GALAHAD CRO package#

purpose#

The cro package provides a crossover from a primal-dual interior-point solution to given convex quadratic program to a basic one in which the matrix of defining active constraints/variables is of full rank. This applies to the problem of minimizing the quadratic objective function

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

subject to the general linear constraints and simple bounds

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

where $H$ and $A$ are, respectively, given $n$ by $n$ symmetric postive-semi-definite and $m$ by $n$ matrices, $g$ is a vector, $f$ 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 is most suitable for problems involving a large number of unknowns $x$.

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

terminology#

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

\[A x = c\]

and

\[c_l \leq c \leq c_u, \;\; x_l \leq x \leq x_u,\]

the dual optimality conditions

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

and

\[y_l \geq 0, \;\; y_u \leq 0, \;\; z_l \geq 0 \;\;\mbox{and}\;\; z_u \leq 0,\;\;\mbox{(1)}\]

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#

Denote the active constraints by $A_A x = c_A$ and the active bounds by $I_A x = x_A$. Then any optimal solution satisfies the linear system

\[\begin{split}\left(\begin{array}{ccc} H & - A_A^T & - I^T_A \\ A_A & 0 & 0 \\ I_A & 0 & 0 \end{array}\right) \left(\begin{array}{c} x \\ y_A \\ z_A \end{array}\right) = \left(\begin{array}{c} - g \\ c_A \\ x_A \end{array}\right), \end{split}\]

where $y_A$ and $z_A$ are the corresponding active Lagrange multipliers and dual variables respectively. Consequently the difference between any two solutions $(\Delta x, \Delta y, \Delta z)$ must satisfy

\[\begin{split}\left(\begin{array}{ccc} H & - A_A^T & - I^T_A \\ A_A & 0 & 0 \\ I_A & 0 & 0 \end{array}\right) \left(\begin{array}{c} \Delta x \\ \Delta y_A \\ \Delta z_A \end{array}\right) = 0.\;\;\mbox{(2)}\end{split}\]

Thus there can only be multiple solution if the coefficient matrix $K$ of (2) is singular. The algorithm used in \fullpackagename\ exploits this. The matrix $K$ is checked for singularity using the package ULS. If $K$ is non singular, the solution is unique and the solution input by the user provides a linearly independent active set. Otherwise $K$ is singular, and partitions $A_A^T = ( A_{AB}^T \;\; A_{AN}^T)$ and $I_A^T = ( I_{AB}^T \;\; I_{AN}^T)$ are found so that

\[\begin{split}\left(\begin{array}{ccc} H & - A_{AB}^T & - I^T_{AB} \\ A_{AB} & 0 & 0 \\ I_{AB} & 0 & 0 \end{array}\right)\end{split}\]

is non-singular and the non-basic constraints $A_{AN}^T$ and $I_{AN}^T$ are linearly dependent on the basic ones $( A_{AB}^T \;\; I_{AB}^T)$. In this case (2) is equivalent to

\[\begin{split}\left(\begin{array}{ccc} H & - A_{AB}^T & - I^T_{AB} \\ A_{AB} & 0 & 0 \\ I_{AB} & 0 & 0 \end{array}\right) \left(\begin{array}{c} \Delta x \\ \Delta y_{AB} \\ \Delta z_{AB} \end{array}\right) = \left(\begin{array}{c} A_{AN}^T \\ 0 \\ 0 \end{array}\right) \Delta y_{AN} + \left(\begin{array}{c} I^T_{AN} \\ 0 \\ 0 \end{array}\right) \Delta z_{AN} .\;\;\mbox{(3)}\end{split}\]

Thus, starting from the user’s $(x, y, z)$ and with a factorization of the coefficient matrix of (3) found by the package SLS, the alternative solution $(x + \alpha x, y + \alpha y, z + \alpha z)$, featuring $(\Delta x, \Delta y_{AB}, \Delta z_{AB})$ from (3) in which successively one of the components of $\Delta y_{AN}$ and $\Delta z_{AN}$ in turn is non zero, is taken. The scalar $\alpha$ at each stage is chosen to be the largest possible that guarantees (1); this may happen when a non-basic multiplier/dual variable reaches zero, in which case the corresponding constraint is disregarded, or when this happens for a basic multiplier/dual variable, in which case this constraint is exchanged with the non-basic one under consideration and disregarded. The latter corresponds to changing the basic-non-basic partition in (3), and subsequent solutions may be found by updating the factorization of the coefficient matrix in (3) following the basic-non-basic swap using the package SCU.

matrix storage#

The unsymmetric $m$ by $n$ matrix $A$ must be presented and stored in sparse row-wise storage format. For this, only the nonzero entries are stored, and 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.

