GALAHAD BQPB package#

purpose#

The bqpb package uses a primal-dual interior-point method to solve a given bound-constrained 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 simple bounds

\[x_l \leq x \leq x_u,\]

where $H$ is a given $n$ by $n$ symmetric postive-semi-definite matrix, $g$, $w$ and $x^0$ are vectors, $f$ is a scalar, and any of the components of the vectors $x_l$ 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/bqpb.pdf for additional details.

terminology#

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

\[x_l \leq x \leq x_u,\;\;\mbox{(1)}\]

the dual optimality conditions

\[H x + g = z,\;\;\mbox{and}\;\; z = z_l + z_u,\;\;\mbox{(2a)}\]

and

\[z_l \geq 0 \;\;\mbox{and}\;\; z_u \leq 0,\;\;\mbox{(2b)}\]

and the complementary slackness conditions

\[(x -x_l )^{T} z_l = 0 \;\;\mbox{and}\;\;(x -x_u )^{T} z_u = 0,\;\;\mbox{(3)}\]

where the vector $z$ is known as the dual variables for the bounds, 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#

Primal-dual interior point methods iterate towards a point that satisfies these optimality conditions by ultimately aiming to satisfy (2a) and (3), while ensuring that (1) and (2b) are satisfied as strict inequalities at each stage. Appropriate norms of the amounts by which (2a) and (3) fail to be satisfied are known as the dual infeasibility, and the violation of complementary slackness, respectively. The fact that (1) and (2b) are satisfied as strict inequalities gives such methods their other title, namely interior-point methods.

The method aims at each stage to reduce the overall violation of (2a) and (3), rather than reducing each of the terms individually. Given an estimate $v = (x, \; z, \; z^{l}, \; z^{u})$ of the primal-dual variables, a correction $\Delta v = \Delta (x, \;z,\;z^{l} ,\;z^{u} )$ is obtained by solving a suitable linear system of Newton equations for the nonlinear systems (2a) and a parameterized ``residual trajectory’’ perturbation of (3); residual trajectories proposed by Zhang (1994) and Zhao and Sun (1999) are possibilities. An improved estimate $v + \alpha \Delta v$ is then used, where the step-size $\alpha$ is chosen as close to 1.0 as possible while ensuring both that (1) and (2b) continue to hold and that the individual components which make up the complementary slackness (3) do not deviate too significantly from their average value. The parameter that controls the perturbation of (3) is ultimately driven to zero.

The Newton equations are solved by applying the matrix factorization package SBLS, but there are options to factorize the matrix as a whole (the so-called “augmented system” approach), to perform a block elimination first (the “Schur-complement” approach), or to let the method itself decide which of the two previous options is more appropriate. The “Schur-complement” approach is usually to be preferred when all the weights are nonzero or when every variable is bounded (at least one side).

The package is actually just a front-end to the more-sophisticated package CQP that saves users from setting unnecessary arguments.

references#

The basic algorithm is a generalisation of those of

Y. Zhang, ``On the convergence of a class of infeasible interior-point methods for the horizontal linear complementarity problem’’. SIAM J. Optimization 4(1) (1994) 208-227,

and

G. Zhao and J. Sun, ``On the rate of local convergence of high-order infeasible path-following algorithms for $P_*$ linear complementarity problems’’. Computational Optimization and Applications **14(1)* (1999) 293-307,

with many enhancements described by

N. I. M. Gould, D. Orban and D. P. Robinson, ``Trajectory-following methods for large-scale degenerate convex quadratic programming’’, Mathematical Programming Computation 5(2) (2013) 113-142.

matrix storage#

The symmetric $n$ by $n$ matrix $H$ may 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 bqpb package must be called in the following order:

bqpb_initialize - provide default control parameters and set up initial data structures
bqpb_read_specfile (optional) - override control values by reading replacement values from a file
bqpb_import - set up problem data structures and fixed values
bqpb_reset_control (optional) - possibly change control parameters if a sequence of problems are being solved
solve the problem by calling one of
- bqpb_solve_qp - solve the bound-constrained quadratic program
- bqpb_solve_sldqp - solve the bound-constrained shifted least-distance problem
bqpb_information (optional) - recover information about the solution and solution process
bqpb_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 bqpb_control_type;
struct bqpb_inform_type;
struct bqpb_time_type;

