GALAHAD QPA package#

purpose#

The qpa package uses a working-set method to solve non-convex quadratic programs in various guises. The first is the l${}_1$ quadratic programming problem that aims to minimize

\[f(x;\rho_g,\rho_b) = q(x) + \rho_g v_g(x) + \rho_b v_b(x)\]

involving the quadratic objective

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

and the infeasibilities

\[v_g(x) = \sum_{i=1}^{m} \max(c_i^l - a_i^T x, 0) + \sum_{i=1}^{m} \max(a_i^T x - c_i^u, 0)\]

and

\[v_b(x) = \sum_{j=1}^{n} \max(x_j^l - x_j , 0) + \sum_{j=1}^{n} \max(x_j - x_j^u , 0),\]

where the $n$ by $n$ symmetric matrix $H$, the vectors $g$, $a_i$, $c^l$, $c^u$, $x^l$, $x^u$ and the scalars $f$, $\rho_g$ and $\rho_b$ are given. Full advantage is taken of any zero coefficients in the matrices $H$ or $A$ (whose rows are the vectors $a_i^T$). Any of the constraint bounds $c_i^l$, $c_i^u$, $x_j^l$ and $x_j^u$ may be infinite.

The package may also be used to solve the standard quadratic programming problem whose aim is to minimize $q(x)$ subject to the general linear constraints and simple bounds

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

by automatically adjusting the parameters $\rho_g$ and $\rho_b$ in $f(x;\rho_g,\rho_b)$.

Similarly, the package is capable of solving the bound-constrained l${}_1$ quadratic programming problem whose intention is to minimize $q(x) + \rho_g v_g(x),$ subject to the above simple bound constraints by automatically adjusting $\rho_b$ in $f(x;\rho_g,\rho_b)$.

If the matrix $H$ is positive semi-definite, a global solution is found. However, if $H$ is indefinite, the procedure may find a (weak second-order) critical point that is not the global solution to the given problem.

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

N.B. In many cases, the alternative quadratic programming package qpb is faster, and thus to be preferred.

terminolgy#

Any required solution $x$ for the standard quadratic programming problem 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.

method#

At the $k$-th iteration of the method, an improvement to the value of the merit function $m(x, \rho_g, \rho_b ) = q(x) + \rho_g v_g(x) + \rho_b v_b(x)$ at $x = x^{(k)}$ is sought. This is achieved by first computing a search direction $s^{(k)}$, and then setting $x^{(k+1)} = x^{(k)} + \alpha^{(k)} s^{(k)}$, where the stepsize $\alpha^{(k)}$ is chosen as the first local minimizer of $\phi ( \alpha ) = m( x^{(k)} + \alpha s^{(k)} , \rho_g, \rho_b )$ as $\alpha$ incesases from zero. The stepsize calculation is straightforward, and exploits the fact that $\phi ( \alpha )$ is a piecewise quadratic function of $\alpha$.

The search direction is defined by a subset of the “active” terms in $v(x)$, i.e., those for which $a_i^T x = c_i^l$ or $c_i^u$ (for $i=1,\ldots,m$) or $x_j = x_j^l$ or $x_j^u$ (for ($j=1,\ldots,n$). The “working” set $W^{(k)}$ is chosen from the active terms, and is such that its members have linearly independent gradients. The search direction $s^{(k)}$ is chosen as an approximate solution of the equality-constrained quadratic program

\[\begin{split} {\renewcommand{\arraystretch}{0.8} \begin{array}[t]{c} \mbox{minimize} \\ \mbox{ $s \in R^n $ } \end{array} \;} q(x^{(k)} + s) + \rho_g l_g^{(k)} (s) + \rho_b l_b^{(k)} (s),\;\;\mbox{(1)}\end{split}\]

subject to

\[a_i^T s = 0,\;\; i \in \{ 1, \ldots , m \} \cap W^{(k)}, \;\;\mbox{and}\;\; x_j = 0, \;\; i \in \{1, \ldots , n \} \cap W^{(k)},\;\;\mbox{(2)}\]

where

\[l_g^{(k)} (s) = - \sum_{\stackrel{i=1}{a_i^T x < c_i^l}}^m a_i^T s \; + \sum_{\stackrel{i=1}{a_i^T x > c_i^u}}^m a_i^T s\]

and

\[l_b^{(k)} (s) = - \sum_{\stackrel{j=1}{x_j < x_j^l}}^n s_j \; + \sum_{\stackrel{j=1}{x_j > x_j^u}}^n s_j. \]