The symmetric $n$ by $n$ matrix $H$ must also be presented and stored in sparse row-wise storage format. But, crucially, now symmetry is exploited by only storing values from the lower triangular part (i.e, those entries that lie on or below the leading diagonal). As before, only the nonzero entries of the matrices are stored. Only the nonzero entries from the lower triangle are stored, and these 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.

introduction to function calls#

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

cro_initialize - provide default control parameters and set up initial data structures
cro_read_specfile (optional) - override control values by reading replacement values from a file
cro_crossover_solution - move from a primal-dual soution to a full rank one
cro_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 cro_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 cro_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 cro_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/cro/CRO.template. See also Table 2.1 in the Fortran documentation provided in $GALAHAD/doc/cro.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 cro_control_type)
specfile	is a one-dimensional array of type Vararg{Cchar} that must give the name of the specification file

    function cro_crossover_solution(T, data, control, inform, n, m, m_equal,
                                    h_ne, H_val, H_col, H_ptr,
                                    a_ne, A_val, A_col, A_ptr,
                                    g, c_l, c_u, x_l, x_u,
                                    x, c, y, z, x_stat, c_stat)

Crosover the solution from a primal-dual to a basic one.

Parameters:

control	is a structure whose members provide control parameters for the remaining procedures (see cro_control_type). The parameter .status is as follows:
data	holds private internal data.
inform	is a structure containing output information (see cro_inform_type). The component .status gives the exit status from the package. Possible values are: 0 The crossover 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 >= m_equal >= 0 has been violated. -4 the bound constraints are inconsistent. -5 the general constraints are likely inconsistent. -9 an error has occured in SLS_analyse. -10 an error has occured in SLS_factorize. -11 an error has occured in SLS_solve. -12 an error has occured in ULS_factorize. -14 an error has occured in ULS_solve. -16 the residuals are large; the factorization may be unsatisfactory.
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.
m_equal	is a scalar variable of type Int32 that holds the number of general linear equality constraints. Such constraints must occur first in $A$.
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 type T that holds the values of the entries of the lower triangular part of the Hessian matrix $H$. The entries are stored by consecutive rows, the order within each row is unimportant.
H_col	is a one-dimensional array of type Int32 that holds the column indices of the lower triangular part of $H$, in the same order as those in H_val.
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 $H$. The n+1-st component holds the total number of entries (plus one if fortran indexing is used).
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 type T that holds the values of the entries of the constraint Jacobian matrix $A$. The entries are stored by consecutive rows, the order within each row is unimportant. Equality constraints must be ordered first.
A_col	is a one-dimensional array of size A_ne and type Int32 that holds the column indices of $A$ in the same order as those in A_val.
A_ptr	is a one-dimensional array of size m+1 and type Int32 that holds the starting position of each row of $A$. The m+1-st component holds the total number of entries (plus one if fortran indexing is used).
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 $g_j$.
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 $c(x) = A x$. 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 must be set on entry to give the status of the problem variables. If x_stat(j) is negative, the variable $x_j$ is active on its lower bound, if it is positive, it is active and lies on its upper bound, and if it is zero, it is inactiive and lies between its bounds. On exit, the $j$ -th component of x_stat is -1 if the variable is basic and active on its lower bound, -2 it is non-basic but active on its lower bound, 1 if it is basic and active on its upper bound, 2 it is non-basic but active on its upper bound, and 0 if it is inactive.
c_stat	is a one-dimensional array of size m and type Int32 that must be set on entry to give the status of the general linear constraints. If c_stat(i) is negative, the constraint value $a_i^Tx$ is active on its lower bound, if it is positive, it is active and lies on its upper bound, and if it is zero, it is inactiive and lies between its bounds. On exit, the $i$ -th component of x_stat is -1 if the constraint is basic and active on its lower bound, -2 it is non-basic but active on its lower bound, 1 if it is basic and active on its upper bound, 2 it is non-basic but active on its upper bound, and 0 if it is inactive.

    function cro_terminate(T, data, control, inform)

Deallocate all internal private storage

Parameters:

data	holds private internal data
control	is a structure containing control information (see cro_control_type)
inform	is a structure containing output information (see cro_inform_type)

available structures#

cro_control_type structure#

    struct cro_control_type{T}
      f_indexing::Bool
      error::Int32
      out::Int32
      print_level::Int32
      max_schur_complement::Int32
      infinity::T
      feasibility_tolerance::T
      check_io::Bool
      refine_solution::Bool
      space_critical::Bool
      deallocate_error_fatal::Bool
      symmetric_linear_solver::NTuple{31,Cchar}
      unsymmetric_linear_solver::NTuple{31,Cchar}
      prefix::NTuple{31,Cchar}
      sls_control::sls_control_type{T}
      sbls_control::sbls_control_type{T}
      uls_control::uls_control_type{T}
      ir_control::ir_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 max_schur_complement

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

T infinity

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

T feasibility_tolerance

feasibility tolerance for KKT violation

Bool check_io

if .check_io is true, the input (x,y,z) will be fully tested for consistency

Bool refine_solution

if .refine solution is true, attempt to satisfy the KKT conditions as accurately as possible

