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#

unsymmetric storage#

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

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

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

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

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

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

symmetric storage#

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 $1 \leq j \leq i \leq n$) need be held. In this case the lower triangle should be stored by rows, that is component $(i-1) * i / 2 + j$ of the storage array H_val will hold the value $H_{ij}$ (and, by symmetry, $H_{ji}$) for $1 \leq j \leq i \leq n$. 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, $1 \leq l \leq ne$, of $H$, its row index i, column index j and value $H_{ij}$, $1 \leq j \leq i \leq n$, 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+1) holds the total number of entries plus one. The column indices j, $1 \leq j \leq i$, 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 $1 \leq i \neq j \leq n$) only the diagonals entries $H_{ii}$, $1 \leq i \leq n$ 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.

parametric real type T#

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

callable functions#

    function dqp_initialize(T, data, control, status)

Set default control values and initialize private data

Parameters:

data

holds private internal data

control

is a structure containing control information (see dqp_control_type)

status

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

0
The initialization was succcessful.

    function dqp_read_specfile(T, control, specfile)

Read the content of a specification file, and assign values associated with given keywords to the corresponding control parameters. An in-depth discussion of specification files is available, and a detailed list of keywords with associated default values is provided in $GALAHAD/src/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 structure containing control information (see dqp_control_type)
specfile	is a one-dimensional array of type Vararg{Cchar} that must give the name of the specification file

    function dqp_import(T, control, data, status, n, m,
                        H_type, H_ne, H_row, H_col, H_ptr,
                        A_type, A_ne, A_row, A_col, A_ptr)

Import problem data into internal storage prior to solution.

Parameters:

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

    function dqp_reset_control(T, control, data, status)

Reset control parameters after import if required.

Parameters:

control

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

data

holds private internal data

status

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

0
The import was successful.

    function dqp_solve_qp(T, 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)

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

Parameters:

data	holds private internal data
status	is a scalar variable of type Int32 that gives the entry and exit status from the package. Possible exit values are: 0 The run was successful. -1 An allocation error occurred. A message indicating the offending array is written on unit control.error, and the returned allocation status and a string containing the name of the offending array are held in inform.alloc_status and inform.bad_alloc respectively. -2 A deallocation error occurred. A message indicating the offending array is written on unit control.error and the returned allocation status and a string containing the name of the offending array are held in inform.alloc_status and inform.bad_alloc respectively. -3 The restrictions n > 0 and m > 0 or requirement that a type contains its relevant string ‘dense’, ‘coordinate’, ‘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 Int32 that holds the number of variables
m	is a scalar variable of type Int32 that holds the number of general linear constraints.
h_ne	is a scalar variable of type Int32 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 T 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 T that holds the linear term $g$ of the objective function. The j-th component of `g`, j = 1, … , n, contains $g_j$.
f	is a scalar of type T that holds the constant term $f$ of the objective function.
a_ne	is a scalar variable of type Int32 that holds the number of entries in the constraint Jacobian matrix $A$.
A_val	is a one-dimensional array of size a_ne and type T that holds the values of the entries of the constraint Jacobian matrix $A$ in any of the available storage schemes.
c_l	is a one-dimensional array of size m and type T that holds the lower bounds $c^l$ on the constraints $A x$. The i-th component of `c_l`, i = 1, … , m, contains $c^l_i$.
c_u	is a one-dimensional array of size m and type T that holds the upper bounds $c^l$ on the constraints $A x$. The i-th component of `c_u`, i = 1, … , m, contains $c^u_i$.
x_l	is a one-dimensional array of size n and type T that holds the lower bounds $x^l$ on the variables $x$. The j-th component of `x_l`, j = 1, … , n, contains $x^l_j$.
x_u	is a one-dimensional array of size n and type T that holds the upper bounds $x^l$ on the variables $x$. The j-th component of `x_u`, j = 1, … , n, contains $x^l_j$.
x	is a one-dimensional array of size n and type T that holds the values $x$ of the optimization variables. The j-th component of `x`, j = 1, … , n, contains $x_j$.
c	is a one-dimensional array of size m and type T that holds the residual $c(x)$. The i-th component of `c`, j = 1, … , m, contains $c_j(x)$.
y	is a one-dimensional array of size n and type T that holds the values $y$ of the Lagrange multipliers for the general linear constraints. The j-th component of `y`, j = 1, … , m, contains $y_j$.
z	is a one-dimensional array of size n and type T that holds the values $z$ of the dual variables. The j-th component of `z`, j = 1, … , n, contains $z_j$.
x_stat	is a one-dimensional array of size n and type Int32 that gives the optimal status of the problem variables. If x_stat(j) is negative, the variable $x_j$ most likely lies on its lower bound, if it is positive, it lies on its upper bound, and if it is zero, it lies between its bounds.
c_stat	is a one-dimensional array of size m and type Int32 that gives the optimal status of the general linear constraints. If c_stat(i) is negative, the constraint value $a_i^Tx$ most likely lies on its lower bound, if it is positive, it lies on its upper bound, and if it is zero, it lies between its bounds.