The equality-constrained quadratic program (1)–(2) is solved by a projected preconditioned conjugate gradient method. The method terminates either after a prespecified number of iterations, or if the solution is found, or if a direction of infinite descent, along which $q(x^{(k)} + s) + \rho_g l_g^{(k)} (s) + \rho_b l_b^{(k)} (s)$ decreases without bound within the feasible region (2), is located. Succesively more accurate approximations are required as suspected solutions of the $\ell_1$-QP are approached.

Preconditioning of the conjugate gradient iteration requires the solution of one or more linear systems of the form

\[\begin{split}\left(\begin{array}{cc} M^{(k)} & A^{(k)T} \\ A^{(k)} & 0 \end{array}\right) \left(\begin{array}{c} p \\ u \end{array}\right) = \left(\begin{array}{c} g \\ 0 \end{array}\right),\;\;\mbox{(3)}\end{split}\]

where $M^{(k)}$ is a “suitable” approximation to $H$ and the rows of $A^{(k)}$ comprise the gradients of the terms in the current working set. Rather than recomputing a factorization of the preconditioner at every iteration, a Schur complement method is used, recognising the fact that gradual changes occur to successive working sets. The main iteration is divided into a sequence of “major” iterations. At the start of each major iteration (say, at the overall iteration $l$), a factorization of the current “reference” matrix, that is the matrix

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

is obtained using the matrix factorization package SLS. This reference matrix may be factorized as a whole (the so-called “augmented system” approach), or by performing a block elimination first (the “Schur-complement” approach). The latter is usually to be preferred when a (non-singular) diagonal preconditioner is used, but may be inefficient if any of the columns of $A^{(l)}$ is too dense. Subsequent iterations within the current major iteration obtain solutions to (3) via the factors of $K^{(l)}$ and an appropriate (dense) Schur complement, obtained from the package SCU. The major iteration terminates once the space required to hold the factors of the (growing) Schur complement exceeds a given threshold.

The working set changes by (a) adding an active term encountered during the determination of the stepsize, or (b) the removal of a term if $s = 0$ solves (1)–(2). The decision on which to remove in the latter case is based upon the expected decrease upon the removal of an individual term, and this information is available from the magnitude and sign of the components of the auxiliary vector $u$ computed in (3). At optimality, the components of $u$ for $a_i$ terms will all lie between $0$ and $\rho_g$ — and those for the other terms between $0$ and $\rho_b$ — and any violation of this rule indicates further progress is possible. The components of $u$ corresonding to the terms involving $a_i^T x$ are sometimes known as Lagrange multipliers (or generalized gradients) and denoted by $y$, while those for the remaining $x_j$ terms are dual variables and denoted by $z$.

To solve the standard quadratic programming problem, a sequence of $\ell_1$-quadratic programs are solved, each with a larger value of $\rho_g$ and/or $\rho_b$ than its predecessor. The required solution has been found once the infeasibilities $v_g(x)$ and $v_b(x)$ have been reduced to zero at the solution of the $\ell_1$-problem for the given $\rho_g$ and $\rho_b$.

In order to make the solution as efficient as possible, the variables and constraints are reordered internally by the package QPP prior to solution. In particular, fixed variables and free (unbounded on both sides) constraints are temporarily removed.

reference#

The method is described in detail in

N. I. M. Gould and Ph. L. Toint ``An iterative working-set method for large-scale non-convex quadratic programming’’. Applied Numerical Mathematics 43(1–2) (2002) 109–128.

