GALAHAD WCP package#

purpose#

The wcp package uses a primal-dual interior-point method to find a well-centered point within a polyhedral set. The aim is to find a point that lies interior to the boundary of the polyhedron definied by 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, 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 $x$, since full advantage is taken of “any zero coefficients in the matrix $A$. The package identifies infeasible problems, and problems for which there is no strict interior.

See Section 4 of $GALAHAD/doc/wcp.pdf for a brief description of the method employed and other details.

terminolgy#

More specifically, if possible, the package finds a solution to the system of primal optimality equations

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

the dual optimality equations

\[g = A^{T} y + z,\;\; y = y^{l} + y^{u} \;\;\mbox{and}\;\; z = z^{l} + z^{u},\;\;\mbox{(2)}\]

and perturbed complementary slackness equations

\[( c_i^{} - c^l_i ) y^{l}_i = (\mu_c^{l})_i^{} \;\;\mbox{and}\;\; ( c_i^{} - c_i^u ) y^{u}_i = (\mu_c^{u})_i^{}, \;\;\; i = 1, \ldots , m,\;\;\mbox{(3)}\]

and

\[((x^{}_j - x^l_j ) z_j^{l} = (\mu_x^{l})_j^{} \;\;\mbox{and}\;\; ( x^{}_j - x^u_j ) z_j^{u} = (\mu_x^{u})_j^{}, \;\;\; j = 1, \ldots , n,\;\;\mbox{(4)}\]

for which

\[c^{l} \leq c \leq c^{u}, \;\; x^{l} \leq x \leq x^{u}, \;\; y^{l} \geq 0, \;\; y^{u} \leq 0, \;\; z^{l} \geq 0 \;\; \mbox{and} \;\; z^{u} \leq 0. \;\;\mbox{(5)}\]

Here $\mu_c^{l}$, $\mu_c^{u}$, $\mu_x^{l}$ and $\mu_x^{u}$ are vectors of strictly positive targets, $g$ is another given target vector (which is often zero), $(y^{l}, y^{u})$ and $(z^{l}, z^{u})$ are Lagrange multipliers for the linear constraints and dual variables for the simple bounds respectively, and vector inequalities hold component-wise; $c$ gives the constraint value $A x$. Since the perturbed complementarity equations normally imply that

\[c^{l} < c < c^{u}, \;\; x^{l} < x < x^{u}, \;\; y^{l} > 0, \;\; y^{u} < 0, \;\; z^{l} > 0 \;\; \mbox{and} \;\; z^{u} < 0, \;\;\mbox{(6)}\]

such a primal-dual point $(x, c, y^{l}, y^{u}, z^{l}, z^{l})$ may be used, for example, as a feasible starting point for primal-dual interior-point methods for solving the linear programming problem of minimizing $g^T x$ within the given polyhedral set. method

method#

The algorithm is iterative, and at each major iteration attempts to find a solution to the perturbed system (1), (2),

\[( c_i^{} - c^l_i + (\theta_c^l)_i^{} ) ( y^{l}_i + (\theta_y^l)_i^{} ) = (\mu_c^{l})_i^{} \;\;\mbox{and}\;\; ( c_i^{} - c_i^u - (\theta_c^u)_i^{} ) y^{u}_i - (\theta_y^u)_i^{} ) = (\mu_c^{u})_i^{}, \;\;\; i = 1, \ldots , m,\;\;\mbox{(7)}\]

\[ ( x_j^{} - x^l_j + (\theta_x^l)_j^{} ) ( z^{l}_j + (\theta_z^l)_j^{} ) = (\mu_x^{l})_j^{} \;\;\mbox{and}\;\; ( x_j^{} - x_j^u - (\theta_x^u)_j^{} ) ( z^{u}_j - (\theta_z^u)_j^{} ) = (\mu_x^{u})_j^{}, \;\;\; j = 1, \ldots , n,\;\;\mbox{(8)}\]

