GALAHAD BQP package#
purpose#
The bqp package uses a preconditioned, projected-gradient method to
solve a given bound-constrained convex quadratic program.
The aim is to minimize the quadratic objective function
See Section 4 of $GALAHAD/doc/bqp.pdf for a brief description of the method employed and other details.
terminology#
Any required solution \(x\) necessarily satisfies the primal optimality conditions
method#
Projected-gradient methods iterate towards a point that satisfies these conditions by ultimately aiming to satisfy \(H x + g = z\) and \(z = z_l + z_u\), while satifying the remaining optimality conditions at each stage. Appropriate norms of the amounts by which the optimality conditions fail to be satisfied are known as the primal and dual infeasibility, and the violation of complementary slackness, respectively.
The method is iterative. Each iteration proceeds in two stages. Firstly, the so-called generalized Cauchy point for the quadratic objective is found. (The purpose of this point is to ensure that the algorithm converges and that the set of bounds which are satisfied as equations at the solution is rapidly identified.) Thereafter an improvement to the objective is sought using either a direct-matrix or truncated conjugate-gradient algorithm.
reference#
This is a specialised version of the method presented in
A. R. Conn, N. I. M. Gould and Ph. L. Toint, Global convergence of a class of trust region algorithms for optimization with simple bounds. SIAM Journal on Numerical Analysis 25 (1988) 433-460.
matrix storage#
The symmetric \(n\) by \(n\) matrix \(H\) may 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 bqp package must be called in the following order:
bqp_initialize - provide default control parameters and set up initial data structures
bqp_read_specfile (optional) - override control values by reading replacement values from a file
set up problem data structures and fixed values by caling one of
bqp_import - in the case that \(H\) is explicitly available
bqp_import_without_h - in the case that only the effect of applying \(H\) to a vector is possible
bqp_reset_control (optional) - possibly change control parameters if a sequence of problems are being solved
solve the problem by calling one of
bqp_solve_given_h - solve the problem using values of \(H\)
bqp_solve_reverse_h_prod - solve the problem by returning to the caller for products of \(H\) with specified vectors
bqp_information (optional) - recover information about the solution and solution process
bqp_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 bqp_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 bqp_control_type) |
status |
is a scalar variable of type Int32 that gives the exit status from the package. Possible values are (currently):
|
function bqp_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/bqp/BQP.template. See also Table 2.1 in the Fortran documentation provided in $GALAHAD/doc/bqp.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 bqp_control_type) |
specfile |
is a one-dimensional array of type Vararg{Cchar} that must give the name of the specification file |
function bqp_import(T, control, data, status, n, H_type, ne, H_row, H_col, H_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 bqp_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:
|
n |
is a scalar variable of type Int32 that holds the number of variables. |
H_type |
is a one-dimensional array of type Vararg{Cchar} that specifies the symmetric storage scheme used for the Hessian. It should be one of ‘coordinate’, ‘sparse_by_rows’, ‘dense’, ‘diagonal’ or ‘absent’, the latter if access to the Hessian is via matrix-vector products; lower or upper case variants are allowed. |
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 three schemes. |
H_row |
is a one-dimensional array of size 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 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 or diagonal 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 |
function bqp_import_without_h(T, control, data, status, n)
Import problem data into internal storage prior to solution.
Parameters:
control |
is a structure whose members provide control parameters for the remaining procedures (see bqp_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:
|
n |
is a scalar variable of type Int32 that holds the number of variables. |
function bqp_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 bqp_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:
|
function bqp_solve_given_h(T, data, status, n, h_ne, H_val, g, f, x_l, x_u, x, z, x_stat)
Solve the bound-constrained 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. On initial entry, status must be set to 1. Possible exit values are:
|
n |
is a scalar variable of type Int32 that holds the number of variables |
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 |
f |
is a scalar of type T that holds the constant term \(f\) of the objective function. |
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_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 |
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 |
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 |
x_stat |
is a one-dimensional array of size n and type Int32 that gives the optimal status of the problem variables. If x_stat(j) is negative, the variable \(x_j\) most likely lies on its lower bound, if it is positive, it lies on its upper bound, and if it is zero, it lies between its bounds. |
function bqp_solve_reverse_h_prod(T, data, status, n, g, f, x_l, x_u, x, z, x_stat, v, prod, nz_v, nz_v_start, nz_v_end, nz_prod, nz_prod_end)
Solve the bound-constrained quadratic program when the products of the Hessian \(H\) with specified vectors may be computed by the calling 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:
|
n |
is a scalar variable of type Int32 that holds the number of variables |
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 |
f |
is a scalar of type T that holds the constant term \(f\) of the objective function. |
