GALAHAD DQP package#

purpose#

The dqp package uses a dual gradient-projection method to solve a given stricly-convex 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 constraints and simple bounds

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

where $H$ and $A$ are, respectively, given $n$ by $n$ symmetric postive-definite and $m$ by $n$ matrices, $g$, $w$ and $x^0$ are vectors, $f$ 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/dqp.pdf for additional details.

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

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

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.

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#

Dual gradient-projection methods solve the quadratic programmimg problem by instead solving the dual quadratic program

\[\begin{split}\begin{array}{ll}\mbox{minimize}\;\; q^D(y^{l}, y^{u}, z^{l}, z^{u}) = & \!\!\! \frac{1}{2} [ ( y^{l0} + y^{u} )^T A + ( z^{l} + z^{u} ]^T ) H^{-1} [ A^T ( y^{l} + y^{u} ) + z^{l} + z^{u} ] \\ & - [ ( y^{l} + y^{u} )^T A + ( z^{l} + z^{u} ]^T ) H^{-1} g - ( c^{l T} y^{l} + c^{u T} y^{u} + x^{l T} z^{l} + x^{u T} z^{u}) \\ \mbox{subject to} & ( y^{l}, z^{l} ) \geq 0 \;\;\mbox{and} \;\; (y^{u}, z^{u}) \leq 0,\end{array}\end{split}\]

and then recovering the required solution from the linear system

\[H x = - g + A^T ( y^{l} + y^{u} ) + z^{l} + z^{u}.\]

The dual problem is solved by an accelerated gradient-projection method comprising of alternating phases in which (i) the current projected dual gradient is traced downhill (the ‘arc search’) as far as possible and (ii) the dual variables that are currently on their bounds are temporarily fixed and the unconstrained minimizer of $q^D(y^{l}, y^{u}, z^{l}, z^{u})$ with respect to the remaining variables is sought; the minimizer in the second phase may itself need to be projected back into the dual feasible region (either using a brute-force backtrack or a second arc search).

Both phases require the solution of sparse systems of symmetric linear equations, and these are handled by the matrix factorization package SBLS or the conjugate-gradient package GLTR. The systems are commonly singular, and this leads to a requirement to find the Fredholm Alternative for the given matrix and its right-hand side. In the non-singular case, there is an option to update existing factorizations using the “Schur-complement” approach given by the package SCU.

Optionally, the problem may be pre-processed temporarily to eliminate dependent constraints using the package FDC. This may improve the performance of the subsequent iteration.

reference#

The basic algorithm is described in

N. I. M. Gould and D. P. Robinson, ``A dual gradient-projection method for large-scale strictly-convex quadratic problems’’, Computational Optimization and Applications 67(1) (2017) 1-38.

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.

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, $0 \leq i \leq n-1,$ 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 dqp package must be called in the following order:

dqp_initialize - provide default control parameters and set up initial data structures
dqp_read_specfile (optional) - override control values by reading replacement values from a file
dqp_import - set up problem data structures and fixed values
dqp_reset_control (optional) - possibly change control parameters if a sequence of problems are being solved
solve the problem by calling one of
- dqp_solve_qp - solve the quadratic program
- dqp_solve_sldqp - solve the shifted least-distance problem
dqp_information (optional) - recover information about the solution and solution process
dqp_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 dqp_control_type;
struct dqp_inform_type;
struct dqp_time_type;

// function calls

void dqp_initialize(void **data, struct dqp_control_type* control, ipc_ *status);
void dqp_read_specfile(struct dqp_control_type* control, const char specfile[]);

void dqp_import(
    struct dqp_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 dqp_reset_control(
    struct dqp_control_type* control,
    void **data,
    ipc_ *status
);

void dqp_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[],
    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 dqp_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[],
    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 dqp_information(void **data, struct dqp_inform_type* inform, ipc_ *status);

void dqp_terminate(
    void **data,
    struct dqp_control_type* control,
    struct dqp_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 dqp_initialize(void **data, struct dqp_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 dqp_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 dqp_read_specfile(struct dqp_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/dqp/DQP.template. See also Table 2.1 in the Fortran documentation provided in $GALAHAD/doc/dqp.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 dqp_control_type)
specfile	is a character string containing the name of the specification file

