the CG. Possible values are

0 automatic
1 no preconditioner, i.e, the identity within full factorization
2 full factorization
3 band within full factorization
4 diagonal using the barrier terms within full factorization (OBSOLETE)
5 optionally supplied diagonal, G = D

ipc_ semi_bandwidth

the semi-bandwidth of a band preconditioner, if appropriate (OBSOLETE)

ipc_ new_a

how much has A changed since last problem solved: 0 = not changed, 1 = values changed, 2 = structure changed

ipc_ new_h

how much has H changed since last problem solved: 0 = not changed, 1 = values changed, 2 = structure changed

ipc_ sif_file_device

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

rpc_ pivot_tol

the threshold pivot used by the matrix factorization. See the documentation for SBLS for details (OBSOLETE)

rpc_ pivot_tol_for_basis

the threshold pivot used by the matrix factorization when finding the ba See the documentation for ULS for details (OBSOLETE)

rpc_ zero_pivot

any pivots smaller than zero_pivot in absolute value will be regarded to zero when attempting to detect linearly dependent constraints (OBSOLETE)

rpc_ inner_fraction_opt

the computed solution which gives at least inner_fraction_opt times the optimal value will be found (OBSOLETE)

rpc_ radius

an upper bound on the permitted step (-ve will be reset to an appropriat large value by eqp_solve)

rpc_ min_diagonal

diagonal preconditioners will have diagonals no smaller than min_diagonal (OBSOLETE)

rpc_ max_infeasibility_relative

if the constraints are believed to be rank defficient and the residual at a “typical” feasible point is larger than max( max_infeasibility_relative * norm A, max_infeasibility_absolute ) the problem will be marked as infeasible

rpc_ max_infeasibility_absolute

see max_infeasibility_relative

rpc_ inner_stop_relative

the computed solution is considered as an acceptable approximation to th minimizer of the problem if the gradient of the objective in the preconditioning(inverse) norm is less than max( inner_stop_relative * initial preconditioning(inverse) gradient norm, inner_stop_absolute )

rpc_ inner_stop_absolute

see inner_stop_relative

rpc_ inner_stop_inter

see inner_stop_relative

bool find_basis_by_transpose

if .find_basis_by_transpose is true, implicit factorization precondition will be based on a basis of A found by examining A’s transpose (OBSOLETE)

bool remove_dependencies

if .remove_dependencies is true, the equality constraints will be preprocessed to remove any linear dependencies

bool space_critical

if .space_critical 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

char sif_file_name[31]

name of generated SIF 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’

struct fdc_control_type fdc_control

control parameters for FDC

struct sbls_control_type sbls_control

control parameters for SBLS

struct gltr_control_type gltr_control

control parameters for GLTR

eqp_time_type structure#

#include <galahad_eqp.h>

struct eqp_time_type {
    // components

    rpc_ total;
    rpc_ find_dependent;
    rpc_ factorize;
    rpc_ solve;
    rpc_ solve_inter;
    rpc_ clock_total;
    rpc_ clock_find_dependent;
    rpc_ clock_factorize;
    rpc_ clock_solve;
};

detailed documentation#

time derived type as a C struct

components#

rpc_ total

the total CPU time spent in the package

rpc_ find_dependent

the CPU time spent detecting linear dependencies

rpc_ factorize

the CPU time spent factorizing the required matrices

rpc_ solve

the CPU time spent computing the search direction

rpc_ solve_inter

see solve

rpc_ clock_total

the total clock time spent in the package

rpc_ clock_find_dependent

the clock time spent detecting linear dependencies

rpc_ clock_factorize

the clock time spent factorizing the required matrices

rpc_ clock_solve

the clock time spent computing the search direction

eqp_inform_type structure#

#include <galahad_eqp.h>

struct eqp_inform_type {
    // components

    ipc_ status;
    ipc_ alloc_status;
    char bad_alloc[81];
    ipc_ cg_iter;
    ipc_ cg_iter_inter;
    int64_t factorization_integer;
    int64_t factorization_real;
    rpc_ obj;
    struct eqp_time_type time;
    struct fdc_inform_type fdc_inform;
    struct sbls_inform_type sbls_inform;
    struct gltr_inform_type gltr_inform;
};