// function calls

void bqpb_initialize(
    void **data,
    struct bqpb_control_type* control,
    ipc_ *status
);

void bqpb_read_specfile(
    struct bqpb_control_type* control,
    const char specfile[]
);

void bqpb_import(
    struct bqpb_control_type* control,
    void **data,
    ipc_ *status,
    ipc_ n,
    const char H_type[],
    ipc_ H_ne,
    const ipc_ H_row[],
    const ipc_ H_col[],
    const ipc_ H_ptr[]
);

void bqpb_reset_control(
    struct bqpb_control_type* control,
    void **data,
    ipc_ *status
);

void bqpb_solve_qp(
    void **data,
    ipc_ *status,
    ipc_ n,
    ipc_ h_ne,
    const rpc_ H_val[],
    const rpc_ g[],
    const rpc_ f,
    const rpc_ x_l[],
    const rpc_ x_u[],
    rpc_ x[],
    rpc_ z[],
    ipc_ x_stat[]
);

void bqpb_solve_sldqp(
    void **data,
    ipc_ *status,
    ipc_ n,
    const rpc_ w[],
    const rpc_ x0[],
    const rpc_ g[],
    const rpc_ f,
    const rpc_ x_l[],
    const rpc_ x_u[],
    rpc_ x[],
    rpc_ z[],
    ipc_ x_stat[]
);

void bqpb_information(void **data, struct bqpb_inform_type* inform, ipc_ *status);

void bqpb_terminate(
    void **data,
    struct bqpb_control_type* control,
    struct bqpb_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 SINGLE.

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 bqpb_initialize(
    void **data,
    struct bqpb_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 bqpb_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 bqpb_read_specfile(
    struct bqpb_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/bqpb/BQPB.template. See also Table 2.1 in the Fortran documentation provided in $GALAHAD/doc/bqpb.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 bqpb_control_type)
specfile	is a character string containing the name of the specification file

void bqpb_import(
    struct bqpb_control_type* control,
    void **data,
    ipc_ *status,
    ipc_ n,
    const char H_type[],
    ipc_ H_ne,
    const ipc_ H_row[],
    const ipc_ H_col[],
    const ipc_ H_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 bqpb_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 restriction n > 0 or requirement that a H_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.

void bqpb_reset_control(
    struct bqpb_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 bqpb_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 bqpb_solve_qp(
    void **data,
    ipc_ *status,
    ipc_ n,
    ipc_ h_ne,
    const rpc_ H_val[],
    const rpc_ g[],
    const rpc_ f,
    const rpc_ x_l[],
    const rpc_ x_u[],
    rpc_ x[],
    rpc_ z[],
    ipc_ x_stat[]
)

Solve the bound-constrained 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 restriction n > 0 or requirement that a H_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. -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
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.
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 $x_j$.
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.

void bqpb_solve_sldqp(
    void **data,
    ipc_ *status,
    ipc_ n,
    const rpc_ w[],
    const rpc_ x0[],
    const rpc_ g[],
    const rpc_ f,
    const rpc_ x_l[],
    const rpc_ x_u[],
    rpc_ x[],
    rpc_ z[],
    ipc_ x_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 restriction n > 0 or requirement that h_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. -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
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.
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 $x_j$.
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.

void bqpb_information(void **data, struct bqpb_inform_type* inform, ipc_ *status)

Provides output information

Parameters:

data

holds private internal data

inform

is a struct containing output information (see bqpb_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 bqpb_terminate(
    void **data,
    struct bqpb_control_type* control,
    struct bqpb_inform_type* inform
)

Deallocate all internal private storage

Parameters:

data	holds private internal data
control	is a struct containing control information (see bqpb_control_type)
inform	is a struct containing output information (see bqpb_inform_type)