and

\[c^{l} - \theta_c^l < c < c^{u} + \theta_c^u, \;\; x^{l} - \theta_x^l < x < x^{u} + \theta_x^u, \;\; y^{l} > - \theta_y^l , \;\; y^{u} < \theta_y^u , \;\; z^{l} > - \theta_z^l \;\; \mbox{and} \;\; z^{u} < \theta_z^u ,\;\;\mbox{(9)}\]

where the vectors of perturbations $\theta^l_c$, $\theta^u_c$, $\theta^l_x$, $\theta^u_x$, $\theta^l_x$, $\theta^u_x$, $\theta^l_y$, $\theta^u_y$, $\theta^l_z$ and $\theta^u_z$, are non-negative. Rather than solve (1), (2) and (7)–(9) exactly, we instead seek a feasible point for the easier relaxation (1), (2) and

\[\begin{split}\begin{array}{cccccl} \gamma (\mu_c^{l})_i^{} & \leq & ( c_i^{} - c^l_i + (\theta_c^l)_i^{} ) ( y^{l}_i + (\theta_y^l)_i^{} ) & \leq & (\mu_c^{l})_i^{} / \gamma & \;\;\mbox{and}\;\; \\ \gamma (\mu_c^{u})_i^{} & \leq & ( c_i^{} - c_i^u - (\theta_c^u)_i^{} ) ( y^{u}_i - (\theta_y^u)_i^{} ) & \leq & (\mu_c^{u})_i^{}, /\gamma & \;\;\; i = 1, \ldots , m, \;\;\mbox{and}\;\; \\ \gamma (\mu_x^{l})_j^{} & \leq & ( x_j^{} - x^l_j + (\theta_x^l)_j^{} ) ( z^{l}_j + (\theta_z^l)_j^{} ) & \leq & (\mu_x^{l})_j^{} /\gamma & \;\;\mbox{and}\;\; \\ \gamma (\mu_x^{u})_j^{} & \leq & ( x_j^{} - x_j^u - (\theta_x^u)_j^{} ) ( z^{u}_j - (\theta_z^u)_j^{} ) & \leq & (\mu_x^{u})_j^{} /\gamma , & \;\;\; j = 1, \ldots , n, \end{array} \;\;\mbox{(10)}\end{split}\]

for some $\gamma \in (0,1]$ which is allowed to be smaller than one if there is a nonzero perturbation.

Given any solution to (1)–(2) and (10) satisfying (9), the perturbations are reduced (sometimes to zero) so as to ensure that the current solution is feasible for the next perturbed problem. Specifically, the perturbation $(\theta^l_c)_i^{}$ for the constraint $c_i^{} \geq c^l_i$ is set to zero if $c_i$ is larger than some given parameter $\epsilon > 0$. If not, but $c_i$ is strictly positive, the perturbation will be reduced by a multiplier $\rho \in (0,1)$. Otherwise, the new perturbation will be set to $\xi (\theta^l_c)_i^{} + ( 1 - \xi ) ( c_i^l - c_i^{} )$ for some factor $\xi \in (0,1)$. Identical rules are used to reduce the remaining primal and dual perturbations. The targets $\mu_c^{l}$, $\mu_c^{u}$, $\mu_x^{l}$ and $\mu_x^{u}$ will also be increased by the factor $\beta \geq 1$ for those (primal and/or dual) variables with strictly positive perturbations so as to try to accelerate the convergence.

Ultimately the intention is to drive all the perturbations to zero. It can be shown that if the original problem (1)–(4) and (6) has a solution, the perturbations will be zero after a finite number of major iterations. Equally, if there is no interior solution (6), the sets of (primal and dual) variables that are necessarily at (one of) their bounds for all feasible points—we refer to these as implicit equalities—will be identified, as will the possibility that there is no point (interior or otherwise) in the primal and/or dual feasible regions.

