GALAHAD EQP package

purpose

The eqp package uses an iterative method to solve a given equality-constrained quadratic program. The aim is to minimize the quadratic objective function

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

or the shifted-least-distance objective function

\[s(x) = f + g^T x + \frac{1}{2} \sum_{j=1}^n w_j^2 (x_j - x_j^0)^2,\]

subject to the general linear equality constraints

\[A x + c = 0,\]

where \(H\) and \(A\) are, respectively, given \(n\) by \(n\) symmetric and \(m\) by \(n\) general matrices, \(g\), \(w\), \(x^0\) and \(c\) are vectors, and \(f\) is a scalar. The method is most suitable for problems involving a large number of unknowns \(x\).

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

terminology

Any required solution \(x\) necessarily satisfies the primal optimality conditions

\[A x + c = 0 \;\;\mbox{(1)}\]

and the dual optimality conditions

\[H x + g = A^T y,\]

where the vector \(y\) is known as the Lagrange multipliers for the general linear constraints.

In the shifted-least-distance case, \(g\) is shifted by \(-W^2 x^0\), and \(H = W^2\), where \(W\) is the diagonal matrix whose entries are the \(w_j\).

method

A solution to the problem is found in two phases. In the first, a point \(x_F\) satisfying (1) is found. In the second, the required solution \(x = x_F + s\) is determined by finding \(s\) to minimize \(q(s) = \frac{1}{2} s^T H s + g_F^T s + f_F^{}\) subject to the homogeneous constraints \(A s = 0\), where \(g_F^{} = H x_F^{} + g\) and \(f_F^{} = \frac{1}{2} x_F^T H x_F^{} + g^T x_F^{} + f\). The required constrained minimizer of \(q(s)\) is obtained by implictly applying the preconditioned conjugate-gradient method in the null space of \(A\). Any preconditioner of the form

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

is suitable, and the package SBLS provides a number of possibilities. In order to ensure that the minimizer obtained is finite, an additional, precautionary trust-region constraint \(\|s\| \leq \Delta\) for some suitable positive radius \(\Delta\) is imposed, and the package GLTR is used to solve this additionally-constrained problem.

references

The preconditioning aspcets are described in detail in

H. S. Dollar, N. I. M. Gould and A. J. Wathen. ``On implicit-factorization constraint preconditioners’’. In Large Scale Nonlinear Optimization (G. Di Pillo and M. Roma, eds.) Springer Series on Nonconvex Optimization and Its Applications, Vol. 83, Springer Verlag (2006) 61–82

and

H. S. Dollar, N. I. M. Gould, W. H. A. Schilders and A. J. Wathen ``On iterative methods and implicit-factorization preconditioners for regularized saddle-point systems’’. SIAM Journal on Matrix Analysis and Applications 28(1) (2006) 170–189,

while the constrained conjugate-gradient method is discussed in

N. I. M. Gould, S. Lucidi, M. Roma and Ph. L. Toint, ``Solving the trust-region subproblem using the Lanczos method’’. SIAM Journal on Optimization 9(2) (1999), 504–525.

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 eqp package must be called in the following order:

• eqp_initialize - provide default control parameters and set up initial data structures

• eqp_read_specfile (optional) - override control values by reading replacement values from a file

• eqp_import - set up problem data structures and fixed values

• eqp_reset_control (optional) - possibly change control parameters if a sequence of problems are being solved

• solve the problem by calling one of

  • eqp_solve_qp - solve the quadratic program

  • eqp_solve_sldqp - solve the shifted least-distance problem

• eqp_resolve_qp (optional) - resolve the problem with the same Hessian and Jacobian, but different \(g\), \(f\) and/or \(c\)

• eqp_information (optional) - recover information about the solution and solution process

• eqp_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 eqp_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 eqp_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 eqp_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/eqp/EQP.template. See also Table 2.1 in the Fortran documentation provided in $GALAHAD/doc/eqp.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 eqp_control_type)
specfile	is a one-dimensional array of type Vararg{Cchar} that must give the name of the specification file

    function eqp_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 eqp_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 eqp_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 eqp_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 eqp_solve_qp(T, data, status, n, m, h_ne, H_val, g, f,
                          a_ne, A_val, c, x, y)

Solve the quadratic program when the Hessian \(H\) is available.