available structures#

bqpb_control_type structure#

#include <galahad_bqpb.h>

struct bqpb_control_type {
    // components

    bool f_indexing;
    ipc_ error;
    ipc_ out;
    ipc_ print_level;
    ipc_ start_print;
    ipc_ stop_print;
    ipc_ maxit;
    ipc_ infeas_max;
    ipc_ muzero_fixed;
    ipc_ restore_problem;
    ipc_ indicator_type;
    ipc_ arc;
    ipc_ series_order;
    ipc_ sif_file_device;
    ipc_ qplib_file_device;
    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_ perturb_h;
    rpc_ prfeas;
    rpc_ dufeas;
    rpc_ muzero;
    rpc_ tau;
    rpc_ gamma_c;
    rpc_ gamma_f;
    rpc_ reduce_infeas;
    rpc_ obj_unbounded;
    rpc_ potential_unbounded;
    rpc_ identical_bounds_tol;
    rpc_ mu_pounce;
    rpc_ indicator_tol_p;
    rpc_ indicator_tol_pd;
    rpc_ indicator_tol_tapia;
    rpc_ cpu_time_limit;
    rpc_ clock_time_limit;
    bool remove_dependencies;
    bool treat_zero_bounds_as_general;
    bool treat_separable_as_general;
    bool just_feasible;
    bool getdua;
    bool puiseux;
    bool every_order;
    bool feasol;
    bool balance_initial_complentarity;
    bool crossover;
    bool space_critical;
    bool deallocate_error_fatal;
    bool generate_sif_file;
    bool generate_qplib_file;
    char sif_file_name[31];
    char qplib_file_name[31];
    char prefix[31];
    struct fdc_control_type fdc_control;
    struct sbls_control_type sbls_control;
    struct fit_control_type fit_control;
    struct roots_control_type roots_control;
    struct cro_control_type cro_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

$\leq$ 0 gives no output,
= 1 gives a one-line summary for every iteration,
= 2 gives a summary of the inner iteration for each iteration,
$\geq$ 3 gives increasingly verbose (debugging) output

ipc_ start_print

any printing will start on this iteration

ipc_ stop_print

any printing will stop on this iteration

ipc_ maxit

at most maxit inner iterations are allowed

ipc_ infeas_max

the number of iterations for which the overall infeasibility of the problem is not reduced by at least a factor .reduce_infeas before the problem is flagged as infeasible (see reduce_infeas)

ipc_ muzero_fixed

the initial value of the barrier parameter will not be changed for the first muzero_fixed iterations

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_ indicator_type

specifies the type of indicator function used. Pssible values are

1 primal indicator: a constraint is active if and only if the distance to its nearest bound $\leq$.indicator_p_tol
2 primal-dual indicator: a constraint is active if and only if the distance to its nearest bound $\leq$.indicator_tol_pd \* size of corresponding multiplier
3 primal-dual indicator: a constraint is active if and only if the distance to its nearest bound $\leq$.indicator_tol_tapia \* distance to same bound at previous iteration

ipc_ arc

which residual trajectory should be used to aim from the current iterate to the solution. Possible values are

1 the Zhang linear residual trajectory
2 the Zhao-Sun quadratic residual trajectory
3 the Zhang arc ultimately switching to the Zhao-Sun residual trajectory
4 the mixed linear-quadratic residual trajectory
5 the Zhang arc ultimately switching to the mixed linear-quadratic residual trajectory

ipc_ series_order

the order of (Taylor/Puiseux) series to fit to the path data

ipc_ sif_file_device

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

ipc_ qplib_file_device

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

rpc_ infinity

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

rpc_ stop_abs_p

the required absolute and relative accuracies for the primal infeasibility

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_ perturb_h

.perturb_h will be added to the Hessian

rpc_ prfeas

initial primal variables will not be closer than .prfeas from their bounds

rpc_ dufeas

initial dual variables will not be closer than .dufeas from their bounds

rpc_ muzero

the initial value of the barrier parameter. If muzero is not positive, it will be reset to an appropriate value

rpc_ tau

the weight attached to primal-dual infeasibility compared to complementa when assessing step acceptance

rpc_ gamma_c

individual complementarities will not be allowed to be smaller than gamma_c times the average value

rpc_ gamma_f

the average complementarity will not be allowed to be smaller than gamma_f times the primal/dual infeasibility

rpc_ reduce_infeas

if the overall infeasibility of the problem is not reduced by at least a factor .reduce_infeas over .infeas_max iterations, the problem is flagged as infeasible (see infeas_max)

rpc_ obj_unbounded

if the objective function value is smaller than obj_unbounded, it will be flagged as unbounded from below.

rpc_ potential_unbounded

if W=0 and the potential function value is smaller than .potential_unbounded $\ast$ number of one-sided bounds, the analytic center will be flagged as unbounded