void dqp_import(
    struct dqp_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 dqp_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’ or ‘identity’ 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’, or ‘identity’; 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 dqp_reset_control(
    struct dqp_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 dqp_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 dqp_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[],
    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 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. -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. -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_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, j = 0, … , n-1, contains $c_j(x)$.
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, j = 0, … , n-1, contains $y_j$.
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 dqp_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[],
    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 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. -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. -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_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, j = 0, … , n-1, contains $c_j(x)$.
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, j = 0, … , n-1, contains $y_j$.
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 dqp_information(void **data, struct dqp_inform_type* inform, ipc_ *status)

Provides output information

Parameters:

data

holds private internal data

inform

is a struct containing output information (see dqp_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 dqp_terminate(
    void **data,
    struct dqp_control_type* control,
    struct dqp_inform_type* inform
)

Deallocate all internal private storage

Parameters:

data	holds private internal data
control	is a struct containing control information (see dqp_control_type)
inform	is a struct containing output information (see dqp_inform_type)

available structures#

dqp_control_type structure#

#include <galahad_dqp.h>

struct dqp_control_type {
    // components

    bool f_indexing;
    ipc_ error;
    ipc_ out;
    ipc_ print_level;
    ipc_ start_print;
    ipc_ stop_print;
    ipc_ print_gap;
    ipc_ dual_starting_point;
    ipc_ maxit;
    ipc_ max_sc;
    ipc_ cauchy_only;
    ipc_ arc_search_maxit;
    ipc_ cg_maxit;
    ipc_ explore_optimal_subspace;
    ipc_ restore_problem;
    ipc_ sif_file_device;
    ipc_ qplib_file_device;
    rpc_ rho;
    rpc_ infinity;
    rpc_ stop_abs_p;
    rpc_ stop_rel_p;
    rpc_ stop_abs_d;
    rpc_ stop_rel_d;
    rpc_ stop_abs_c;
    rpc_ stop_rel_c;
    rpc_ stop_cg_relative;
    rpc_ stop_cg_absolute;
    rpc_ cg_zero_curvature;
    rpc_ max_growth;
    rpc_ identical_bounds_tol;
    rpc_ cpu_time_limit;
    rpc_ clock_time_limit;
    rpc_ initial_perturbation;
    rpc_ perturbation_reduction;
    rpc_ final_perturbation;
    bool factor_optimal_matrix;
    bool remove_dependencies;
    bool treat_zero_bounds_as_general;
    bool exact_arc_search;
    bool subspace_direct;
    bool subspace_alternate;
    bool subspace_arc_search;
    bool space_critical;
    bool deallocate_error_fatal;
    bool generate_sif_file;
    bool generate_qplib_file;
    char symmetric_linear_solver[31];
    char definite_linear_solver[31];
    char unsymmetric_linear_solver[31];
    char sif_file_name[31];
    char qplib_file_name[31];
    char prefix[31];
    struct fdc_control_type fdc_control;
    struct sls_control_type sls_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_ start_print

any printing will start on this iteration

ipc_ stop_print

any printing will stop on this iteration

ipc_ print_gap

printing will only occur every print_gap iterations

ipc_ dual_starting_point

which starting point should be used for the dual problem

-1 user supplied comparing primal vs dual variables
0 user supplied
1 minimize linearized dual
2 minimize simplified quadratic dual
3 all free (= all active primal costraints)
4 all fixed on bounds (= no active primal costraints)

ipc_ maxit

at most maxit inner iterations are allowed

ipc_ max_sc

the maximum permitted size of the Schur complement before a refactorization is performed (used in the case where there is no Fredholm Alternative, 0 = refactor every iteration)

ipc_ cauchy_only

a subspace step will only be taken when the current Cauchy step has changed no more than than cauchy_only active constraints; the subspace step will always be taken if cauchy_only < 0

ipc_ arc_search_maxit

how many iterations are allowed per arc search (-ve = as many as require

ipc_ cg_maxit

how many CG iterations to perform per DQP iteration (-ve reverts to n+1)

ipc_ explore_optimal_subspace

once a potentially optimal subspace has been found, investigate it

0 as per an ordinary subspace
1 by increasing the maximum number of allowed CG iterations
2 by switching to a direct method

ipc_ restore_problem

indicate whether and how much of the input problem should be restored on output. Possible values are