Parameters:

data	holds private internal data
status	is a scalar variable of type Int32 that gives the entry and exit status from the package. Possible exit values are: 0 The run was successful. -1 An allocation error occurred. A message indicating the offending array is written on unit control.error, and the returned allocation status and a string containing the name of the offending array are held in inform.alloc_status and inform.bad_alloc respectively. -2 A deallocation error occurred. A message indicating the offending array is written on unit control.error and the returned allocation status and a string containing the name of the offending array are held in inform.alloc_status and inform.bad_alloc respectively. -3 The restrictions n > 0 and m > 0 or requirement that a type contains its relevant string ‘dense’, ‘coordinate’, ‘sparse_by_rows’, ‘diagonal’, ‘scaled_identity’, ‘identity’, ‘zero’ or ‘none’ has been violated. -7 The constraints appear to have no feasible point. -9 The analysis phase of the factorization failed; the return status from the factorization package is given in the component inform.factor_status -10 The factorization failed; the return status from the factorization package is given in the component inform.factor_status. -11 The solution of a set of linear equations using factors from the factorization package failed; the return status from the factorization package is given in the component inform.factor_status. -16 The problem is so ill-conditioned that further progress is impossible. -17 The step is too small to make further impact. -18 Too many iterations have been performed. This may happen if control.maxit is too small, but may also be symptomatic of a badly scaled problem. -19 The CPU time limit has been reached. This may happen if control.cpu_time_limit is too small, but may also be symptomatic of a badly scaled problem. -23 An entry from the strict upper triangle of \(H\) has been specified.
n	is a scalar variable of type Int32 that holds the number of variables
m	is a scalar variable of type Int32 that holds the number of general linear constraints.
h_ne	is a scalar variable of type Int32 that holds the number of entries in the lower triangular part of the Hessian matrix \(H\).
H_val	is a one-dimensional array of size h_ne and type T that holds the values of the entries of the lower triangular part of the Hessian matrix \(H\) in any of the available storage schemes.
g	is a one-dimensional array of size n and type T that holds the linear term \(g\) of the objective function. The j-th component of g, j = 1, … , n, contains \(g_j\).
f	is a scalar of type T that holds the constant term \(f\) of the objective function.
a_ne	is a scalar variable of type Int32 that holds the number of entries in the constraint Jacobian matrix \(A\).
A_val	is a one-dimensional array of size a_ne and type T that holds the values of the entries of the constraint Jacobian matrix \(A\) in any of the available storage schemes.
c	is a one-dimensional array of size m and type T that holds the linear term \(c\) in the constraints. The i-th component of c, i = 1, … , m, contains \(c_i\).
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 \(x_j\).
y	is a one-dimensional array of size n and type T that holds the values \(y\) of the Lagrange multipliers for the linear constraints. The j-th component of y, i = 1, … , m, contains \(y_i\).

    function eqp_solve_sldqp(T, data, status, n, m, w, x0, g, f,
                             a_ne, A_val, c, x, y)

Solve the shifted least-distance quadratic program

Parameters:

data	holds private internal data
status	is a scalar variable of type Int32 that gives the entry and exit status from the package. Possible exit values are: 0 The run was successful -1 An allocation error occurred. A message indicating the offending array is written on unit control.error, and the returned allocation status and a string containing the name of the offending array are held in inform.alloc_status and inform.bad_alloc respectively. -2 A deallocation error occurred. A message indicating the offending array is written on unit control.error and the returned allocation status and a string containing the name of the offending array are held in inform.alloc_status and inform.bad_alloc respectively. -3 The restrictions n > 0 and m > 0 or requirement that a type contains its relevant string ‘dense’, ‘coordinate’, ‘sparse_by_rows’, ‘diagonal’, ‘scaled_identity’, ‘identity’, ‘zero’ or ‘none’ has been violated. -7 The constraints appear to have no feasible point. -9 The analysis phase of the factorization failed; the return status from the factorization package is given in the component inform.factor_status -10 The factorization failed; the return status from the factorization package is given in the component inform.factor_status. -11 The solution of a set of linear equations using factors from the factorization package failed; the return status from the factorization package is given in the component inform.factor_status. -16 The problem is so ill-conditioned that further progress is impossible. -17 The step is too small to make further impact. -18 Too many iterations have been performed. This may happen if control.maxit is too small, but may also be symptomatic of a badly scaled problem. -19 The CPU time limit has been reached. This may happen if control.cpu_time_limit is too small, but may also be symptomatic of a badly scaled problem. -23 An entry from the strict upper triangle of \(H\) has been specified.
n	is a scalar variable of type Int32 that holds the number of variables
m	is a scalar variable of type Int32 that holds the number of general linear constraints.
w	is a one-dimensional array of size n and type T that holds the values of the weights \(w\).
x0	is a one-dimensional array of size n and type T that holds the values of the shifts \(x^0\).
g	is a one-dimensional array of size n and type T that holds the linear term \(g\) of the objective function. The j-th component of g, j = 1, … , n, contains \(g_j\).
f	is a scalar of type T that holds the constant term \(f\) of the objective function.
a_ne	is a scalar variable of type Int32 that holds the number of entries in the constraint Jacobian matrix \(A\).
A_val	is a one-dimensional array of size a_ne and type T that holds the values of the entries of the constraint Jacobian matrix \(A\) in any of the available storage schemes.
c	is a one-dimensional array of size m and type T that holds the linear term \(c\) in the constraints. The i-th component of c, i = 1, … , m, contains \(c_i\).
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 \(x_j\).
y	is a one-dimensional array of size n and type T that holds the values \(y\) of the Lagrange multipliers for the linear constraints. The j-th component of y, i = 1, … , m, contains \(y_i\).

    function eqp_resolve_qp(T, data, status, n, m, g, f, c, x, y)

Resolve the quadratic program or shifted least-distance quadratic program when some or all of the data \(g\), \(f\) and \(c\) has changed

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. -7 The constraints appear to have no feasible point. -11 The solution of a set of linear equations using factors from the factorization package failed; the return status from the factorization package is given in the component inform.factor_status. -16 The problem is so ill-conditioned that further progress is impossible. -17 The step is too small to make further impact. -18 Too many iterations have been performed. This may happen if control.maxit is too small, but may also be symptomatic of a badly scaled problem. -19 The CPU time limit has been reached. This may happen if control.cpu_time_limit is too small, but may also be symptomatic of a badly scaled problem. -23 An entry from the strict upper triangle of \(H\) has been specified.
n	is a scalar variable of type Int32 that holds the number of variables
m	is a scalar variable of type Int32 that holds the number of general linear constraints.
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.
c	is a one-dimensional array of size m and type T that holds the linear term \(c\) in the constraints. The i-th component of c, i = 1, … , m, contains \(c_i\).
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 \(x_j\).
y	is a one-dimensional array of size n and type T that holds the values \(y\) of the Lagrange multipliers for the linear constraints. The j-th component of y, i = 1, … , m, contains \(y_i\).

    function eqp_information(T, data, inform, status)

Provides output information

Parameters:

data	holds private internal data
inform	is a structure containing output information (see eqp_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 eqp_terminate(T, data, control, inform)

Deallocate all internal private storage

Parameters:

data	holds private internal data
control	is a structure containing control information (see eqp_control_type)
inform	is a structure containing output information (see eqp_inform_type)

available structures

eqp_control_type structure

    struct eqp_control_type{T}
      f_indexing::Bool
      error::Int32
      out::Int32
      print_level::Int32
      factorization::Int32
      max_col::Int32
      indmin::Int32
      valmin::Int32
      len_ulsmin::Int32
      itref_max::Int32
      cg_maxit::Int32
      preconditioner::Int32
      semi_bandwidth::Int32
      new_a::Int32
      new_h::Int32
      sif_file_device::Int32
      pivot_tol::T
      pivot_tol_for_basis::T
      zero_pivot::T
      inner_fraction_opt::T
      radius::T
      min_diagonal::T
      max_infeasibility_relative::T
      max_infeasibility_absolute::T
      inner_stop_relative::T
      inner_stop_absolute::T
      inner_stop_inter::T
      find_basis_by_transpose::Bool
      remove_dependencies::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}
      gltr_control::gltr_control_type{T}

detailed documentation

control derived type as a Julia structure

components

Bool f_indexing

use C or Fortran sparse matrix indexing

Int32 error

error and warning diagnostics occur on stream error

Int32 out

general output occurs on stream out

Int32 print_level

the level of output required is specified by print_level

Int32 factorization

the factorization to be used. Possible values are /li 0 automatic /li 1 Schur-complement factorization /li 2 augmented-system factorization (OBSOLETE)