detailed documentation#

inform derived type as a C struct

components#

ipc_ status

return status. See EQP_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_ cg_iter

the total number of conjugate gradient iterations required

ipc_ cg_iter_inter

see cg_iter

int64_t factorization_integer

the total integer workspace required for the factorization

int64_t factorization_real

the total real workspace required for the factorization

rpc_ obj

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

struct eqp_time_type time

timings (see above)

struct fdc_inform_type fdc_inform

inform parameters for FDC

struct sbls_inform_type sbls_inform

inform parameters for SBLS

struct gltr_inform_type gltr_inform

return information from GLTR

Index

example calls#

This is an example of how to use the package to solve an equality-constrained quadratic program; the code is available in $GALAHAD/src/eqp/C/eqpt.c . A variety of supported Hessian and 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.

/* eqpt.c */
/* Full test for the EQP C interface using C sparse matrix indexing */

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

int main(void) {

    // Derived types
    void *data;
    struct eqp_control_type control;
    struct eqp_inform_type inform;

    // Set problem data
    ipc_ n = 3; // dimension
    ipc_ m = 2; // number of general constraints
    ipc_ H_ne = 3; // Hesssian elements
    ipc_ H_row[] = {0, 1, 2 };   // row indices, NB lower triangle
    ipc_ H_col[] = {0, 1, 2};    // column indices, NB lower triangle
    ipc_ H_ptr[] = {0, 1, 2, 3}; // row pointers
    rpc_ H_val[] = {1.0, 1.0, 1.0 };   // values
    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[] = {3.0, 0.0};   // rhs of the constraints

    // Set output storage
    char st = ' ';
    ipc_ status;

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

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

    for( ipc_ d=1; d <= 6; d++){

        // Initialize EQP
        eqp_initialize( &data, &control, &status );

        // Set user-defined control options
        control.f_indexing = false; // C sparse matrix indexing
        control.fdc_control.use_sls = true ;
        strcpy(control.fdc_control.symmetric_linear_solver, "sytr ") ;
        strcpy(control.sbls_control.symmetric_linear_solver, "sytr ") ;
        strcpy(control.sbls_control.definite_linear_solver, "sytr ") ;

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

        switch(d){
            case 1: // sparse co-ordinate storage
                st = 'C';
                eqp_import( &control, &data, &status, n, m,
                           "coordinate", H_ne, H_row, H_col, NULL,
                           "coordinate", A_ne, A_row, A_col, NULL );
                eqp_solve_qp( &data, &status, n, m, H_ne, H_val, g, f,
                              A_ne, A_val, c, x, y );
                break;
            case 2: // sparse by rows
                st = 'R';
                eqp_import( &control, &data, &status, n, m,
                            "sparse_by_rows", H_ne, NULL, H_col, H_ptr,
                            "sparse_by_rows", A_ne, NULL, A_col, A_ptr );
                eqp_solve_qp( &data, &status, n, m, H_ne, H_val, g, f,
                              A_ne, A_val, c, x, y );
                break;
            case 3: // dense
                st = 'D';
                ipc_ H_dense_ne = 6; // number of elements of H
                ipc_ A_dense_ne = 6; // number of elements of A
                rpc_ H_dense[] = {1.0, 0.0, 1.0, 0.0, 0.0, 1.0};
                rpc_ A_dense[] = {2.0, 1.0, 0.0, 0.0, 1.0, 1.0};
                eqp_import( &control, &data, &status, n, m,
                            "dense", H_ne, NULL, NULL, NULL,
                            "dense", A_ne, NULL, NULL, NULL );
                eqp_solve_qp( &data, &status, n, m, H_dense_ne, H_dense, g, f,
                              A_dense_ne, A_dense, c, x, y );
                break;
            case 4: // diagonal
                st = 'L';
                eqp_import( &control, &data, &status, n, m,
                            "diagonal", H_ne, NULL, NULL, NULL,
                            "sparse_by_rows", A_ne, NULL, A_col, A_ptr );
                eqp_solve_qp( &data, &status, n, m, H_ne, H_val, g, f,
                              A_ne, A_val, c, x, y );
                break;