Bool space_critical

if .space_critical is 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

char symmetric_linear_solver[31]

indefinite linear equation solver

char unsymmetric_linear_solver[31]

unsymmetric linear equation solver

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 sls_control_type sls_control

control parameters for SLS

struct sbls_control_type sbls_control

control parameters for SBLS

struct uls_control_type uls_control

control parameters for ULS

struct ir_control_type ir_control

control parameters for iterative refinement

cro_time_type structure#

    struct cro_time_type{T}
      total::Float32
      analyse::Float32
      factorize::Float32
      solve::Float32
      clock_total::T
      clock_analyse::T
      clock_factorize::T
      clock_solve::T

detailed documentation#

time derived type as a Julia structure

components#

Float32 total

the total CPU time spent in the package

Float32 analyse

the CPU time spent reordering the matrix prior to factorization

Float32 factorize

the CPU time spent factorizing the required matrices

Float32 solve

the CPU time spent computing corrections

T clock_total

the total clock time spent in the package

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 corrections

cro_inform_type structure#

    struct cro_inform_type{T}
      status::Int32
      alloc_status::Int32
      bad_alloc::NTuple{81,Cchar}
      dependent::Int32
      time::cro_time_type{T}
      sls_inform::sls_inform_type{T}
      sbls_inform::sbls_inform_type{T}
      uls_inform::uls_inform_type{T}
      scu_status::Int32
      scu_inform::scu_inform_type
      ir_inform::ir_inform_type{T}

detailed documentation#

inform derived type as a Julia structure

components#

Int32 status

return status. See CRO_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 dependent

the number of dependent active constraints

struct cro_time_type time

timings (see above)

struct sls_inform_type sls_inform

information from SLS

struct sbls_inform_type sbls_inform

information from SBLS

struct uls_inform_type uls_inform

information from ULS

Int32 scu_status

information from SCU

struct scu_inform_type scu_inform

see scu_status

struct ir_inform_type ir_inform

information from IR

Index

example calls#

This is an example of how to use the package to crossover from a primal-dual QP solution to a basic one; the code is available in $GALAHAD/src/cro/Julia/test_cro.jl . A variety of supported Hessian and constraint matrix storage formats are shown.

# test_cro.jl
# Simple code to test the Julia interface to CRO

using GALAHAD
using Test
using Printf
using Accessors
using Quadmath

function test_cro(::Type{T}) where T
  # Derived types
  data = Ref{Ptr{Cvoid}}()
  control = Ref{cro_control_type{T}}()
  inform = Ref{cro_inform_type{T}}()

  # Set problem dimensions
  n = 11 # dimension
  m = 3 # number of general constraints
  m_equal = 1 # number of equality constraints

  #  describe the objective function

  H_ne = 21
  H_val = T[1.0, 0.5, 1.0, 0.5, 1.0, 0.5, 1.0, 0.5, 1.0, 0.5, 1.0, 0.5, 1.0, 0.5, 1.0,
                  0.5, 1.0, 0.5, 1.0, 0.5, 1.0]
  H_col = Cint[1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11]
  H_ptr = Cint[1, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22]
  g = T[0.5, -0.5, -1.0, -1.0, -1.0, -1.0, -1.0, -1.0, -1.0, -1.0, -0.5]

  #  describe constraints

  A_ne = 30
  A_val = T[1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0,
                  1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0]
  A_col = Cint[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 3, 4, 5, 6, 7, 8, 9, 10, 11, 2, 3, 4, 5,
               6, 7, 8, 9, 10, 11]
  A_ptr = Cint[1, 12, 21, 31]
  c_l = T[10.0, 9.0, -Inf]
  c_u = T[10.0, Inf, 10.0]
  x_l = T[0.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0]
  x_u = T[Inf, Inf, Inf, Inf, Inf, Inf, Inf, Inf, Inf, Inf, Inf]

  # provide optimal variables, Lagrange multipliers and dual variables
  x = T[0.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0]
  c = T[10.0, 9.0, 10.0]
  y = T[-1.0, 1.5, -2.0]
  z = T[2.0, 4.0, 2.5, 2.5, 2.5, 2.5, 2.5, 2.5, 2.5, 2.5, 2.5]

  # provide interior-point constraint and variable status
  c_stat = Cint[-1, -1, 1]
  x_stat = Cint[-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1]

  # Set output storage
  status = Ref{Cint}()
  @printf(" Fortran sparse matrix indexing\n\n")

  # Initialize CRO
  cro_initialize(T, data, control, status)

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

  # crossover the solution
  cro_crossover_solution(T, data, control, inform, n, m, m_equal, H_ne, H_val, H_col, H_ptr,
                         A_ne, A_val, A_col, A_ptr, g, c_l, c_u, x_l, x_u, x, c, y, z,
                         x_stat, c_stat)

  @printf(" CRO_crossover exit status = %1i\n", inform[].status)

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

  return 0
end

@testset "CRO" begin
  @test test_cro(Float32) == 0
  @test test_cro(Float64) == 0
  @test test_cro(Float128) == 0
end