Each major iteration requires the solution $v = (x,c,z^l,z^u,y^l,y^u)$ of the nonlinear system (1), (2) and (7)–(9) for fixed perturbations, using a minor iteration. The minor iteration uses a stabilized (predictor-corrector) Newton method, in which the arc $v(\alpha) = v + \alpha \dot{v} + \alpha^2 \ddot{v}$, $\alpha \in [0,1]$, involving the standard Newton step $\dot{v}$ for the equations (1), (2), (7) and (8), optionally augmented by a corrector $\ddot{v}$ to account for the nonlinearity in (7) and (8), is truncated so as to ensure that

\[( c_i^{}(\alpha) - c^l_i + (\theta_c^l)_i^{} ) ( y^{l}_i(\alpha) + (\theta_y^l)_i^{} ) \geq \tau (\mu_c^{l})_i^{}, \;\;\; i = 1, \ldots , m,\]

\[( c_i^{}(\alpha) - c_i^u - (\theta_c^u)_i^{} ) ( y^{u}_i(\alpha) - (\theta_y^u)_i^{} ) \geq \tau (\mu_c^{u})_i^{}, \;\;\; i = 1, \ldots , m,\]

\[( x_j^{}(\alpha) - x^l_j + (\theta_x^l)_j^{} ) ( z^{l}_j(\alpha) + (\theta_z^l)_j^{} ) \geq \tau (\mu_x^{l})_j^{}, \;\;\; j = 1, \ldots , n,\]

and

\[( x_j^{}(\alpha) - x_j^u - (\theta_x^u)_j^{} ) ( z^{u}_j(\alpha) - (\theta_z^u)_j^{} ) \geq \tau (\mu_x^{u})_j^{}, \;\;\; j = 1, \ldots , n,\]

for some $\tau \in (0,1)$, always holds, and also so that the norm of the residuals to (1), (2), (7) and (8) is reduced as much as possible. The Newton and corrector systems are solved using a factorization of the Jacobian of its defining functions (the so-called “augmented system” approach) or of a reduced system in which some of the trivial equations are eliminated (the “Schur-complement” approach). The factors are obtained using the package SBLS.

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. In addition, an attempt to identify and remove linearly dependent equality constraints may be made by factorizing

\[\begin{split}\left(\begin{array}{cc}\alpha I & A^T_{\cal E} \\ A^{}_{\cal E} & 0 \end{array}\right),\end{split}\]

where $A_{\cal E}$ denotes the gradients of the equality constraints and $\alpha > 0$ is a given scaling factor, using SBLS, and examining small pivot blocks.

reference#

The basic algorithm, its convergence analysis and results of numerical experiments are given in

C. Cartis and N. I. M. Gould, Finding a point n the relative interior of a polyhedron. Technical Report TR-2006-016, Rutherford Appleton Laboratory (2006).

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 wcp package must be called in the following order:

wcp_initialize - provide default control parameters and set up initial data structures
wcp_read_specfile (optional) - override control values by reading replacement values from a file
wcp_import - set up problem data structures and fixed values
wcp_reset_control (optional) - possibly change control parameters if a sequence of problems are being solved
wcp_find_wcp - find a well-centered point
wcp_information (optional) - recover information about the solution and solution process
wcp_terminate - deallocate data structures

See the examples section for illustrations of use.

parametric real type T and integer type INT#

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). The symbol INT refers to a parametric integer type that may be Int32 (32-bit integer) or Int64 (64-bit integer).

callable functions#

    function wcp_initialize(T, INT, 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 wcp_control_type)

status

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

0
The initialization was successful.

    function wcp_read_specfile(T, INT, 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/wcp/WCP.template. See also Table 2.1 in the Fortran documentation provided in $GALAHAD/doc/wcp.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 wcp_control_type)
specfile	is a one-dimensional array of type Vararg{Cchar} that must give the name of the specification file

    function wcp_import(T, INT, 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 wcp_control_type)
data	holds private internal data
status	is a scalar variable of type INT 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 INT that holds the number of variables.
m	is a scalar variable of type INT 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 INT 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 INT 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 INT 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 INT 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 wcp_reset_control(T, INT, 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 wcp_control_type)

data

holds private internal data

status

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

0
The import was successful.

     function wcp_find_wcp(T, INT, data, status, n, m, g, a_ne, A_val, c_l, c_u,
                           x_l, x_u, x, c, y_l, y_u, z_l, z_u, x_stat, c_stat)