0 nothing restored
1 scalar and vector parameters
2 all parameters

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_ rho

the penalty weight, rho. The general constraints are not enforced explicitly, but instead included in the objective as a penalty term weighted by rho when rho > 0. If rho <= 0, the general constraints are explicit (that is, there is no penalty term in the objective function)

rpc_ infinity

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

rpc_ stop_abs_p

the required absolute and relative accuracies for the primal infeasibilies

rpc_ stop_rel_p

see stop_abs_p

rpc_ stop_abs_d

the required absolute and relative accuracies for the dual infeasibility

rpc_ stop_rel_d

see stop_abs_d

rpc_ stop_abs_c

the required absolute and relative accuracies for the complementarity

rpc_ stop_rel_c

see stop_abs_c

rpc_ stop_cg_relative

the CG iteration will be stopped as soon as the current norm of the preconditioned gradient is smaller than max( stop_cg_relative * initial preconditioned gradient, stop_cg_absolute )

rpc_ stop_cg_absolute

see stop_cg_relative

rpc_ cg_zero_curvature

threshold below which curvature is regarded as zero if CG is used

rpc_ max_growth

maximum growth factor allowed without a refactorization

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)

rpc_ initial_perturbation

the initial penalty weight (for DLP only)

rpc_ perturbation_reduction

the penalty weight reduction factor (for DLP only)

rpc_ final_perturbation

the final penalty weight (for DLP only)

bool factor_optimal_matrix

are the factors of the optimal augmented matrix required? (for DLP only)

bool remove_dependencies

the equality constraints will be preprocessed to remove any linear dependencies if true

bool treat_zero_bounds_as_general

any problem bound with the value zero will be treated as if it were a general value if true

bool exact_arc_search

if .exact_arc_search is true, an exact piecewise arc search will be performed. Otherwise an ineaxt search using a backtracing Armijo strategy will be employed

bool subspace_direct

if .subspace_direct is true, the subspace step will be calculated using a direct (factorization) method, while if it is false, an iterative (conjugate-gradient) method will be used.

bool subspace_alternate

if .subspace_alternate is true, the subspace step will alternate between a direct (factorization) method and an iterative (GLTR conjugate-gradient) method. This will override .subspace_direct

bool subspace_arc_search

if .subspace_arc_search is true, a piecewise arc search will be performed along the subspace step. Otherwise the search will stop at the firstconstraint encountered

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

bool generate_qplib_file

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

char symmetric_linear_solver[31]

the name of the symmetric-indefinite linear equation solver used. Possible choices are currently: ‘sils’, ‘ma27’, ‘ma57’, ‘ma77’, ‘ma86’, ‘ma97’, ‘ssids’, ‘mumps’, ‘pardiso’, ‘mkl_pardiso’, ‘pastix’, ‘wsmp’, and ‘sytr’, although only ‘sytr’ and, for OMP 4.0-compliant compilers, ‘ssids’ are installed by default; others are easily installed (see README.external). More details of the capabilities of each solver are provided in the documentation for galahad_sls.

char definite_linear_solver[31]

the name of the definite linear equation solver used. Possible choices are currently: ‘sils’, ‘ma27’, ‘ma57’, ‘ma77’, ‘ma86’, ‘ma87’, ‘ma97’, ‘ssids’, ‘mumps’, ‘pardiso’, ‘mkl_pardiso’, ‘pastix’, ‘wsmp’, ‘potr’, ‘sytr’ and ‘pbtr’, although only ‘potr’, ‘sytr’, ‘pbtr’ and, for OMP 4.0-compliant compilers, ‘ssids’ are installed by default; others are easily installed (see README.external). More details of the capabilities of each solver are provided in the documentation for galahad_sls.

char unsymmetric_linear_solver[31]

the name of the unsymmetric linear equation solver used. Possible choices are currently: ‘gls’, ‘ma48’ and ‘getr’, although only ‘getr’ is installed by default; others are easily installed (see README.external). More details of the capabilities of each solver are provided in the documentation for galahad_uls.

