GALAHAD LSQP package#

purpose#

The lsqp package uses an interior-point trust-region method to solve a given linear or separable convex quadratic program. The aim is to minimize the separable quadratic 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 $A$ is a given $m$ by $n$ matrix, $g$, $w$ 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 $x$.

In the special case where $w = 0$, $g = 0$ and $f = 0$, the so-called analytic center of the feasible set will be found, while linear programming, or constrained least distance, problems may be solved by picking $w = 0$, or $g = 0$ and $f = 0$, respectively.

See Section 4 of $GALAHAD/doc/lsqp.pdf for additiional details.

The more-modern package cqp offers similar functionality, and is often to be preferred.

terminology#

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

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

and

\[c_l \leq c \leq c_u, \;\; x_l \leq x \leq x_u,\;\;\mbox{(1b)}\]

the dual optimality conditions

\[W^2 ( x - x^0 ) + g = A^{T} y + z,\;\; y = y_l + y_u \;\;\mbox{and}\;\; z = z_l + z_u,\;\;\mbox{(2a)}\]

and

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

and the complementary slackness conditions

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

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#

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

When $w \neq 0$ or $g \neq 0$, the method aims at each stage to reduce the overall violation of (1a), (2a) and (3), rather than reducing each of the terms individually. Given an estimate $v = (x, \; c, \; y, \; y^{l}, \; y^{u}, \; z, \; z^{l}, \; z^{u})$ of the primal-dual variables, a correction $\Delta v = \Delta (x, \; c, \; y, \; y^{l}, \; y^{u} ,\;z,\;z^{l} ,\;z^{u} )$ is obtained by solving a suitable linear system of Newton equations for the nonlinear systems (1a), (2a) and a parameterized perturbation of (3). An improved estimate $v + \alpha \Delta v$ is then used, where the step-size $\alpha$ is chosen as close to 1.0 as possible while ensuring both that (1b) and (2b) continue to hold and that the individual components which make up the complementary slackness (3) do not deviate too significantly from their average value. The parameter that controls the perturbation of (3) is ultimately driven to zero.

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

When $w = 0$ and $g = 0$, the method aims instead firstly to find an interior primal feasible point, that is to ensure that (1a) is satisfied. One this has been achieved, attention is switched to mninizing the potential function

\[\phi (x,\;c) = - \sum_{i=1}^{m} \log ( c_{i} - c_{i}^{l} ) - \sum_{i=1}^{m} \log ( c_{i}^{u} - c_{i} ) - \sum_{j=1}^{n} \log ( x_{j} - x_{j}^{l} ) - \sum_{j=1}^{n} \log ( x_{j}^{u} - x_{j} ) ,\]

while ensuring that (1a) remain satisfied and that $x$ and $c$ are strictly interior points for (1b). The global minimizer of this minimization problem is known as the analytic center of the feasible region, and may be viewed as a feasible point that is as far from the boundary of the constraints as possible. Note that terms in the above sumations corresponding to infinite bounds are ignored, and that equality constraints are treated specially. Appropriate “primal” Newton corrections are used to generate a sequence of improving points converging to the analytic center, while the iteration is stabilized by performing inesearches along these corrections with respect to $\phi (x,\;c)$.

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.

references#

The basic algorithm is that of

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

with a number of enhancements described by

A. R. Conn, N. I. M. Gould, D. Orban and Ph. L. Toint, ``A primal-dual trust-region algorithm for minimizing a non-convex function subject to general inequality and linear equality constraints’’. Mathematical Programming **87* (1999) 215-249.

matrix storage#

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

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

introduction to function calls#

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

lsqp_initialize - provide default control parameters and set up initial data structures
lsqp_read_specfile (optional) - override control values by reading replacement values from a file
lsqp_import - set up problem data structures and fixed values
lsqp_reset_control (optional) - possibly change control parameters if a sequence of problems are being solved
lsqp_solve_qp - solve the quadratic program
lsqp_information (optional) - recover information about the solution and solution process
lsqp_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 lsqp_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 lsqp_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 lsqp_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/lsqp/LSQP.template. See also Table 2.1 in the Fortran documentation provided in $GALAHAD/doc/lsqp.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 lsqp_control_type)
specfile	is a one-dimensional array of type Vararg{Cchar} that must give the name of the specification file

    function lsqp_import(T, control, data, status, n, m,
                         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 lsqp_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.
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.
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 lsqp_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 lsqp_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 lsqp_solve_qp(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 separable convex 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.
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`, i = 1, … , m, contains $c_i(x)$.
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`, i = 1, … , m, contains $y_i$.
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 $a_i^T x$ 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 lsqp_information(T, data, inform, status)

Provides output information.

Parameters:

data

holds private internal data

inform

is a structure containing output information (see lsqp_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 lsqp_terminate(T, data, control, inform)

Deallocate all internal private storage.

Parameters:

data	holds private internal data
control	is a structure containing control information (see lsqp_control_type)
inform	is a structure containing output information (see lsqp_inform_type)

available structures#

lsqp_control_type structure#

    struct lsqp_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
      indmin::Int32
      valmin::Int32
      itref_max::Int32
      infeas_max::Int32
      muzero_fixed::Int32
      restore_problem::Int32
      indicator_type::Int32
      extrapolate::Int32
      path_history::Int32
      path_derivatives::Int32
      fit_order::Int32
      sif_file_device::Int32
      infinity::T
      stop_p::T
      stop_d::T
      stop_c::T
      prfeas::T
      dufeas::T
      muzero::T
      reduce_infeas::T
      potential_unbounded::T
      pivot_tol::T
      pivot_tol_for_dependencies::T
      zero_pivot::T
      identical_bounds_tol::T
      mu_min::T
      indicator_tol_p::T
      indicator_tol_pd::T
      indicator_tol_tapia::T
      cpu_time_limit::T
      clock_time_limit::T
      remove_dependencies::Bool
      treat_zero_bounds_as_general::Bool
      just_feasible::Bool
      getdua::Bool
      puiseux::Bool
      feasol::Bool
      balance_initial_complentarity::Bool
      use_corrector::Bool
      array_syntax_worse_than_do_loop::Bool
      space_critical::Bool
      deallocate_error_fatal::Bool
      generate_sif_file::Bool
      sif_file_name::NTuple{31,Cchar}
      prefix::NTuple{31,Cchar}
      fdc_control::fdc_control_type{T}
      sbls_control::sbls_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 indmin