            case 5: // scaled identity
                st = 'S';
                eqp_import( &control, &data, &status, n, m,
                            "scaled_identity", H_ne, NULL, NULL, NULL,
                            "sparse_by_rows", A_ne, NULL, A_col, A_ptr );
                eqp_solve_qp( &data, &status, n, m, H_ne, H_val, g, f,
                              A_ne, A_val, c, x, y );
                break;
            case 6: // identity
                st = 'I';
                eqp_import( &control, &data, &status, n, m,
                            "identity", H_ne, NULL, NULL, NULL,
                            "sparse_by_rows", A_ne, NULL, A_col, A_ptr );
                eqp_solve_qp( &data, &status, n, m, H_ne, H_val, g, f,
                              A_ne, A_val, c, x, y );
                break;
            case 7: // zero
                st = 'Z';
                eqp_import( &control, &data, &status, n, m,
                            "zero", H_ne, NULL, NULL, NULL,
                            "sparse_by_rows", A_ne, NULL, A_col, A_ptr );
                eqp_solve_qp( &data, &status, n, m, H_ne, H_val, g, f,
                              A_ne, A_val, c, x, y );
                break;
            }
        eqp_information( &data, &inform, &status );

        if(inform.status == 0){
#ifdef REAL_128
// interim replacement for quad output: $GALAHAD/include/galahad_pquad_if.h
#include "galahad_pquad_if.h"
#else
            printf("%c:%6" i_ipc_ " cg iterations. Optimal objective "
                   "value = %.2f status = %1" i_ipc_ "\n",
                   st, inform.cg_iter, inform.obj, inform.status);
#endif
        }else{
            printf("%c: EQP_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
        eqp_terminate( &data, &control, &inform );
    }

    // test shifted least-distance interface
    for( ipc_ d=1; d <= 1; d++){

        // Initialize EQP
        eqp_initialize( &data, &control, &status );

        // Set user-defined control options
        control.f_indexing = false; // C sparse matrix indexing
        control.fdc_control.use_sls = true ;
        strcpy(control.fdc_control.symmetric_linear_solver, "sytr ") ;
        strcpy(control.sbls_control.symmetric_linear_solver, "sytr ") ;
        strcpy(control.sbls_control.definite_linear_solver, "sytr ") ;

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

        // Set shifted least-distance data

        rpc_ w[] = {1.0,1.0,1.0};
        rpc_ x_0[] = {0.0,0.0,0.0};

        switch(d){
            case 1: // sparse co-ordinate storage
                st = 'W';
                eqp_import( &control, &data, &status, n, m,
                           "shifted_least_distance", H_ne, NULL, NULL, NULL,
                           "coordinate", A_ne, A_row, A_col, NULL );
                eqp_solve_sldqp( &data, &status, n, m, w, x_0, g, f,
                                 A_ne, A_val, c, x, y );
                break;

            }
        eqp_information( &data, &inform, &status );

        if(inform.status == 0){
#ifdef REAL_128
// interim replacement for quad output: $GALAHAD/include/galahad_pquad_if.h
#include "galahad_pquad_if.h"
#else
            printf("%c:%6" i_ipc_ " cg iterations. Optimal objective "
                   "value = %.2f status = %1" i_ipc_ "\n",
                   st, inform.cg_iter, inform.obj, inform.status);
#endif
        }else{
            printf("%c: EQP_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
        eqp_terminate( &data, &control, &inform );
    }

}

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

/* eqptf.c */
/* Full test for the EQP C interface using Fortran sparse matrix indexing */

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

int main(void) {

    // Derived types
    void *data;
    struct eqp_control_type control;
    struct eqp_inform_type inform;

    // Set problem data
    ipc_ n = 3; // dimension
    ipc_ m = 2; // number of general constraints
    ipc_ H_ne = 3; // Hesssian elements
    ipc_ H_row[] = {1, 2, 3 };   // row indices, NB lower triangle
    ipc_ H_col[] = {1, 2, 3};    // column indices, NB lower triangle
    ipc_ H_ptr[] = {1, 2, 3, 4}; // row pointers
    rpc_ H_val[] = {1.0, 1.0, 1.0 };   // values
    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[] = {3.0, 0.0};   // rhs of the constraints