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’

struct fdc_control_type fdc_control

control parameters for FDC

struct sls_control_type sls_control

control parameters for SLS

struct sbls_control_type sbls_control

control parameters for SBLS

struct gltr_control_type gltr_control

control parameters for GLTR

dqp_time_type structure#

#include <galahad_dqp.h>

struct dqp_time_type {
    // components

    rpc_ total;
    rpc_ preprocess;
    rpc_ find_dependent;
    rpc_ analyse;
    rpc_ factorize;
    rpc_ solve;
    rpc_ search;
    rpc_ clock_total;
    rpc_ clock_preprocess;
    rpc_ clock_find_dependent;
    rpc_ clock_analyse;
    rpc_ clock_factorize;
    rpc_ clock_solve;
    rpc_ clock_search;
};

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_ find_dependent

the CPU time spent detecting linear dependencies

rpc_ analyse

the CPU time spent analysing the required matrices prior to factorization

rpc_ factorize

the CPU time spent factorizing the required matrices

rpc_ solve

the CPU time spent computing the search direction

rpc_ search

the CPU time spent in the linesearch

rpc_ clock_total

the total clock time spent in the package

rpc_ clock_preprocess

the clock time spent preprocessing the problem

rpc_ clock_find_dependent

the clock time spent detecting linear dependencies

rpc_ clock_analyse

the clock time spent analysing the required matrices prior to factorization

rpc_ clock_factorize

the clock time spent factorizing the required matrices

rpc_ clock_solve

the clock time spent computing the search direction

rpc_ clock_search

the clock time spent in the linesearch

dqp_inform_type structure#

#include <galahad_dqp.h>

struct dqp_inform_type {
    // components

    ipc_ status;
    ipc_ alloc_status;
    char bad_alloc[81];
    ipc_ iter;
    ipc_ cg_iter;
    ipc_ factorization_status;
    int64_t factorization_integer;
    int64_t factorization_real;
    ipc_ nfacts;
    ipc_ threads;
    rpc_ obj;
    rpc_ primal_infeasibility;
    rpc_ dual_infeasibility;
    rpc_ complementary_slackness;
    rpc_ non_negligible_pivot;
    bool feasible;
    ipc_ checkpointsIter[16];
    rpc_ checkpointsTime[16];
    struct dqp_time_type time;
    struct fdc_inform_type fdc_inform;
    struct sls_inform_type sls_inform;
    struct sbls_inform_type sbls_inform;
    struct gltr_inform_type gltr_inform;
    ipc_ scu_status;
    struct scu_inform_type scu_inform;
    struct rpd_inform_type rpd_inform;
};

detailed documentation#

inform derived type as a C struct

components#

ipc_ status

return status. See DQP_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_ cg_iter

the total number of iterations required

ipc_ factorization_status

the return status from the factorization

int64_t factorization_integer

the total integer workspace required for the factorization

int64_t factorization_real

the total real workspace required for the factorization

ipc_ nfacts

the total number of factorizations performed

ipc_ threads

the number of threads used

rpc_ obj

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

rpc_ primal_infeasibility

the value of the primal infeasibility

rpc_ dual_infeasibility

the value of the dual infeasibility

rpc_ complementary_slackness

the value of the complementary slackness

rpc_ non_negligible_pivot

the smallest pivot that was not judged to be zero when detecting linearly dependent constraints

bool feasible

is the returned “solution” feasible?

ipc_ checkpointsIter[16]

checkpoints(i) records the iteration at which the criticality measures first fall below $10^{-i-1}$, i = 0, …, 15 (-1 means not achieved)

rpc_ checkpointsTime[16]

see checkpointsIter

struct dqp_time_type time

timings (see above)

struct fdc_inform_type fdc_inform

inform parameters for FDC

struct sls_inform_type sls_inform

inform parameters for SLS

struct sbls_inform_type sbls_inform

inform parameters for SBLS

struct gltr_inform_type gltr_inform

return information from GLTR

ipc_ scu_status

inform parameters for SCU

struct scu_inform_type scu_inform

see scu_status

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 given convex quadratic program; the code is available in $GALAHAD/src/dqp/C/dqpt.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.