Find a well-centered point in the feasible region

Parameters:

data	holds private internal data
status	is a scalar variable of type INT 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. -4 The constraint bounds are inconsistent. -5 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 INT that holds the number of variables
m	is a scalar variable of type INT that holds the number of general linear constraints.
g	is a one-dimensional array of size n and type T that holds the target vector $g$. The j-th component of `g`, j = 1, … , n, contains $g_j$.
a_ne	is a scalar variable of type INT 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_l	is a one-dimensional array of size n and type T that holds the values $y^l$ of the Lagrange multipliers for the lower bounds on the general linear constraints. The j-th component of `y_l`, i = 1, … , m, contains $y^l_i$.
y_u	is a one-dimensional array of size n and type T that holds the values $y^u$ of the Lagrange multipliers for the upper bounds on the general linear constraints. The j-th component of `y_u`, i = 1, … , m, contains $y^u_i$.
z_l	is a one-dimensional array of size n and type T that holds the values $z^l$ of the dual variables for the lower bounds on the variables. The j-th component of `z_l`, j = 1, … , n, contains $z^l_j$.
z_u	is a one-dimensional array of size n and type T that holds the values $z^u$ of the dual variables for the upper bounds on the variables. The j-th component of `z_u`, j = 1, … , n, contains $z^u_j$.
x_stat	is a one-dimensional array of size n and type INT 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 INT 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 wcp_information(T, INT, data, inform, status)

Provides output information.

Parameters:

data

holds private internal data

inform

is a structure containing output information (see wcp_inform_type)

status

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

0
The values were recorded successfully

    function wcp_terminate_s(data, control, inform)

Deallocate all internal private storage.

Parameters:

data	holds private internal data
control	is a structure containing control information (see wcp_control_type)
inform	is a structure containing output information (see wcp_inform_type)

available structures#

wcp_control_type structure#

    struct wcp_control_type{T,INT}
      f_indexing::Bool
      error::INT
      out::INT
      print_level::INT
      start_print::INT
      stop_print::INT
      maxit::INT
      initial_point::INT
      factor::INT
      max_col::INT
      indmin::INT
      valmin::INT
      itref_max::INT
      infeas_max::INT
      perturbation_strategy::INT
      restore_problem::INT
      infinity::T
      stop_p::T
      stop_d::T
      stop_c::T
      prfeas::T
      dufeas::T
      mu_target::T
      mu_accept_fraction::T
      mu_increase_factor::T
      required_infeas_reduction::T
      implicit_tol::T
      pivot_tol::T
      pivot_tol_for_dependencies::T
      zero_pivot::T
      perturb_start::T
      alpha_scale::T
      identical_bounds_tol::T
      reduce_perturb_factor::T
      reduce_perturb_multiplier::T
      insufficiently_feasible::T
      perturbation_small::T
      cpu_time_limit::T
      clock_time_limit::T
      remove_dependencies::Bool
      treat_zero_bounds_as_general::Bool
      just_feasible::Bool
      balance_initial_complementarity::Bool
      use_corrector::Bool
      space_critical::Bool
      deallocate_error_fatal::Bool
      record_x_status::Bool
      record_c_status::Bool
      prefix::NTuple{31,Cchar}
      fdc_control::fdc_control_type{T,INT}
      sbls_control::sbls_control_type{T,INT}

detailed documentation#

control derived type as a Julia structure

components#

Bool f_indexing

use C or Fortran sparse matrix indexing

INT error

error and warning diagnostics occur on stream error

INT out

general output occurs on stream out

INT print_level

the level of output required is specified by print_level

INT start_print

any printing will start on this iteration

INT stop_print

any printing will stop on this iteration

INT maxit

at most maxit inner iterations are allowed

INT initial_point

how to choose the initial point. Possible values are

0 the values input in X, shifted to be at least prfeas from their nearest bound, will be used
1 the nearest point to the “bound average” 0.5(X_l+X_u) that satisfies the linear constraints will be used