    function dqp_solve_sldqp(T, data, status, n, m, w, x0, g, f,
                             a_ne, A_val, c_l, c_u, x_l, x_u,
                             x, c, y, z, x_stat, c_stat)

Solve the shifted least-distance quadratic program

Parameters:

data	holds private internal data
status	is a scalar variable of type Int32 that gives the entry and exit status from the package. Possible exit values are: 0 The run was successful -1 An allocation error occurred. A message indicating the offending array is written on unit control.error, and the returned allocation status and a string containing the name of the offending array are held in inform.alloc_status and inform.bad_alloc respectively. -2 A deallocation error occurred. A message indicating the offending array is written on unit control.error and the returned allocation status and a string containing the name of the offending array are held in inform.alloc_status and inform.bad_alloc respectively. -3 The restrictions n > 0 and m > 0 or requirement that a type contains its relevant string ‘dense’, ‘coordinate’, ‘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 Int32 that holds the number of variables
m	is a scalar variable of type Int32 that holds the number of general linear constraints.
w	is a one-dimensional array of size n and type T that holds the values of the weights $w$.
x0	is a one-dimensional array of size n and type T that holds the values of the shifts $x^0$.
g	is a one-dimensional array of size n and type T that holds the linear term $g$ of the objective function. The j-th component of `g`, j = 1, … , n, contains $g_j$.
f	is a scalar of type T that holds the constant term $f$ of the objective function.
a_ne	is a scalar variable of type Int32 that holds the number of entries in the constraint Jacobian matrix $A$.
A_val	is a one-dimensional array of size a_ne and type T that holds the values of the entries of the constraint Jacobian matrix $A$ in any of the available storage schemes.
c_l	is a one-dimensional array of size m and type T that holds the lower bounds $c^l$ on the constraints $A x$. The i-th component of `c_l`, i = 1, … , m, contains $c^l_i$.
c_u	is a one-dimensional array of size m and type T that holds the upper bounds $c^l$ on the constraints $A x$. The i-th component of `c_u`, i = 1, … , m, contains $c^u_i$.
x_l	is a one-dimensional array of size n and type T that holds the lower bounds $x^l$ on the variables $x$. The j-th component of `x_l`, j = 1, … , n, contains $x^l_j$.
x_u	is a one-dimensional array of size n and type T that holds the upper bounds $x^l$ on the variables $x$. The j-th component of `x_u`, j = 1, … , n, contains $x^l_j$.
x	is a one-dimensional array of size n and type T that holds the values $x$ of the optimization variables. The j-th component of `x`, j = 1, … , n, contains $x_j$.
c	is a one-dimensional array of size m and type T that holds the residual $c(x)$. The i-th component of `c`, j = 1, … , m, contains $c_j(x)$.
y	is a one-dimensional array of size n and type T that holds the values $y$ of the Lagrange multipliers for the general linear constraints. The j-th component of `y`, j = 1, … , m, contains $y_j$.
z	is a one-dimensional array of size n and type T that holds the values $z$ of the dual variables. The j-th component of `z`, j = 1, … , n, contains $z_j$.
x_stat	is a one-dimensional array of size n and type Int32 that gives the optimal status of the problem variables. If x_stat(j) is negative, the variable $x_j$ most likely lies on its lower bound, if it is positive, it lies on its upper bound, and if it is zero, it lies between its bounds.
c_stat	is a one-dimensional array of size m and type Int32 that gives the optimal status of the general linear constraints. If c_stat(i) is negative, the constraint value $a_i^Tx$ most likely lies on its lower bound, if it is positive, it lies on its upper bound, and if it is zero, it lies between its bounds.