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 qpa package must be called in the following order:

qpa_initialize - provide default control parameters and set up initial data structures
qpa_read_specfile (optional) - override control values by reading replacement values from a file
qpa_import - set up problem data structures and fixed values
qpa_reset_control (optional) - possibly change control parameters if a sequence of problems are being solved
solve the problem by calling one of
- qpa_solve_qp - solve the quadratic program (2)-(4)
- qpa_solve_l1qp - solve the l1 quadratic program (1)
- qpa_solve_bcl1qp - solve the bound constrained l1 quadratic program (4)-(5)
qpa_information (optional) - recover information about the solution and solution process
qpa_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), Float64 (double precision) or, if supported, Float128 (quadruple precision).

callable functions#

    function qpa_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 qpa_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 successful.

    function qpa_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/qpa/QPA.template. See also Table 2.1 in the Fortran documentation provided in $GALAHAD/doc/qpa.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 qpa_control_type)
specfile	is a one-dimensional array of type Vararg{Cchar} that must give the name of the specification file

    function qpa_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 qpa_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’, ‘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 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’, ‘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 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 qpa_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 qpa_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 qpa_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 (2)-(4).

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. -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 current 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. On entry, if control.cold_start = 0, x_stat should be set as above to provide a guide to the initial working set.
c_stat	is a one-dimensional array of size m and type Int32 that gives the current 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. On entry, if control.cold_start = 0, c_stat should be set as above to provide a guide to the initial working set.

    function qpa_solve_l1qp(T, data, status, n, m, h_ne, H_val, g, f,
                             rho_g, rho_b, a_ne, A_val, c_l, c_u,
                             x_l, x_u, x, c, y, z, x_stat, c_stat)

Solve the l_1 quadratic program (1).

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. -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.
rho_g	is a scalar of type T that holds the parameter $\rho_g$ associated with the linear constraints.
rho_b	is a scalar of type T that holds the parameter $\rho_b$ associated with the simple bound constraints.
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 current 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. On entry, if control.cold_start = 0, x_stat should be set as above to provide a guide to the initial working set.
c_stat	is a one-dimensional array of size m and type Int32 that gives the current 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. On entry, if control.cold_start = 0, c_stat should be set as above to provide a guide to the initial working set.

    function qpa_solve_bcl1qp(T, data, status, n, m, h_ne, H_val, g, f,
                              rho_g, a_ne, A_val, c_l, c_u, x_l, x_u,
                              x, c, y, z, x_stat, c_stat)

Solve the bound-constrained l_1 quadratic program (4)-(5)

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. -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.
rho_g	is a scalar of type T that holds the parameter $\rho_g$ associated with the linear constraints.
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 current 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. On entry, if control.cold_start = 0, x_stat should be set as above to provide a guide to the initial working set.
c_stat	is a one-dimensional array of size m and type Int32 that gives the current 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. On entry, if control.cold_start = 0, c_stat should be set as above to provide a guide to the initial working set.

    function qpa_information(T, data, inform, status)

Provides output information

Parameters:

data

holds private internal data

inform

is a structure containing output information (see qpa_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 qpa_terminate(T, data, control, inform)

Deallocate all internal private storage

Parameters:

data	holds private internal data
control	is a structure containing control information (see qpa_control_type)
inform	is a structure containing output information (see qpa_inform_type)

available structures#

qpa_control_type structure#

    struct qpa_control_type{T}
      f_indexing::Bool
      error::Int32
      out::Int32
      print_level::Int32
      start_print::Int32
      stop_print::Int32
      maxit::Int32
      factor::Int32
      max_col::Int32
      max_sc::Int32
      indmin::Int32
      valmin::Int32
      itref_max::Int32
      infeas_check_interval::Int32
      cg_maxit::Int32
      precon::Int32
      nsemib::Int32
      full_max_fill::Int32
      deletion_strategy::Int32
      restore_problem::Int32
      monitor_residuals::Int32
      cold_start::Int32
      sif_file_device::Int32
      infinity::T
      feas_tol::T
      obj_unbounded::T
      increase_rho_g_factor::T
      infeas_g_improved_by_factor::T
      increase_rho_b_factor::T
      infeas_b_improved_by_factor::T
      pivot_tol::T
      pivot_tol_for_dependencies::T
      zero_pivot::T
      inner_stop_relative::T
      inner_stop_absolute::T
      multiplier_tol::T
      cpu_time_limit::T
      clock_time_limit::T
      treat_zero_bounds_as_general::Bool
      solve_qp::Bool
      solve_within_bounds::Bool
      randomize::Bool
      array_syntax_worse_than_do_loop::Bool
      space_critical::Bool
      deallocate_error_fatal::Bool
      generate_sif_file::Bool
      symmetric_linear_solver::NTuple{31,Cchar}
      sif_file_name::NTuple{31,Cchar}
      prefix::NTuple{31,Cchar}
      each_interval::Bool
      sls_control::sls_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 maxit