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_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 |
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 |
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 |
x_stat |
is a one-dimensional array of size n and type Int32 that gives the optimal status of the problem variables. If x_stat(j) is negative, the variable \(x_j\) most likely lies on its lower bound, if it is positive, it lies on its upper bound, and if it is zero, it lies between its bounds. |
v |
is a one-dimensional array of size n and type T that is used for reverse communication (see status=2-4 above for details) |
prod |
is a one-dimensional array of size n and type T that is used for reverse communication (see status=2-4 above for details) |
nz_v |
is a one-dimensional array of size n and type Int32 that is used for reverse communication (see status=3-4 above for details) |
nz_v_start |
is a scalar of type Int32 that is used for reverse communication (see status=3-4 above for details) |
nz_v_end |
is a scalar of type Int32 that is used for reverse communication (see status=3-4 above for details) |
nz_prod |
is a one-dimensional array of size n and type Int32 that is used for reverse communication (see status=4 above for details) |
nz_prod_end |
is a scalar of type Int32 that is used for reverse communication (see status=4 above for details) |
function bqp_information(T, data, inform, status)
Provides output information
Parameters:
data |
holds private internal data |
inform |
is a structure containing output information (see bqp_inform_type) |
status |
is a scalar variable of type Int32 that gives the exit status from the package. Possible values are (currently):
|
function bqp_terminate(T, data, control, inform)
Deallocate all internal private storage
Parameters:
data |
holds private internal data |
control |
is a structure containing control information (see bqp_control_type) |
inform |
is a structure containing output information (see bqp_inform_type) |
available structures#
bqp_control_type structure#
struct bqp_control_type{T} f_indexing::Bool error::Int32 out::Int32 print_level::Int32 start_print::Int32 stop_print::Int32 print_gap::Int32 maxit::Int32 cold_start::Int32 ratio_cg_vs_sd::Int32 change_max::Int32 cg_maxit::Int32 sif_file_device::Int32 infinity::T stop_p::T stop_d::T stop_c::T identical_bounds_tol::T stop_cg_relative::T stop_cg_absolute::T zero_curvature::T cpu_time_limit::T exact_arcsearch::Bool space_critical::Bool deallocate_error_fatal::Bool generate_sif_file::Bool sif_file_name::NTuple{31,Cchar} prefix::NTuple{31,Cchar} sbls_control::sbls_control_type{T}
detailed documentation#
control derived type as a Julia structure
components#
Bool f_indexing
use C or Fortran sparse matrix indexing
Int32 error
unit number for error and warning diagnostics
Int32 out
general output unit number
Int32 print_level
the level of output required
Int32 start_print
on which iteration to start printing
Int32 stop_print
on which iteration to stop printing
Int32 print_gap
how many iterations between printing
Int32 maxit
how many iterations to perform (-ve reverts to HUGE(1)-1)
Int32 cold_start
cold_start should be set to 0 if a warm start is required (with variable assigned according to B_stat, see below), and to any other value if the values given in prob.X suffice
Int32 ratio_cg_vs_sd
the ratio of how many iterations use CG rather steepest descent
Int32 change_max
the maximum number of per-iteration changes in the working set permitted when allowing CG rather than steepest descent
Int32 cg_maxit
how many CG iterations to perform per BQP iteration (-ve reverts to n+1)
Int32 sif_file_device
the unit number to write generated SIF file describing the current problem
T infinity
any bound larger than infinity in modulus will be regarded as infinite
T stop_p
the required accuracy for the primal infeasibility
T stop_d
the required accuracy for the dual infeasibility
T stop_c
the required accuracy for the complementary slackness
T identical_bounds_tol
any pair of constraint bounds (x_l,x_u) that are closer than i dentical_bounds_tol will be reset to the average of their values
T stop_cg_relative
the CG iteration will be stopped as soon as the current norm of the preconditioned gradient is smaller than max( stop_cg_relative * initial preconditioned gradient, stop_cg_absolute)
T stop_cg_absolute
see stop_cg_relative
T zero_curvature
threshold below which curvature is regarded as zero
T cpu_time_limit
the maximum CPU time allowed (-ve = no limit)
Bool exact_arcsearch
exact_arcsearch is true if an exact arcsearch is required, and false if approximation suffices
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 times
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, a SIF file describing the current problem will be generated
NTuple{31,Cchar} sif_file_name
name (max 30 characters) of generated SIF file containing input problem
NTuple{31,Cchar} prefix
all output lines will be prefixed by a string (max 30 characters) prefix(2:LEN(TRIM(.prefix))-1) where prefix contains the required string enclosed in quotes, e.g. “string” or ‘string’
struct sbls_control_type sbls_control
control parameters for SBLS
bqp_time_type structure#
struct bqp_time_type total::Float32 analyse::Float32 factorize::Float32 solve::Float32
detailed documentation#
time derived type as a Julia structure
components#
Float32 total
total time
Float32 analyse
time for the analysis phase
Float32 factorize
time for the factorization phase
Float32 solve
time for the linear solution phase
bqp_inform_type structure#
struct bqp_inform_type{T} status::Int32 alloc_status::Int32 factorization_status::Int32 iter::Int32 cg_iter::Int32 obj::T norm_pg::T bad_alloc::NTuple{81,Cchar} time::bqp_time_type sbls_inform::sbls_inform_type{T}
detailed documentation#
inform derived type as a Julia structure
components#
Int32 status
reported return status:
0
success
-1
allocation error
-2
deallocation error
-3
matrix data faulty (.n < 1, .ne < 0)
-20
alegedly +ve definite matrix is not
Int32 alloc_status
Fortran STAT value after allocate failure.