    function dqp_information(T, data, inform, status)

Provides output information

Parameters:

data

holds private internal data

inform

is a structure containing output information (see dqp_inform_type)

status

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

0
The values were recorded successfully

    function dqp_terminate(T, data, control, inform)

Deallocate all internal private storage

Parameters:

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

available structures#

dqp_control_type structure#

    struct dqp_control_type{T}
      f_indexing::Bool
      error::Int32
      out::Int32
      print_level::Int32
      start_print::Int32
      stop_print::Int32
      print_gap::Int32
      dual_starting_point::Int32
      maxit::Int32
      max_sc::Int32
      cauchy_only::Int32
      arc_search_maxit::Int32
      cg_maxit::Int32
      explore_optimal_subspace::Int32
      restore_problem::Int32
      sif_file_device::Int32
      qplib_file_device::Int32
      rho::T
      infinity::T
      stop_abs_p::T
      stop_rel_p::T
      stop_abs_d::T
      stop_rel_d::T
      stop_abs_c::T
      stop_rel_c::T
      stop_cg_relative::T
      stop_cg_absolute::T
      cg_zero_curvature::T
      max_growth::T
      identical_bounds_tol::T
      cpu_time_limit::T
      clock_time_limit::T
      initial_perturbation::T
      perturbation_reduction::T
      final_perturbation::T
      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::Bool
      symmetric_linear_solver::NTuple{31,Cchar}
      definite_linear_solver::NTuple{31,Cchar}
      unsymmetric_linear_solver::NTuple{31,Cchar}
      sif_file_name::NTuple{31,Cchar}
      qplib_file_name::NTuple{31,Cchar}
      prefix::NTuple{31,Cchar}
      fdc_control::fdc_control_type{T}
      sls_control::sls_control_type{T}
      sbls_control::sbls_control_type{T}
      gltr_control::gltr_control_type{T}

detailed documentation#

control derived type as a Julia structure

components#

Bool f_indexing

use C or Fortran sparse matrix indexing

Int32 error

error and warning diagnostics occur on stream error

Int32 out

general output occurs on stream out

Int32 print_level

the level of output required is specified by print_level

Int32 start_print

any printing will start on this iteration

Int32 stop_print

any printing will stop on this iteration

Int32 print_gap

printing will only occur every print_gap iterations

Int32 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)

Int32 maxit

at most maxit inner iterations are allowed

Int32 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)

Int32 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

Int32 arc_search_maxit

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

Int32 cg_maxit

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

Int32 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

Int32 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

Int32 sif_file_device

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

Int32 qplib_file_device

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

T 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)

T infinity

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

T stop_abs_p

the required absolute and relative accuracies for the primal infeasibilies

T stop_rel_p

see stop_abs_p

T stop_abs_d

the required absolute and relative accuracies for the dual infeasibility

T stop_rel_d

see stop_abs_d

T stop_abs_c

the required absolute and relative accuracies for the complementarity

T stop_rel_c

see stop_abs_c

T 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 )

T stop_cg_absolute

see stop_cg_relative

T cg_zero_curvature

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

T max_growth

maximum growth factor allowed without a refactorization

T identical_bounds_tol

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

T cpu_time_limit

the maximum CPU time allowed (-ve means infinite)

T clock_time_limit

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

T initial_perturbation

the initial penalty weight (for DLP only)

T perturbation_reduction

the penalty weight reduction factor (for DLP only)