    // Set output storage
    char st = ' ';
    ipc_ status;

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

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

    for( ipc_ d=1; d <= 6; d++){

        // Initialize EQP
        eqp_initialize( &data, &control, &status );

        // Set user-defined control options
        control.f_indexing = true; // Fortran sparse matrix indexing
        control.fdc_control.use_sls = true ;
        strcpy(control.fdc_control.symmetric_linear_solver, "sytr ") ;
        strcpy(control.sbls_control.symmetric_linear_solver, "sytr ") ;
        strcpy(control.sbls_control.definite_linear_solver, "sytr ") ;

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

        switch(d){
            case 1: // sparse co-ordinate storage
                st = 'C';
                eqp_import( &control, &data, &status, n, m,
                           "coordinate", H_ne, H_row, H_col, NULL,
                           "coordinate", A_ne, A_row, A_col, NULL );
                eqp_solve_qp( &data, &status, n, m, H_ne, H_val, g, f,
                              A_ne, A_val, c, x, y );
                break;
            case 2: // sparse by rows
                st = 'R';
                eqp_import( &control, &data, &status, n, m,
                            "sparse_by_rows", H_ne, NULL, H_col, H_ptr,
                            "sparse_by_rows", A_ne, NULL, A_col, A_ptr );
                eqp_solve_qp( &data, &status, n, m, H_ne, H_val, g, f,
                              A_ne, A_val, c, x, y );
                break;
            case 3: // dense
                st = 'D';
                ipc_ H_dense_ne = 6; // number of elements of H
                ipc_ A_dense_ne = 6; // number of elements of A
                rpc_ H_dense[] = {1.0, 0.0, 1.0, 0.0, 0.0, 1.0};
                rpc_ A_dense[] = {2.0, 1.0, 0.0, 0.0, 1.0, 1.0};
                eqp_import( &control, &data, &status, n, m,
                            "dense", H_ne, NULL, NULL, NULL,
                            "dense", A_ne, NULL, NULL, NULL );
                eqp_solve_qp( &data, &status, n, m, H_dense_ne, H_dense, g, f,
                              A_dense_ne, A_dense, c, x, y );
                break;
            case 4: // diagonal
                st = 'L';
                eqp_import( &control, &data, &status, n, m,
                            "diagonal", H_ne, NULL, NULL, NULL,
                            "sparse_by_rows", A_ne, NULL, A_col, A_ptr );
                eqp_solve_qp( &data, &status, n, m, H_ne, H_val, g, f,
                              A_ne, A_val, c, x, y );
                break;

            case 5: // scaled identity
                st = 'S';
                eqp_import( &control, &data, &status, n, m,
                            "scaled_identity", H_ne, NULL, NULL, NULL,
                            "sparse_by_rows", A_ne, NULL, A_col, A_ptr );
                eqp_solve_qp( &data, &status, n, m, H_ne, H_val, g, f,
                              A_ne, A_val, c, x, y );
                break;
            case 6: // identity
                st = 'I';
                eqp_import( &control, &data, &status, n, m,
                            "identity", H_ne, NULL, NULL, NULL,
                            "sparse_by_rows", A_ne, NULL, A_col, A_ptr );
                eqp_solve_qp( &data, &status, n, m, H_ne, H_val, g, f,
                              A_ne, A_val, c, x, y );
                break;
            case 7: // zero
                st = 'Z';
                eqp_import( &control, &data, &status, n, m,
                            "zero", H_ne, NULL, NULL, NULL,
                            "sparse_by_rows", A_ne, NULL, A_col, A_ptr );
                eqp_solve_qp( &data, &status, n, m, H_ne, H_val, g, f,
                              A_ne, A_val, c, x, y );
                break;
            }
        eqp_information( &data, &inform, &status );

        if(inform.status == 0){
#ifdef REAL_128
#include "galahad_pquad_if.h"
#else
            printf("%c:%6" i_ipc_ " cg iterations. Optimal objective "
                   "value = %.2f status = %1" i_ipc_ "\n",
                   st, inform.cg_iter, inform.obj, inform.status);
#endif
        }else{
            printf("%c: EQP_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
        eqp_terminate( &data, &control, &inform );
    }