at most maxit inner iterations are allowed

Int32 factor

the factorization to be used. Possible values are 0 automatic 1 Schur-complement factorization 2 augmented-system factorization

Int32 max_col

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

Int32 max_sc

the maximum permitted size of the Schur complement before a refactorization is performed

Int32 indmin

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

Int32 valmin

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

Int32 itref_max

the maximum number of iterative refinements allowed (OBSOLETE)

Int32 infeas_check_interval

the infeasibility will be checked for improvement every infeas_check_interval iterations (see infeas_g_improved_by_factor and infeas_b_improved_by_factor below)

Int32 cg_maxit

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

Int32 precon

the preconditioner to be used for the CG is defined by precon. Possible values are 0 automatic 1 no preconditioner, i.e, the identity within full factorization 2 full factorization 3 band within full factorization 4 diagonal using the barrier terms within full factorization

Int32 nsemib

the semi-bandwidth of a band preconditioner, if appropriate

Int32 full_max_fill

if the ratio of the number of nonzeros in the factors of the reference matrix to the number of nonzeros in the matrix itself exceeds full_max_fill, and the preconditioner is being selected automatically (precon = 0), a banded approximation will be used instead

Int32 deletion_strategy

the constraint deletion strategy to be used. Possible values are:

0 most violated of all 1 LIFO (last in, first out) k LIFO(k) most violated of the last k in LIFO

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 monitor_residuals

the frequency at which residuals will be monitored

Int32 cold_start

indicates whether a cold or warm start should be made. Possible values are

0 warm start - the values set in C_stat and B_stat indicate which constraints will be included in the initial working set. 1 cold start from the value set in X; constraints active at X will determine the initial working set. 2 cold start with no active constraints 3 cold start with only equality constraints active 4 cold start with as many active constraints as possible

Int32 sif_file_device

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

T infinity

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

T feas_tol

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

T obj_unbounded

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

T increase_rho_g_factor

if the problem is currently infeasible and solve_qp (see below) is .TRUE. the current penalty parameter for the general constraints will be increased by increase_rho_g_factor when needed

T infeas_g_improved_by_factor

if the infeasibility of the general constraints has not dropped by a fac of infeas_g_improved_by_factor over the previous infeas_check_interval iterations, the current corresponding penalty parameter will be increase

T increase_rho_b_factor

if the problem is currently infeasible and solve_qp or solve_within_boun (see below) is .TRUE., the current penalty parameter for the simple boun constraints will be increased by increase_rho_b_factor when needed

T infeas_b_improved_by_factor

if the infeasibility of the simple bounds has not dropped by a factor of infeas_b_improved_by_factor over the previous infeas_check_interval iterations, the current corresponding penalty parameter will be increase

T pivot_tol

the threshold pivot used by the matrix factorization. See the documentation for SLS for details (OBSOLE

T pivot_tol_for_dependencies

the threshold pivot used by the matrix factorization when attempting to detect linearly dependent constraints.