/* dqpt.c */
/* Full test for the DQP 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_dqp.h"
#ifdef REAL_128
#include <quadmath.h>
#endif

int main(void) {

    // Derived types
    void *data;
    struct dqp_control_type control;
    struct dqp_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
//    ipc_ H_ne = 4; // Hesssian elements
//    ipc_ H_row[] = {0, 1, 2, 2 };   // row indices, NB lower triangle
//    ipc_ H_col[] = {0, 1, 1, 2};    // column indices, NB lower triangle
//    ipc_ H_ptr[] = {0, 1, 3, 4}; // row pointers
//    rpc_ H_val[] = {1.0, 2.0, 1.0, 3.0 };   // 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_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 qp storage formats\n\n");

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

        // Initialize DQP
        dqp_initialize( &data, &control, &status );

        // Set user-defined control options
        control.f_indexing = false; // C sparse matrix indexing
        strcpy(control.symmetric_linear_solver, "sytr ") ;
        strcpy(control.definite_linear_solver, "sytr ") ;
        strcpy(control.fdc_control.symmetric_linear_solver, "sytr ") ;
        control.fdc_control.use_sls = true;
        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};
        rpc_ z[] = {0.0,0.0,0.0};

        switch(d){
            case 1: // sparse co-ordinate storage
                st = 'C';
                dqp_import( &control, &data, &status, n, m,
                           "coordinate", H_ne, H_row, H_col, NULL,
                           "coordinate", A_ne, A_row, A_col, NULL );
                dqp_solve_qp( &data, &status, n, m, H_ne, H_val, 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';
                dqp_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 );
                dqp_solve_qp( &data, &status, n, m, H_ne, H_val, 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_ 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};
                dqp_import( &control, &data, &status, n, m,
                            "dense", H_ne, NULL, NULL, NULL,
                            "dense", A_ne, NULL, NULL, NULL );
                dqp_solve_qp( &data, &status, n, m, H_dense_ne, H_dense, g, f,
                              A_dense_ne, A_dense, c_l, c_u, x_l, x_u,
                              x, c, y, z, x_stat, c_stat );
                break;
            case 4: // diagonal
                st = 'L';
                dqp_import( &control, &data, &status, n, m,
                            "diagonal", H_ne, NULL, NULL, NULL,
                            "sparse_by_rows", A_ne, NULL, A_col, A_ptr );
                dqp_solve_qp( &data, &status, n, m, H_ne, H_val, g, f,
                              A_ne, A_val, c_l, c_u, x_l, x_u, x, c, y, z,
                              x_stat, c_stat );
                break;

            case 5: // scaled identity
                st = 'S';
                dqp_import( &control, &data, &status, n, m,
                            "scaled_identity", H_ne, NULL, NULL, NULL,
                            "sparse_by_rows", A_ne, NULL, A_col, A_ptr );
                dqp_solve_qp( &data, &status, n, m, H_ne, H_val, g, f,
                              A_ne, A_val, c_l, c_u, x_l, x_u, x, c, y, z,
                              x_stat, c_stat );
                break;
            case 6: // identity
                st = 'I';
                dqp_import( &control, &data, &status, n, m,
                            "identity", H_ne, NULL, NULL, NULL,
                            "sparse_by_rows", A_ne, NULL, A_col, A_ptr );
                dqp_solve_qp( &data, &status, n, m, H_ne, H_val, g, f,
                              A_ne, A_val, c_l, c_u, x_l, x_u, x, c, y, z,
                              x_stat, c_stat );
                break;
            }
        dqp_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: DQP_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
        dqp_terminate( &data, &control, &inform );
    }

    printf("\n separate test of sldqp\n\n");

    // test shifted least-distance interface
//  for( ipc_ d=1; d <= 0; d++){
    for( ipc_ d=1; d <= 1; d++){

        // Initialize DQP
        dqp_initialize( &data, &control, &status );

        // Set user-defined control options
        control.f_indexing = false; // C sparse matrix indexing
        strcpy(control.symmetric_linear_solver, "sytr ") ;
        strcpy(control.definite_linear_solver, "sytr ") ;
        strcpy(control.fdc_control.symmetric_linear_solver, "sytr ") ;
        control.fdc_control.use_sls = true;
        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};
        rpc_ z[] = {0.0,0.0,0.0};