Int32 max_col

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

Int32 indmin

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

Int32 valmin

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

Int32 len_ulsmin

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

Int32 itref_max

the maximum number of iterative refinements allowed (OBSOLETE)

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 preconditioner

the preconditioner to be used for the CG. 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 (OBSOLETE)

• 5 optionally supplied diagonal, G = D

Int32 semi_bandwidth

the semi-bandwidth of a band preconditioner, if appropriate (OBSOLETE)

Int32 new_a

how much has A changed since last problem solved: 0 = not changed, 1 = values changed, 2 = structure changed

Int32 new_h

how much has H changed since last problem solved: 0 = not changed, 1 = values changed, 2 = structure changed

Int32 sif_file_device

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

T pivot_tol

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

T pivot_tol_for_basis

the threshold pivot used by the matrix factorization when finding the ba See the documentation for ULS for details (OBSOLETE)

T zero_pivot

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

T inner_fraction_opt

the computed solution which gives at least inner_fraction_opt times the optimal value will be found (OBSOLETE)

T radius

an upper bound on the permitted step (-ve will be reset to an appropriat large value by eqp_solve)

T min_diagonal

diagonal preconditioners will have diagonals no smaller than min_diagonal (OBSOLETE)

T max_infeasibility_relative

if the constraints are believed to be rank defficient and the residual at a “typical” feasible point is larger than max( max_infeasibility_relative * norm A, max_infeasibility_absolute ) the problem will be marked as infeasible

T max_infeasibility_absolute

see max_infeasibility_relative

T inner_stop_relative

the computed solution is considered as an acceptable approximation to th minimizer of the problem if the gradient of the objective 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 inner_stop_inter

see inner_stop_relative

Bool find_basis_by_transpose

if .find_basis_by_transpose is true, implicit factorization precondition will be based on a basis of A found by examining A’s transpose (OBSOLETE)

Bool remove_dependencies

if .remove_dependencies is true, the equality constraints will be preprocessed to remove any linear dependencies

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

struct gltr_control_type gltr_control

control parameters for GLTR

eqp_time_type structure

    struct eqp_time_type{T}
      total::T
      find_dependent::T
      factorize::T
      solve::T
      solve_inter::T
      clock_total::T
      clock_find_dependent::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 find_dependent

the CPU time spent detecting linear dependencies

T factorize

the CPU time spent factorizing the required matrices

T solve

the CPU time spent computing the search direction

T solve_inter

see solve

T clock_total

the total clock time spent in the package

T clock_find_dependent

the clock time spent detecting linear dependencies

T clock_factorize

the clock time spent factorizing the required matrices

T clock_solve

the clock time spent computing the search direction

eqp_inform_type structure

    struct eqp_inform_type{T}
      status::Int32
      alloc_status::Int32
      bad_alloc::NTuple{81,Cchar}
      cg_iter::Int32
      cg_iter_inter::Int32
      factorization_integer::Int64
      factorization_real::Int64
      obj::T
      time::eqp_time_type{T}
      fdc_inform::fdc_inform_type{T}
      sbls_inform::sbls_inform_type{T}
      gltr_inform::gltr_inform_type{T}

detailed documentation

inform derived type as a Julia structure

components

Int32 status

return status. See EQP_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 cg_iter

the total number of conjugate gradient iterations required

Int32 cg_iter_inter

see cg_iter

Int64 factorization_integer

the total integer workspace required for the factorization

Int64 factorization_real

the total real workspace required for the factorization

T obj

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

struct eqp_time_type time

timings (see above)

struct fdc_inform_type fdc_inform

inform parameters for FDC

struct sbls_inform_type sbls_inform

inform parameters for SBLS

struct gltr_inform_type gltr_inform

return information from GLTR

Index

example calls

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

# test_eqp.jl
# Simple code to test the Julia interface to EQP

using GALAHAD
using Test
using Printf
using Accessors
using Quadmath

function test_eqp(::Type{T}) where T
  # Derived types
  data = Ref{Ptr{Cvoid}}()
  control = Ref{eqp_control_type{T}}()
  inform = Ref{eqp_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
  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 = T[3.0, 0.0]  # rhs of the constraints