INT factor

the factorization to be used. Possible values are

0 automatic
1 Schur-complement factorization
2 augmented-system factorization

INT max_col

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

INT indmin

an initial guess as to the integer workspace required by SBLS

INT valmin

an initial guess as to the real workspace required by SBLS

INT itref_max

the maximum number of iterative refinements allowed

INT infeas_max

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

INT perturbation_strategy

the strategy used to reduce relaxed constraint bounds. Possible values are

0 do not perturb the constraints
1 reduce all perturbations by the same amount with linear reduction
2 reduce each perturbation as much as possible with linear reduction
3 reduce all perturbations by the same amount with superlinear reduction
4 reduce each perturbation as much as possible with superlinear reduction

INT 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

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 bound

T dufeas

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

T mu_target

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

T mu_accept_fraction

the complemtary slackness x_i.z_i will be judged to lie within an acceptable margin around its target value mu as soon as mu_accept_fraction * mu <= x_i.z_i <= ( 1 / mu_accept_fraction ) * mu; the perturbations will be reduced as soon as all of the complemtary slacknesses x_i.z_i lie within acceptable bounds. mu_accept_fraction will be reset to ensure that it lies in the interval (0,1]

T mu_increase_factor

the target value of the barrier parameter will be increased by mu_increase_factor for infeasible constraints every time the perturbations are adjusted

T required_infeas_reduction

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

T implicit_tol

any primal or dual variable that is less feasible than implicit_tol will be regarded as defining an implicit constraint

T pivot_tol

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

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 (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 perturb_start

the constraint bounds will initially be relaxed by .perturb_start; this perturbation will subsequently be reduced to zero. If perturb_start < 0, the amount by which the bounds are relaxed will be computed automatically

T alpha_scale

the test for rank defficiency will be to factorize ( alpha_scale I A^T ) ( A 0 )

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 reduce_perturb_factor

the constraint perturbation will be reduced as follows:

- if the variable lies outside a bound, the corresponding perturbation will be reduced to reduce_perturb_factor * current pertubation
- ( 1 - reduce_perturb_factor ) * violation
- otherwise, if the variable lies within insufficiently_feasible of its bound the pertubation will be reduced to reduce_perturb_multiplier * current pertubation
- otherwise if will be set to zero

T reduce_perturb_multiplier

see reduce_perturb_factor

T insufficiently_feasible

see reduce_perturb_factor

T perturbation_small

if the maximum constraint pertubation is smaller than perturbation_small and the violation is smaller than implicit_tol, the method will deduce that there is a feasible point but no interior

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 balance_initial_complementarity

if .balance_initial_complementarity is .true. the initial complemetarity will be balanced

Bool use_corrector

if .use_corrector, a corrector step will be used

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 record_x_status

if .record_x_status is true, the array inform.X_status will be allocated and the status of the bound constraints will be reported on exit.

Bool record_c_status

if .record_c_status is true, the array inform.C_status will be allocated and the status of the general constraints will be reported on exit.

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’

fdc_control_type{T,INT} fdc_control_type fdc_control

control parameters for FDC

sbls_control_type{T,INT} sbls_control_type sbls_control

control parameters for SBLS

wcp_time_type structure#

    struct wcp_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

wcp_inform_type structure#

    struct wcp_inform_type{T,INT}
      status::INT
      alloc_status::INT
      bad_alloc::NTuple{81,Cchar}
      iter::INT
      factorization_status::INT
      factorization_integer::Int64
      factorization_real::Int64
      nfacts::INT
      c_implicit::INT
      x_implicit::INT
      y_implicit::INT
      z_implicit::INT
      obj::T
      mu_final_target_max::T
      non_negligible_pivot::T
      feasible::Bool
      time::wcp_time_type{T}
      fdc_inform::fdc_inform_type{T,INT}
      sbls_inform::sbls_inform_type{T,INT}

detailed documentation#

inform derived type as a Julia structure

components#

INT status

return status. See WCP_solve for details

INT 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

INT iter

the total number of iterations required

INT 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

INT nfacts

the total number of factorizations performed

INT c_implicit

the number of general constraints that lie on (one) of their bounds for feasible solutions