        // 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';
                dqp_import( &control, &data, &status, n, m,
                           "shifted_least_distance", H_ne, NULL, NULL, NULL,
                           "coordinate", A_ne, A_row, A_col, NULL );
                dqp_solve_sldqp( &data, &status, n, m, w, x_0, g, f,
                                 A_ne, A_val, c_l, c_u, x_l, x_u, x, c, y, z,
                                 x_stat, c_stat );
                break;

            }
        dqp_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: DQP_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
        dqp_terminate( &data, &control, &inform );
    }

}

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

/* dqptf.c */
/* Full test for the DQP C interface using Fortran sparse matrix indexing */

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

int main(void) {

    // Derived types
    void *data;
    struct dqp_control_type control;
    struct dqp_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_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 qp storage formats\n\n");

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

        // Initialize DQP
        dqp_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';
                dqp_import( &control, &data, &status, n, m,
                           "coordinate", H_ne, H_row, H_col, NULL,
                           "coordinate", A_ne, A_row, A_col, NULL );
                dqp_solve_qp( &data, &status, n, m, H_ne, H_val, 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';
                dqp_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 );
                dqp_solve_qp( &data, &status, n, m, H_ne, H_val, 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_ 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};
                dqp_import( &control, &data, &status, n, m,
                            "dense", H_ne, NULL, NULL, NULL,
                            "dense", A_ne, NULL, NULL, NULL );
                dqp_solve_qp( &data, &status, n, m, H_dense_ne, H_dense, g, f,
                              A_dense_ne, A_dense, c_l, c_u, x_l, x_u,
                              x, c, y, z, x_stat, c_stat );
                break;
            case 4: // diagonal
                st = 'L';
                dqp_import( &control, &data, &status, n, m,
                            "diagonal", H_ne, NULL, NULL, NULL,
                            "sparse_by_rows", A_ne, NULL, A_col, A_ptr );
                dqp_solve_qp( &data, &status, n, m, H_ne, H_val, g, f,
                              A_ne, A_val, c_l, c_u, x_l, x_u, x, c, y, z,
                              x_stat, c_stat );
                break;

            case 5: // scaled identity
                st = 'S';
                dqp_import( &control, &data, &status, n, m,
                            "scaled_identity", H_ne, NULL, NULL, NULL,
                            "sparse_by_rows", A_ne, NULL, A_col, A_ptr );
                dqp_solve_qp( &data, &status, n, m, H_ne, H_val, g, f,
                              A_ne, A_val, c_l, c_u, x_l, x_u, x, c, y, z,
                              x_stat, c_stat );
                break;
            case 6: // identity
                st = 'I';
                dqp_import( &control, &data, &status, n, m,
                            "identity", H_ne, NULL, NULL, NULL,
                            "sparse_by_rows", A_ne, NULL, A_col, A_ptr );
                dqp_solve_qp( &data, &status, n, m, H_ne, H_val, g, f,
                              A_ne, A_val, c_l, c_u, x_l, x_u, x, c, y, z,
                              x_stat, c_stat );
                break;
            }
        dqp_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: DQP_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
        dqp_terminate( &data, &control, &inform );
    }

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

        // Initialize DQP
        dqp_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};

        // 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';
                dqp_import( &control, &data, &status, n, m,
                           "shifted_least_distance", H_ne, NULL, NULL, NULL,
                           "coordinate", A_ne, A_row, A_col, NULL );
                dqp_solve_sldqp( &data, &status, n, m, w, x_0, g, f,
                                 A_ne, A_val, c_l, c_u, x_l, x_u, x, c, y, z,
                                 x_stat, c_stat );
                break;

            }
        dqp_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: DQP_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
        dqp_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. -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. -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_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, j = 0, … , n-1, contains \(c_j(x)\).
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, j = 0, … , n-1, contains \(y_j\).
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.

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. -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. -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_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, j = 0, … , n-1, contains \(c_j(x)\).
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, j = 0, … , n-1, contains \(y_j\).
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.