T 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]

indefinite linear equation solver set in symmetric_linear_solver

char definite_linear_solver[31]

definite linear equation solver

char unsymmetric_linear_solver[31]

unsymmetric linear equation solver

NTuple{31,Cchar} sif_file_name

name of generated SIF file containing input problem

NTuple{31,Cchar} qplib_file_name

name of generated QPLIB file containing input problem

NTuple{31,Cchar} prefix

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

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#

    struct dqp_time_type{T}
      total::T
      preprocess::T
      find_dependent::T
      analyse::T
      factorize::T
      solve::T
      search::T
      clock_total::T
      clock_preprocess::T
      clock_find_dependent::T
      clock_analyse::T
      clock_factorize::T
      clock_solve::T
      clock_search::T

detailed documentation#

time derived type as a Julia structure

components#

T total

the total CPU time spent in the package

T preprocess

the CPU time spent preprocessing the problem

T find_dependent

the CPU time spent detecting linear dependencies

T analyse

the CPU time spent analysing the required matrices prior to factorization

T factorize

the CPU time spent factorizing the required matrices

T solve

the CPU time spent computing the search direction

T search

the CPU time spent in the linesearch

T clock_total

the total clock time spent in the package

T clock_preprocess

the clock time spent preprocessing the problem

T clock_find_dependent

the clock time spent detecting linear dependencies

T clock_analyse

the clock time spent analysing the required matrices prior to factorization

T clock_factorize

the clock time spent factorizing the required matrices

T clock_solve

the clock time spent computing the search direction

T clock_search

the clock time spent in the linesearch

dqp_inform_type structure#

    struct dqp_inform_type{T}
      status::Int32
      alloc_status::Int32
      bad_alloc::NTuple{81,Cchar}
      iter::Int32
      cg_iter::Int32
      factorization_status::Int32
      factorization_integer::Int64
      factorization_real::Int64
      nfacts::Int32
      threads::Int32
      obj::T
      primal_infeasibility::T
      dual_infeasibility::T
      complementary_slackness::T
      non_negligible_pivot::T
      feasible::Bool
      checkpointsIter::NTuple{16,Cint}
      checkpointsTime::NTuple{16,T}
      time::dqp_time_type{T}
      fdc_inform::fdc_inform_type{T}
      sls_inform::sls_inform_type{T}
      sbls_inform::sbls_inform_type{T}
      gltr_inform::gltr_inform_type{T}
      scu_status::Int32
      scu_inform::scu_inform_type
      rpd_inform::rpd_inform_type

detailed documentation#

inform derived type as a Julia structure

components#

Int32 status

return status. See DQP_solve for details

Int32 alloc_status

the status of the last attempted allocation/deallocation

NTuple{81,Cchar} bad_alloc

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

Int32 iter

the total number of iterations required

Int32 cg_iter

the total number of iterations required

Int32 factorization_status

the return status from the factorization

Int64 factorization_integer

the total integer workspace required for the factorization

Int64 factorization_real

the total real workspace required for the factorization

Int32 nfacts

the total number of factorizations performed

Int32 threads

the number of threads used

T obj

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

T primal_infeasibility

the value of the primal infeasibility

T dual_infeasibility

the value of the dual infeasibility

T complementary_slackness

the value of the complementary slackness

T 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?

Int32 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)

T 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

Int32 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/Julia/test_dqp.jl . A variety of supported Hessian and constraint matrix storage formats are shown.

# test_dqp.jl
# Simple code to test the Julia interface to DQP

using GALAHAD
using Test
using Printf
using Accessors

function test_dqp(::Type{T}) where T
  # Derived types
  data = Ref{Ptr{Cvoid}}()
  control = Ref{dqp_control_type{T}}()
  inform = Ref{dqp_inform_type{T}}()

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

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

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

  for d in 1:6

    # Initialize DQP
    dqp_initialize(T, data, control, status)

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

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

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

      dqp_solve_qp(T, 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)
    end

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

      dqp_solve_qp(T, 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)
    end

    # dense
    if d == 3
      st = 'D'
      H_dense_ne = 6 # number of elements of H
      A_dense_ne = 6 # number of elements of A
      H_dense = T[1.0, 0.0, 1.0, 0.0, 0.0, 1.0]
      A_dense = T[2.0, 1.0, 0.0, 0.0, 1.0, 1.0]

      dqp_import(T, control, data, status, n, m,
                 "dense", H_ne, C_NULL, C_NULL, C_NULL,
                 "dense", A_ne, C_NULL, C_NULL, C_NULL)