  # Set output storage
  x_stat = zeros(Cint, n) # variable status
  c_stat = zeros(Cint, m) # constraint status
  st = ' '
  status = Ref{Cint}()

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

  for d in 1:6
    # Initialize EQP
    eqp_initialize(T, data, control, status)

    # Set user-defined control options
    @reset control[].f_indexing = true # Fortran sparse matrix indexing
    @reset control[].fdc_control.use_sls = true
    @reset control[].fdc_control.symmetric_linear_solver = galahad_linear_solver("sytr")
    @reset control[].sbls_control.symmetric_linear_solver = galahad_linear_solver("sytr")
    @reset control[].sbls_control.definite_linear_solver = galahad_linear_solver("sytr")

    # 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'
      eqp_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)

      eqp_solve_qp(T, data, status, n, m, H_ne, H_val, g, f,
                   A_ne, A_val, c, x, y)
    end

    # sparse by rows
    if d == 2
      st = 'R'
      eqp_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)

      eqp_solve_qp(T, data, status, n, m, H_ne, H_val, g, f,
                   A_ne, A_val, c, x, y)
    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]

      eqp_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)

      eqp_solve_qp(T, data, status, n, m, H_dense_ne, H_dense, g, f,
                   A_dense_ne, A_dense, c, x, y)
    end

    # diagonal
    if d == 4
      st = 'L'
      eqp_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)

      eqp_solve_qp(T, data, status, n, m, H_ne, H_val, g, f,
                   A_ne, A_val, c, x, y)
    end

    # scaled identity
    if d == 5
      st = 'S'
      eqp_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)

      eqp_solve_qp(T, data, status, n, m, H_ne, H_val, g, f,
                   A_ne, A_val, c, x, y)
    end

    # identity
    if d == 6
      st = 'I'
      eqp_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)

      eqp_solve_qp(T, data, status, n, m, H_ne, H_val, g, f,
                   A_ne, A_val, c, x, y)
    end

    # zero
    if d == 7
      st = 'Z'
      eqp_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)

      eqp_solve_qp(T, data, status, n, m, H_ne, H_val, g, f,
                   A_ne, A_val, c, x, y)
    end

    eqp_information(T, data, inform, status)

    if inform[].status == 0
      @printf("%c:%6i cg iterations. Optimal objective value = %5.2f status = %1i\n",
              st, inform[].cg_iter, inform[].obj, inform[].status)
    else
      @printf("%c: EQP_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
    eqp_terminate(T, data, control, inform)
  end

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

    # Initialize EQP
    eqp_initialize(T, data, control, status)
    @reset control[].fdc_control.use_sls = true
    @reset control[].fdc_control.symmetric_linear_solver = galahad_linear_solver("sytr")
    @reset control[].sbls_control.symmetric_linear_solver = galahad_linear_solver("sytr")
    @reset control[].sbls_control.definite_linear_solver = galahad_linear_solver("sytr")

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

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

    # Set shifted least-distance data

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

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

      eqp_solve_sldqp(T, data, status, n, m, w, x_0, g, f,
                      A_ne, A_val, c, x, y)
    end

    eqp_information(T, data, inform, status)

    if inform[].status == 0
      @printf("%c:%6i cg iterations. Optimal objective value = %5.2f status = %1i\n",
              st, inform[].cg_iter, inform[].obj, inform[].status)
    else
      @printf("%c: EQP_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
    eqp_terminate(T, data, control, inform)
  end

  return 0
end

@testset "EQP" begin
  @test test_eqp(Float32) == 0
  @test test_eqp(Float64) == 0
  @test test_eqp(Float128) == 0
end