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_ mu_pounce

start terminal extrapolation when mu reaches mu_pounce

rpc_ indicator_tol_p

if .indicator_type = 1, a constraint/bound will be deemed to be active if and only if the distance to its nearest bound $\leq$.indicator_p_tol

rpc_ indicator_tol_pd

if .indicator_type = 2, a constraint/bound will be deemed to be active if and only if the distance to its nearest bound $\leq$.indicator_tol_pd \* size of corresponding multiplier

rpc_ indicator_tol_tapia

if .indicator_type = 3, a constraint/bound will be deemed to be active if and only if the distance to its nearest bound $\leq$.indicator_tol_tapia \* distance to same bound at previous iteration

rpc_ cpu_time_limit

the maximum CPU time allowed (-ve means infinite)

rpc_ clock_time_limit

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

bool 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 treat_separable_as_general

if .just_feasible is true, the algorithm will stop as soon as a feasible point is found. Otherwise, the optimal solution to the problem will be found

bool just_feasible

if .treat_separable_as_general, is true, any separability in the problem structure will be ignored

bool getdua

if .getdua, is true, advanced initial values are obtained for the dual variables

bool puiseux

decide between Puiseux and Taylor series approximations to the arc

bool every_order

try every order of series up to series_order?

bool feasol

if .feasol is true, the final solution obtained will be perturbed so that variables close to their bounds are moved onto these bounds

bool balance_initial_complentarity

if .balance_initial_complentarity is true, the initial complemetarity is required to be balanced

bool crossover

if .crossover is true, cross over the solution to one defined by linearly-independent constraints if possible

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 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 sbls_control_type sbls_control

control parameters for SBLS

struct fit_control_type fit_control

control parameters for FIT

struct roots_control_type roots_control

control parameters for ROOTS

struct cro_control_type cro_control

control parameters for CRO

bqpb_time_type structure#

#include <galahad_bqpb.h>

struct bqpb_time_type {
    // components

    rpc_ total;
    rpc_ preprocess;
    rpc_ find_dependent;
    rpc_ analyse;
    rpc_ factorize;
    rpc_ solve;
    rpc_ clock_total;
    rpc_ clock_preprocess;
    rpc_ clock_find_dependent;
    rpc_ clock_analyse;
    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_ 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_ 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

bqpb_inform_type structure#

#include <galahad_bqpb.h>

struct bqpb_inform_type {
    // components

    ipc_ status;
    ipc_ alloc_status;
    char bad_alloc[81];
    ipc_ iter;
    ipc_ factorization_status;
    int64_t factorization_integer;
    int64_t factorization_real;
    ipc_ nfacts;
    ipc_ nbacts;
    ipc_ threads;
    rpc_ obj;
    rpc_ primal_infeasibility;
    rpc_ dual_infeasibility;
    rpc_ complementary_slackness;
    rpc_ init_primal_infeasibility;
    rpc_ init_dual_infeasibility;
    rpc_ init_complementary_slackness;
    rpc_ potential;
    rpc_ non_negligible_pivot;
    bool feasible;
    ipc_ checkpointsIter[16];
    rpc_ checkpointsTime[16];
    struct bqpb_time_type time;
    struct fdc_inform_type fdc_inform;
    struct sbls_inform_type sbls_inform;
    struct fit_inform_type fit_inform;
    struct roots_inform_type roots_inform;
    struct cro_inform_type cro_inform;
    struct rpd_inform_type rpd_inform;
};

detailed documentation#

inform derived type as a C struct

components#

ipc_ status

return status. See BQPB_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_ 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_ nbacts

the total number of “wasted” function evaluations during the linesearch

ipc_ threads

the number of threads used

rpc_ obj

the value of the objective function at the best estimate of the solution determined by BQPB_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_ potential

the value of the logarithmic potential function sum -log(distance to constraint boundary)

rpc_ non_negligible_pivot

the smallest pivot which was not judged to be zero when detecting linear 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 bqpb_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 fit_inform_type fit_inform

return information from FIT

struct roots_inform_type roots_inform

return information from ROOTS

struct cro_inform_type cro_inform

inform parameters for CRO

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 bound-constrained QP; the code is available in $GALAHAD/src/bqpb/C/bqpbt.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.