an initial guess as to the integer workspace required by SBLS

Int32 valmin

an initial guess as to the real workspace required by SBLS

Int32 itref_max

the maximum number of iterative refinements allowed

Int32 infeas_max

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

Int32 muzero_fixed

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

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 indicator_type

specifies the type of indicator function used. Possible values are

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

Int32 extrapolate

should extrapolation be used to track the central path? Possible values

0 never
1 after the final major iteration
2 at each major iteration (unused at present)

Int32 path_history

the maximum number of previous path points to use when fitting the data (unused at present)

Int32 path_derivatives

the maximum order of path derivative to use (unused at present)

Int32 fit_order

the order of (Puiseux) series to fit to the path data: $

\[\]

to fit all data (unused at present)

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 stop_p

the required accuracy for the primal infeasibility

T stop_d

the required accuracy for the dual infeasibility

T stop_c

the required accuracy for the complementarity

T prfeas

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

T dufeas

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

T muzero

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

T reduce_infeas

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

T potential_unbounded

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

T pivot_tol

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

T pivot_tol_for_dependencies

the threshold pivot used by the matrix factorization when attempting to detect linearly dependent constraints. See the documentation for SBLS for details

T zero_pivot

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

T identical_bounds_tol

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

T mu_min

start terminal extrapolation when mu reaches mu_min

T indicator_tol_p

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

T indicator_tol_pd

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

T indicator_tol_tapia

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

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 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 just_feasible

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

Bool getdua

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

Bool puiseux

If extrapolation is to be used, decide between Puiseux and Taylor series.

Bool feasol

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

Bool balance_initial_complentarity

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

Bool use_corrector

if .use_corrector, a corrector step will be used

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

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

NTuple{31,Cchar} sif_file_name

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’

struct fdc_control_type fdc_control

control parameters for FDC

struct sbls_control_type sbls_control

control parameters for SBLS

lsqp_time_type structure#

    struct lsqp_time_type{T}
      total::T
      preprocess::T
      find_dependent::T
      analyse::T
      factorize::T
      solve::T
      clock_total::T
      clock_preprocess::T
      clock_find_dependent::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 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 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

lsqp_inform_type structure#

    struct lsqp_inform_type{T}
      status::Int32
      alloc_status::Int32
      bad_alloc::NTuple{81,Cchar}
      iter::Int32
      factorization_status::Int32
      factorization_integer::Int64
      factorization_real::Int64
      nfacts::Int32
      nbacts::Int32
      obj::T
      potential::T
      non_negligible_pivot::T
      feasible::Bool
      time::lsqp_time_type{T}
      fdc_inform::fdc_inform_type{T}
      sbls_inform::sbls_inform_type{T}

detailed documentation#

inform derived type as a Julia structure

components#

Int32 status

return status. See LSQP_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 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 nbacts

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

T obj

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

T potential

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

T non_negligible_pivot

the smallest pivot which was not judged to be zero when detecting linear dependent constraints

Bool feasible

is the returned “solution” feasible?

struct lsqp_time_type time

timings (see above)

struct fdc_inform_type fdc_inform

inform parameters for FDC

struct sbls_inform_type sbls_inform

inform parameters for SBLS

Index

example calls#

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

# test_lsqp.jl
# Simple code to test the Julia interface to LSQP

using GALAHAD
using Test
using Printf
using Accessors

function test_lsqp(::Type{T}) where T
  # Derived types
  data = Ref{Ptr{Cvoid}}()
  control = Ref{lsqp_control_type{T}}()
  inform = Ref{lsqp_inform_type{T}}()

  # Set problem data
  n = 3 # dimension
  m = 2 # number of general constraints
  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
  w = T[1.0, 1.0, 1.0]
  x_0 = T[0.0, 0.0, 0.0]

  # 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 1:3
    # Initialize LSQP
    lsqp_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'
      lsqp_import(T, control, data, status, n, m,
                  "coordinate", A_ne, A_row, A_col, C_NULL)

      lsqp_solve_qp(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

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

      lsqp_solve_qp(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

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

      lsqp_import(T, control, data, status, n, m,
                  "dense", A_dense_ne, C_NULL, C_NULL, C_NULL)

      lsqp_solve_qp(T, data, status, n, m, w, x_0, g, f,
                    A_dense_ne, A_dense, c_l, c_u, x_l, x_u,
                    x, c, y, z, x_stat, c_stat)
    end

    lsqp_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: LSQP_solve exit status = %1i\n", st, inform[].status)
    end
    # @printf("x: ")
    # for i = 1:n
    #   @printf("%f ", x[i])
    # @printf("\n")
    # @printf("gradient: ")
    # for i = 1:n
    #   @printf("%f ", g[i])
    # @printf("\n")

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

  return 0
end

@testset "LSQP" begin
  @test test_lsqp(Float32) == 0
  @test test_lsqp(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.
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`, i = 1, … , m, contains \(c_i(x)\).
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`, i = 1, … , m, contains \(y_i\).
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 \(a_i^T x\) 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.