T zero_pivot

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

T inner_stop_relative

the search direction is considered as an acceptable approximation to the minimizer of the model if the gradient of the model in the preconditioning(inverse) norm is less than max( inner_stop_relative * initial preconditioning(inverse) gradient norm, inner_stop_absolute )

T inner_stop_absolute

see inner_stop_relative

T multiplier_tol

any dual variable or Lagrange multiplier which is less than multiplier_t outside its optimal interval will be regarded as being acceptable when checking for optimality

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)

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 solve_qp

if solve_qp is .TRUE., the value of prob.rho_g and prob.rho_b will be increased as many times as are needed to ensure that the output solution is feasible, and thus aims to solve the quadratic program (2)-(4)

Bool solve_within_bounds

if solve_within_bounds is .TRUE., the value of prob.rho_b will be increased as many times as are needed to ensure that the output solution is feasible with respect to the simple bounds, and thus aims to solve the bound-constrained quadratic program (4)-(5)

Bool randomize

if randomize is .TRUE., the constraint bounds will be perturbed by small random quantities during the first stage of the solution process. Any randomization will ultimately be removed. Randomization helps when solving degenerate problems

Bool array_syntax_worse_than_do_loop

if .array_syntax_worse_than_do_loop is true, f77-style do loops will be used rather than f90-style array syntax for vector operations (OBSOLETE)

Bool space_critical

if .space_critical true, every effort will be made to use as little space as possible. This may result in longer computation time

Bool deallocate_error_fatal

if .deallocate_error_fatal is true, any array/pointer deallocation error will terminate execution. Otherwise, computation will continue

Bool generate_sif_file

if .generate_sif_file is .true. if a SIF file describing the current problem is to be generated

char symmetric_linear_solver[31]

indefinite linear equation solver

NTuple{31,Cchar} sif_file_name

definite linear equation solver

name of generated SIF 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’

Bool each_interval

component specifically for parametric problems (not used at present)

struct sls_control_type sls_control

control parameters for SLS

qpa_time_type structure#

    struct qpa_time_type{T}
      total::T
      preprocess::T
      analyse::T
      factorize::T
      solve::T
      clock_total::T
      clock_preprocess::T
      clock_analyse::T
      clock_factorize::T
      clock_solve::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 analyse

the CPU time spent analysing the required matrices prior to factorizatio

T factorize

the CPU time spent factorizing the required matrices

T solve

the CPU time spent computing the search direction

T clock_total

the total clock time spent in the package

T clock_preprocess

the clock time spent preprocessing the problem

T clock_analyse

the clock time spent analysing the required matrices prior to factorizat

T clock_factorize

the clock time spent factorizing the required matrices

T clock_solve

the clock time spent computing the search direction

qpa_inform_type structure#

    struct qpa_inform_type{T}
      status::Int32
      alloc_status::Int32
      bad_alloc::NTuple{81,Cchar}
      major_iter::Int32
      iter::Int32
      cg_iter::Int32
      factorization_status::Int32
      factorization_integer::Int64
      factorization_real::Int64
      nfacts::Int32
      nmods::Int32
      num_g_infeas::Int32
      num_b_infeas::Int32
      obj::T
      infeas_g::T
      infeas_b::T
      merit::T
      time::qpa_time_type{T}
      sls_inform::sls_inform_type{T}

detailed documentation#

inform derived type as a Julia structure

components#

Int32 status

return status. See QPA_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 major_iter

the total number of major iterations required

Int32 iter

the total number of iterations required

Int32 cg_iter

the total number of conjugate gradient 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 nmods

the total number of factorizations which were modified to ensure that th matrix was an appropriate preconditioner

Int32 num_g_infeas

the number of infeasible general constraints

Int32 num_b_infeas

the number of infeasible simple-bound constraints

T obj

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

T infeas_g

the 1-norm of the infeasibility of the general constraints

T infeas_b

the 1-norm of the infeasibility of the simple-bound constraints

T merit

the merit function value = obj + rho_g * infeas_g + rho_b * infeas_b

struct qpa_time_type time

timings (see above)

struct sls_inform_type sls_inform

inform parameters for SLS

Index

example calls#

This is an example of how to use the package to solve a quadratic program; the code is available in $GALAHAD/src/qpa/Julia/test_qpa.jl . A variety of supported Hessian and constraint matrix storage formats are shown.