INT x_implicit

the number of variables that lie on (one) of their bounds for all feasible solutions

INT y_implicit

the number of Lagrange multipliers for general constraints that lie on (one) of their bounds for all feasible solutions

INT z_implicit

the number of dual variables that lie on (one) of their bounds for all feasible solutions

T obj

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

T mu_final_target_max

the largest target value on termination

T non_negligible_pivot

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

Bool feasible

is the returned primal-dual “solution” strictly feasible?

wcp_time_type{T} wcp_time_type time

timings (see above)

fdc_inform_type{T,INT} fdc_inform_type fdc_inform

inform parameters for FDC

sbls_inform_type{T,INT} sbls_inform_type sbls_inform

inform parameters for SBLS

Index

example calls#

This is an example of how to use the package to find a well-centred point; the code is available in $GALAHAD/src/wcp/Julia/test_wcp.jl . A variety of supported Hessian and constraint matrix storage formats are shown.

# test_wcp.jl
# Simple code to test the Julia interface to WCP

using GALAHAD
using Test
using Printf
using Accessors
using Quadmath

function test_wcp(::Type{T}, ::Type{INT}) where {T,INT}
  # Derived types
  data = Ref{Ptr{Cvoid}}()
  control = Ref{wcp_control_type{T,INT}}()
  inform = Ref{wcp_inform_type{T,INT}}()

  # Set problem data
  n = INT(3)  # dimension
  m = INT(2)  # number of general constraints
  g = T[0.0, 2.0, 0.0]  # linear term in the objective
  A_ne = INT(4)  # Jacobian elements
  A_row = INT[1, 1, 2, 2]  # row indices
  A_col = INT[1, 2, 2, 3]  # column indices
  A_ptr = INT[1, 3, 5]  # row pointers
  A_val = T[2.0, 1.0, 1.0, 1.0]  # values
  c_l = T[1.0, 2.0]  # constraint lower bound
  c_u = T[2.0, 2.0]  # constraint upper bound
  x_l = T[-1.0, -Inf, -Inf]  # variable lower bound
  x_u = T[1.0, Inf, 2.0]  # variable upper bound

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

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

  for d in 1:3

    # Initialize WCP
    wcp_initialize(T, INT, 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_l = T[0.0, 0.0]
    y_u = T[0.0, 0.0]
    z_l = T[0.0, 0.0, 0.0]
    z_u = T[0.0, 0.0, 0.0]

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

      wcp_find_wcp(T, INT, data, status, n, m, g, A_ne, A_val,
                   c_l, c_u, x_l, x_u, x, c, y_l, y_u, z_l, z_u,
                   x_stat, c_stat)
    end

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

      wcp_find_wcp(T, INT, data, status, n, m, g, A_ne, A_val,
                   c_l, c_u, x_l, x_u, x, c, y_l, y_u, z_l, z_u,
                   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]

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

      wcp_find_wcp(T, INT, data, status, n, m, g, A_dense_ne, A_dense,
                   c_l, c_u, x_l, x_u, x, c, y_l, y_u, z_l, z_u,
                   x_stat, c_stat)
    end

    wcp_information(T, INT, 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: WCP_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
    wcp_terminate(T, INT, data, control, inform)
  end

  return 0
end

for (T, INT, libgalahad) in ((Float32 , Int32, GALAHAD.libgalahad_single      ),
                             (Float32 , Int64, GALAHAD.libgalahad_single_64   ),
                             (Float64 , Int32, GALAHAD.libgalahad_double      ),
                             (Float64 , Int64, GALAHAD.libgalahad_double_64   ),
                             (Float128, Int32, GALAHAD.libgalahad_quadruple   ),
                             (Float128, Int64, GALAHAD.libgalahad_quadruple_64))
  if isfile(libgalahad)
    @testset "WCP -- $T -- $INT" begin
      @test test_wcp(T, INT) == 0
    end
  end
end