      dqp_solve_qp(T, 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)
    end

    # diagonal
    if d == 4
      st = 'L'
      dqp_import(T, control, data, status, n, m,
                 "diagonal", H_ne, C_NULL, C_NULL, C_NULL,
                 "sparse_by_rows", A_ne, C_NULL, A_col, A_ptr)

      dqp_solve_qp(T, 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)
    end

    # scaled identity
    if d == 5
      st = 'S'
      dqp_import(T, control, data, status, n, m,
                 "scaled_identity", H_ne, C_NULL, C_NULL, C_NULL,
                 "sparse_by_rows", A_ne, C_NULL, A_col, A_ptr)

      dqp_solve_qp(T, 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)
    end

    # identity
    if d == 6
      st = 'I'
      dqp_import(T, control, data, status, n, m,
                 "identity", H_ne, C_NULL, C_NULL, C_NULL,
                 "sparse_by_rows", A_ne, C_NULL, A_col, A_ptr)

      dqp_solve_qp(T, 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)
    end

    dqp_information(T, data, inform, status)

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

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

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

  # test shifted least-distance interface
  for d in 1:1

    # Initialize DQP
    dqp_initialize(T, data, control, status)

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

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

    # Set shifted least-distance data

    w = T[1.0, 1.0, 1.0]
    x_0 = T[0.0, 0.0, 0.0]

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

      dqp_solve_sldqp(T, 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)
    end

    dqp_information(T, data, inform, status)

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

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

    # Delete internal workspace
    dqp_terminate(T, data, control, inform)
  end
  return 0
end

@testset "DQP" begin
  @test test_dqp(Float32) == 0
  @test test_dqp(Float64) == 0
end

data	holds private internal data
status	is a scalar variable of type Int32 that gives the entry and exit status from the package. Possible exit values are: 0 The run was successful. -1 An allocation error occurred. A message indicating the offending array is written on unit control.error, and the returned allocation status and a string containing the name of the offending array are held in inform.alloc_status and inform.bad_alloc respectively. -2 A deallocation error occurred. A message indicating the offending array is written on unit control.error and the returned allocation status and a string containing the name of the offending array are held in inform.alloc_status and inform.bad_alloc respectively. -3 The restrictions n > 0 and m > 0 or requirement that a type contains its relevant string ‘dense’, ‘coordinate’, ‘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 Int32 that holds the number of variables
m	is a scalar variable of type Int32 that holds the number of general linear constraints.
h_ne	is a scalar variable of type Int32 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 T 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 T that holds the linear term \(g\) of the objective function. The j-th component of `g`, j = 1, … , n, contains \(g_j\).
f	is a scalar of type T that holds the constant term \(f\) of the objective function.
a_ne	is a scalar variable of type Int32 that holds the number of entries in the constraint Jacobian matrix \(A\).
A_val	is a one-dimensional array of size a_ne and type T that holds the values of the entries of the constraint Jacobian matrix \(A\) in any of the available storage schemes.
c_l	is a one-dimensional array of size m and type T that holds the lower bounds \(c^l\) on the constraints \(A x\). The i-th component of `c_l`, i = 1, … , m, contains \(c^l_i\).
c_u	is a one-dimensional array of size m and type T that holds the upper bounds \(c^l\) on the constraints \(A x\). The i-th component of `c_u`, i = 1, … , m, contains \(c^u_i\).
x_l	is a one-dimensional array of size n and type T that holds the lower bounds \(x^l\) on the variables \(x\). The j-th component of `x_l`, j = 1, … , n, contains \(x^l_j\).
x_u	is a one-dimensional array of size n and type T that holds the upper bounds \(x^l\) on the variables \(x\). The j-th component of `x_u`, j = 1, … , n, contains \(x^l_j\).
x	is a one-dimensional array of size n and type T that holds the values \(x\) of the optimization variables. The j-th component of `x`, j = 1, … , n, contains \(x_j\).
c	is a one-dimensional array of size m and type T that holds the residual \(c(x)\). The i-th component of `c`, j = 1, … , m, contains \(c_j(x)\).
y	is a one-dimensional array of size n and type T that holds the values \(y\) of the Lagrange multipliers for the general linear constraints. The j-th component of `y`, j = 1, … , m, contains \(y_j\).
z	is a one-dimensional array of size n and type T that holds the values \(z\) of the dual variables. The j-th component of `z`, j = 1, … , n, contains \(z_j\).
x_stat	is a one-dimensional array of size n and type Int32 that gives the optimal status of the problem variables. If x_stat(j) is negative, the variable \(x_j\) most likely lies on its lower bound, if it is positive, it lies on its upper bound, and if it is zero, it lies between its bounds.
c_stat	is a one-dimensional array of size m and type Int32 that gives the optimal status of the general linear constraints. If c_stat(i) is negative, the constraint value \(a_i^Tx\) most likely lies on its lower bound, if it is positive, it lies on its upper bound, and if it is zero, it lies between its bounds.