# test_qpa.jl
# Simple code to test the Julia interface to QPA

using GALAHAD
using Test
using Printf
using Accessors
using Quadmath

function test_qpa(::Type{T}) where T
  # Derived types
  data = Ref{Ptr{Cvoid}}()
  control = Ref{qpa_control_type{T}}()
  inform = Ref{qpa_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 = one(T)  # constant term in the objective
  rho_g = T(0.1)  # penalty paramter for general constraints
  rho_b = T(0.1)  # penalty paramter for simple bound constraints
  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 1:7
    # Initialize QPA
    qpa_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'
      qpa_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)

      qpa_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'
      qpa_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)

      qpa_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]

      qpa_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)

      qpa_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'
      qpa_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)

      qpa_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'
      qpa_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)

      qpa_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'
      qpa_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)

      qpa_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

    # zero
    if d == 7
      st = 'Z'
      qpa_import(T, control, data, status, n, m,
                 "zero", H_ne, C_NULL, C_NULL, C_NULL,
                 "sparse_by_rows", A_ne, C_NULL, A_col, A_ptr)

      qpa_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

    qpa_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: QPA_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
    qpa_terminate(T, data, control, inform)
  end

  @printf("\n basic tests of l_1 qp storage formats\n\n")
  qpa_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]

  # solve the l_1qp problem
  qpa_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)

  qpa_solve_l1qp(T, data, status, n, m, H_ne, H_val, g, f, rho_g, rho_b,
                 A_ne, A_val, c_l, c_u, x_l, x_u, x, c, y, z,
                 x_stat, c_stat)

  qpa_information(T, data, inform, status)

  if inform[].status == 0
    @printf("%s %6i iterations. Optimal objective value = %5.2f status = %1i\n",
            "l1qp  ", inform[].iter, inform[].obj, inform[].status)
  else
    @printf("%c: QPA_solve exit status = %1i\n", st, inform[].status)
  end

  # Start from 0
  for i in 1:n
    x[i] = 0.0
    z[i] = 0.0
  end
  for i in 1:m
    y[i] = 0.0
  end

  # solve the bound constrained l_1qp problem
  qpa_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)

  qpa_solve_bcl1qp(T, data, status, n, m, H_ne, H_val, g, f, rho_g,
                   A_ne, A_val, c_l, c_u, x_l, x_u, x, c, y, z,
                   x_stat, c_stat)

  qpa_information(T, data, inform, status)

  if inform[].status == 0
    @printf("%s %6i iterations. Optimal objective value = %5.2f status = %1i\n",
            "bcl1qp", inform[].iter, inform[].obj, inform[].status)
  else
    @printf("%c: QPA_solve exit status = %1i\n", st, inform[].status)
  end

  # Delete internal workspace
  qpa_terminate(T, data, control, inform)

  return 0
end

@testset "QPA" begin
  @test test_qpa(Float32) == 0
  @test test_qpa(Float64) == 0
  @test test_qpa(Float128) == 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. -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 current 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. On entry, if control.cold_start = 0, x_stat should be set as above to provide a guide to the initial working set.
c_stat	is a one-dimensional array of size m and type Int32 that gives the current 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. On entry, if control.cold_start = 0, c_stat should be set as above to provide a guide to the initial working set.

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. -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.
rho_g	is a scalar of type T that holds the parameter \(\rho_g\) associated with the linear constraints.
rho_b	is a scalar of type T that holds the parameter \(\rho_b\) associated with the simple bound constraints.
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 current 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. On entry, if control.cold_start = 0, x_stat should be set as above to provide a guide to the initial working set.
c_stat	is a one-dimensional array of size m and type Int32 that gives the current 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. On entry, if control.cold_start = 0, c_stat should be set as above to provide a guide to the initial working set.

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. -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.
rho_g	is a scalar of type T that holds the parameter \(\rho_g\) associated with the linear constraints.
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 current 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. On entry, if control.cold_start = 0, x_stat should be set as above to provide a guide to the initial working set.
c_stat	is a one-dimensional array of size m and type Int32 that gives the current 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. On entry, if control.cold_start = 0, c_stat should be set as above to provide a guide to the initial working set.