/* bqpbt.c */
/* Full test for the BQPB C interface using C sparse matrix indexing */

#include <stdio.h>
#include <math.h>
#include "galahad_precision.h"
#include "galahad_cfunctions.h"
#include "galahad_bqpb.h"

int main(void) {

    // Derived types
    void *data;
    struct bqpb_control_type control;
    struct bqpb_inform_type inform;

    // Set problem data
    ipc_ n = 3; // dimension
    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[] = {2.0, 0.0, 0.0};   // linear term in the objective
    rpc_ f = 1.0;  // constant term in the objective
    rpc_ x_l[] = {-1.0, - INFINITY, - INFINITY}; // variable lower bound
    rpc_ x_u[] = {1.0, INFINITY, 2.0}; // variable upper bound

    // Set output storage
    ipc_ x_stat[n]; // variable 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 <= 7; d++){

        // Initialize BQPB
        bqpb_initialize( &data, &control, &status );

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

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

        switch(d){
            case 1: // sparse co-ordinate storage
                st = 'C';
                bqpb_import( &control, &data, &status, n,
                            "coordinate", H_ne, H_row, H_col, NULL );
                bqpb_solve_qp( &data, &status, n, H_ne, H_val, g, f,
                               x_l, x_u, x, z, x_stat );
                break;
            printf(" case %1" i_ipc_ " break\n",d);
            case 2: // sparse by rows
                st = 'R';
                bqpb_import( &control, &data, &status, n,
                             "sparse_by_rows", H_ne, NULL, H_col, H_ptr );
                bqpb_solve_qp( &data, &status, n, H_ne, H_val, g, f,
                               x_l, x_u, x, z, x_stat );
                break;
            case 3: // dense
                st = 'D';
                ipc_ H_dense_ne = 6; // number of elements of H
                rpc_ H_dense[] = {1.0, 0.0, 1.0, 0.0, 0.0, 1.0};
                bqpb_import( &control, &data, &status, n,
                             "dense", H_ne, NULL, NULL, NULL );
                bqpb_solve_qp( &data, &status, n, H_dense_ne, H_dense, g, f,
                               x_l, x_u, x, z, x_stat );
                break;
            case 4: // diagonal
                st = 'L';
                bqpb_import( &control, &data, &status, n,
                             "diagonal", H_ne, NULL, NULL, NULL );
                bqpb_solve_qp( &data, &status, n, H_ne, H_val, g, f,
                               x_l, x_u, x, z, x_stat );
                break;

            case 5: // scaled identity
                st = 'S';
                bqpb_import( &control, &data, &status, n,
                             "scaled_identity", H_ne, NULL, NULL, NULL );
                bqpb_solve_qp( &data, &status, n, H_ne, H_val, g, f,
                               x_l, x_u, x, z, x_stat );
                break;
            case 6: // identity
                st = 'I';
                bqpb_import( &control, &data, &status, n,
                             "identity", H_ne, NULL, NULL, NULL );
                bqpb_solve_qp( &data, &status, n, H_ne, H_val, g, f,
                               x_l, x_u, x, z, x_stat );
                break;
            case 7: // zero
                st = 'Z';
                bqpb_import( &control, &data, &status, n,
                             "zero", H_ne, NULL, NULL, NULL );
                bqpb_solve_qp( &data, &status, n, H_ne, H_val, g, f,
                               x_l, x_u, x, z, x_stat );
                break;



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

        if(inform.status == 0){
            printf("%c:%6" i_ipc_ " iterations. Optimal objective value = %5.2f status = %1" i_ipc_ "\n",
                   st, inform.iter, inform.obj, inform.status);
        }else{
            printf("%c: BQPB_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
        bqpb_terminate( &data, &control, &inform );
    }

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

        // Initialize BQPB
        bqpb_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_ 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';
                bqpb_import( &control, &data, &status, n,
                           "shifted_least_distance", H_ne, NULL, NULL, NULL );
                bqpb_solve_sldqp( &data, &status, n, w, x_0, g, f,
                                 x_l, x_u, x, z, x_stat );
                break;

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

        if(inform.status == 0){
            printf("%c:%6" i_ipc_ " iterations. Optimal objective value = %5.2f status = %1" i_ipc_ "\n",
                   st, inform.iter, inform.obj, inform.status);
        }else{
            printf("%c: BQPB_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
        bqpb_terminate( &data, &control, &inform );
    }

}

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

/* bqpbtf.c */
/* Full test for the BQPB C interface using Fortran sparse matrix indexing */