    // test shifted least-distance interface
    for( ipc_ d=1; d <= 1; d++){

        // Initialize EQP
        eqp_initialize( &data, &control, &status );
        control.fdc_control.use_sls = true ;
        strcpy(control.fdc_control.symmetric_linear_solver, "sytr ") ;
        strcpy(control.sbls_control.symmetric_linear_solver, "sytr ") ;
        strcpy(control.sbls_control.definite_linear_solver, "sytr ") ;

        // 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};

        // Set shifted least-distance data

        rpc_ w[] = {1.0,1.0,1.0};
        rpc_ x_0[] = {0.0,0.0,0.0};

        switch(d){
            case 1: // sparse co-ordinate storage
                st = 'W';
                eqp_import( &control, &data, &status, n, m,
                           "shifted_least_distance", H_ne, NULL, NULL, NULL,
                           "coordinate", A_ne, A_row, A_col, NULL );
                eqp_solve_sldqp( &data, &status, n, m, w, x_0, g, f,
                                 A_ne, A_val, c, x, y );
                break;

            }
        eqp_information( &data, &inform, &status );

        if(inform.status == 0){
#ifdef REAL_128
#include "galahad_pquad_if.h"
#else
            printf("%c:%6" i_ipc_ " cg iterations. Optimal objective "
                   "value = %.2f status = %1" i_ipc_ "\n",
                   st, inform.cg_iter, inform.obj, inform.status);
#endif
        }else{
            printf("%c: EQP_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
        eqp_terminate( &data, &control, &inform );
    }

}

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’, ‘sparse_by_rows’, ‘diagonal’, ‘scaled_identity’, ‘identity’, ‘zero’ or ‘none’ has been violated. -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. -23 An entry from the strict upper triangle of \(H\) has been specified.
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.
h_ne	is a scalar variable of type ipc_, that holds the number of entries in the lower triangular part of the Hessian matrix \(H\).
H_val	is a one-dimensional array of size h_ne and type rpc_, that holds the values of the entries of the lower triangular part of the Hessian matrix \(H\) in any of the available storage schemes.
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	is a one-dimensional array of size m and type rpc_, that holds the linear term \(c\) in the constraints. The i-th component of c, i = 0, … , m-1, contains \(c_i\).
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 \(x_j\).
y	is a one-dimensional array of size n and type rpc_, that holds the values \(y\) of the Lagrange multipliers for the linear constraints. The j-th component of y, i = 0, … , m-1, contains \(y_i\).

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’, ‘sparse_by_rows’, ‘diagonal’, ‘scaled_identity’, ‘identity’, ‘zero’ or ‘none’ has been violated. -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. -23 An entry from the strict upper triangle of \(H\) has been specified.
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.
w	is a one-dimensional array of size n and type rpc_, that holds the values of the weights \(w\).
x0	is a one-dimensional array of size n and type rpc_, that holds the values of the shifts \(x^0\).
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	is a one-dimensional array of size m and type rpc_, that holds the linear term \(c\) in the constraints. The i-th component of c, i = 0, … , m-1, contains \(c_i\).
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 \(x_j\).
y	is a one-dimensional array of size n and type rpc_, that holds the values \(y\) of the Lagrange multipliers for the linear constraints. The j-th component of y, i = 0, … , m-1, contains \(y_i\).

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’, ‘sparse_by_rows’, ‘diagonal’, ‘scaled_identity’, ‘identity’, ‘zero’ or ‘none’ has been violated. -7 The constraints appear to have no feasible point. -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. -23 An entry from the strict upper triangle of \(H\) has been specified.
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.
c	is a one-dimensional array of size m and type rpc_, that holds the linear term \(c\) in the constraints. The i-th component of c, i = 0, … , m-1, contains \(c_i\).
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 \(x_j\).
y	is a one-dimensional array of size n and type rpc_, that holds the values \(y\) of the Lagrange multipliers for the linear constraints. The j-th component of y, i = 0, … , m-1, contains \(y_i\).