data	holds private internal data
status	is a scalar variable of type Int32 that gives the entry and exit status from the package. Possible exit values are: 0 The run was successful -1 An allocation error occurred. A message indicating the offending array is written on unit control.error, and the returned allocation status and a string containing the name of the offending array are held in inform.alloc_status and inform.bad_alloc respectively. -2 A deallocation error occurred. A message indicating the offending array is written on unit control.error and the returned allocation status and a string containing the name of the offending array are held in inform.alloc_status and inform.bad_alloc respectively. -3 The restrictions n > 0 and m > 0 or requirement that a type contains its relevant string ‘dense’, ‘coordinate’, ‘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 Int32 that holds the number of variables
m	is a scalar variable of type Int32 that holds the number of general linear constraints.
w	is a one-dimensional array of size n and type T that holds the values of the weights \(w\).
x0	is a one-dimensional array of size n and type T that holds the values of the shifts \(x^0\).
g	is a one-dimensional array of size n and type T that holds the linear term \(g\) of the objective function. The j-th component of `g`, j = 1, … , n, contains \(g_j\).
f	is a scalar of type T that holds the constant term \(f\) of the objective function.
a_ne	is a scalar variable of type Int32 that holds the number of entries in the constraint Jacobian matrix \(A\).
A_val	is a one-dimensional array of size a_ne and type T that holds the values of the entries of the constraint Jacobian matrix \(A\) in any of the available storage schemes.
c_l	is a one-dimensional array of size m and type T that holds the lower bounds \(c^l\) on the constraints \(A x\). The i-th component of `c_l`, i = 1, … , m, contains \(c^l_i\).
c_u	is a one-dimensional array of size m and type T that holds the upper bounds \(c^l\) on the constraints \(A x\). The i-th component of `c_u`, i = 1, … , m, contains \(c^u_i\).
x_l	is a one-dimensional array of size n and type T that holds the lower bounds \(x^l\) on the variables \(x\). The j-th component of `x_l`, j = 1, … , n, contains \(x^l_j\).
x_u	is a one-dimensional array of size n and type T that holds the upper bounds \(x^l\) on the variables \(x\). The j-th component of `x_u`, j = 1, … , n, contains \(x^l_j\).
x	is a one-dimensional array of size n and type T that holds the values \(x\) of the optimization variables. The j-th component of `x`, j = 1, … , n, contains \(x_j\).
c	is a one-dimensional array of size m and type T that holds the residual \(c(x)\). The i-th component of `c`, j = 1, … , m, contains \(c_j(x)\).
y	is a one-dimensional array of size n and type T that holds the values \(y\) of the Lagrange multipliers for the general linear constraints. The j-th component of `y`, j = 1, … , m, contains \(y_j\).
z	is a one-dimensional array of size n and type T that holds the values \(z\) of the dual variables. The j-th component of `z`, j = 1, … , n, contains \(z_j\).
x_stat	is a one-dimensional array of size n and type Int32 that gives the optimal status of the problem variables. If x_stat(j) is negative, the variable \(x_j\) most likely lies on its lower bound, if it is positive, it lies on its upper bound, and if it is zero, it lies between its bounds.
c_stat	is a one-dimensional array of size m and type Int32 that gives the optimal status of the general linear constraints. If c_stat(i) is negative, the constraint value \(a_i^Tx\) most likely lies on its lower bound, if it is positive, it lies on its upper bound, and if it is zero, it lies between its bounds.