#include <stdio.h>
#include <math.h>
#include "galahad_precision.h"
#include "galahad_cfunctions.h"
#include "galahad_bqpb.h"

int main(void) {

    // Derived types
    void *data;
    struct bqpb_control_type control;
    struct bqpb_inform_type inform;

    // Set problem data
    ipc_ n = 3; // dimension
    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[] = {2.0, 0.0, 0.0};   // linear term in the objective
    rpc_ f = 1.0;  // constant term in the objective
    rpc_ x_l[] = {-1.0, - INFINITY, - INFINITY}; // variable lower bound
    rpc_ x_u[] = {1.0, INFINITY, 2.0}; // variable upper bound

    // Set output storage
    ipc_ x_stat[n]; // variable 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 <= 7; d++){

        // Initialize BQPB
        bqpb_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_ z[] = {0.0,0.0,0.0};

        switch(d){
            case 1: // sparse co-ordinate storage
                st = 'C';
                bqpb_import( &control, &data, &status, n,
                            "coordinate", H_ne, H_row, H_col, NULL );
                bqpb_solve_qp( &data, &status, n, H_ne, H_val, g, f,
                               x_l, x_u, x, z, x_stat );
                break;
            printf(" case %1" i_ipc_ " break\n",d);
            case 2: // sparse by rows
                st = 'R';
                bqpb_import( &control, &data, &status, n,
                             "sparse_by_rows", H_ne, NULL, H_col, H_ptr );
                bqpb_solve_qp( &data, &status, n, H_ne, H_val, g, f,
                               x_l, x_u, x, z, x_stat );
                break;
            case 3: // dense
                st = 'D';
                ipc_ H_dense_ne = 6; // number of elements of H
                rpc_ H_dense[] = {1.0, 0.0, 1.0, 0.0, 0.0, 1.0};
                bqpb_import( &control, &data, &status, n,
                             "dense", H_ne, NULL, NULL, NULL );
                bqpb_solve_qp( &data, &status, n, H_dense_ne, H_dense, g, f,
                               x_l, x_u, x, z, x_stat );
                break;
            case 4: // diagonal
                st = 'L';
                bqpb_import( &control, &data, &status, n,
                             "diagonal", H_ne, NULL, NULL, NULL );
                bqpb_solve_qp( &data, &status, n, H_ne, H_val, g, f,
                               x_l, x_u, x, z, x_stat );
                break;

            case 5: // scaled identity
                st = 'S';
                bqpb_import( &control, &data, &status, n,
                             "scaled_identity", H_ne, NULL, NULL, NULL );
                bqpb_solve_qp( &data, &status, n, H_ne, H_val, g, f,
                               x_l, x_u, x, z, x_stat );
                break;
            case 6: // identity
                st = 'I';
                bqpb_import( &control, &data, &status, n,
                             "identity", H_ne, NULL, NULL, NULL );
                bqpb_solve_qp( &data, &status, n, H_ne, H_val, g, f,
                               x_l, x_u, x, z, x_stat );
                break;
            case 7: // zero
                st = 'Z';
                bqpb_import( &control, &data, &status, n,
                             "zero", H_ne, NULL, NULL, NULL );
                bqpb_solve_qp( &data, &status, n, H_ne, H_val, g, f,
                               x_l, x_u, x, z, x_stat );
                break;



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

        if(inform.status == 0){
            printf("%c:%6" i_ipc_ " iterations. Optimal objective value = %5.2f status = %1" i_ipc_ "\n",
                   st, inform.iter, inform.obj, inform.status);
        }else{
            printf("%c: BQPB_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
        bqpb_terminate( &data, &control, &inform );
    }

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

        // Initialize BQPB
        bqpb_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_ 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';
                bqpb_import( &control, &data, &status, n,
                           "shifted_least_distance", H_ne, NULL, NULL, NULL );
                bqpb_solve_sldqp( &data, &status, n, w, x_0, g, f,
                                 x_l, x_u, x, z, x_stat );
                break;

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

        if(inform.status == 0){
            printf("%c:%6" i_ipc_ " iterations. Optimal objective value = %5.2f status = %1" i_ipc_ "\n",
                   st, inform.iter, inform.obj, inform.status);
        }else{
            printf("%c: BQPB_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
        bqpb_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 restriction n > 0 or requirement that a H_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. -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
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.
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 \(x_j\).
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.

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 restriction n > 0 or requirement that h_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. -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
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.
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 \(x_j\).
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.