Int32 factorization_status
status return from factorization
Int32 iter
number of iterations required
Int32 cg_iter
number of CG iterations required
T obj
current value of the objective function
T norm_pg
current value of the projected gradient
NTuple{81,Cchar} bad_alloc
name of array which provoked an allocate failure
struct bqp_time_type time
times for various stages
struct sbls_inform_type sbls_inform
inform values from SBLS
example calls#
This is an example of how to use the package to solve a bound-constrained QP; the code is available in $GALAHAD/src/bqp/C/bqpt.c . A variety of supported Hessian and constraint matrix storage formats are shown.
/* bqpt.c */
/* Full test for the BQP C interface using C sparse matrix indexing */
#include <stdio.h>
#include <math.h>
#include "galahad_precision.h"
#include "galahad_cfunctions.h"
#include "galahad_bqp.h"
#ifdef REAL_128
#include <quadmath.h>
#endif
int main(void) {
// Derived types
void *data;
struct bqp_control_type control;
struct bqp_inform_type inform;
// Set problem data
ipc_ n = 10; // dimension
ipc_ H_ne = 2 * n - 1; // Hesssian elements, NB lower triangle
ipc_ H_dense_ne = n * ( n + 1 ) / 2; // dense Hessian elements
ipc_ H_row[H_ne]; // row indices,
ipc_ H_col[H_ne]; // column indices
ipc_ H_ptr[n+1]; // row pointers
rpc_ H_val[H_ne]; // values
rpc_ H_dense[H_dense_ne]; // dense values
rpc_ H_diag[n]; // diagonal values
rpc_ g[n]; // linear term in the objective
rpc_ f = 1.0; // constant term in the objective
rpc_ x_l[n]; // variable lower bound
rpc_ x_u[n]; // variable upper bound
rpc_ x[n]; // variables
rpc_ z[n]; // dual variables
// Set output storage
ipc_ x_stat[n]; // variable status
char st = ' ';
ipc_ i, l, status;
g[0] = 2.0;
for( ipc_ i = 1; i < n; i++) g[i] = 0.0;
x_l[0] = -1.0;
for( ipc_ i = 1; i < n; i++) x_l[i] = - INFINITY;
x_u[0] = 1.0;
x_u[1] = INFINITY;
for( ipc_ i = 2; i < n; i++) x_u[i] = 2.0;
// H = tridiag(2,1), H_dense = diag(2)
l = 0 ;
H_ptr[0] = l;
H_row[l] = 0; H_col[l] = 0; H_val[l] = 2.0;
for( ipc_ i = 1; i < n; i++)
{
l = l + 1;
H_ptr[i] = l;
H_row[l] = i; H_col[l] = i - 1; H_val[l] = 1.0;
l = l + 1;
H_row[l] = i; H_col[l] = i; H_val[l] = 2.0;
}
H_ptr[n] = l + 1;
l = - 1 ;
for( ipc_ i = 0; i < n; i++)
{
H_diag[i] = 2.0;
for( ipc_ j = 0; j <= i; j++)
{
l = l + 1;
if ( j < i - 1 ) {
H_dense[l] = 0.0;
}
else if ( j == i - 1 ) {
H_dense[l] = 1.0;
}
else {
H_dense[l] = 2.0;
}
}
}
printf(" C sparse matrix indexing\n\n");
printf(" basic tests of bqp storage formats\n\n");
for( ipc_ d=1; d <= 4; d++){
// Initialize BQP
bqp_initialize( &data, &control, &status );
// Set user-defined control options
control.f_indexing = false; // C sparse matrix indexing
// Start from 0
for( ipc_ i = 0; i < n; i++) x[i] = 0.0;
for( ipc_ i = 0; i < n; i++) z[i] = 0.0;
switch(d){
case 1: // sparse co-ordinate storage
st = 'C';
bqp_import( &control, &data, &status, n,
"coordinate", H_ne, H_row, H_col, NULL );
bqp_solve_given_h( &data, &status, n, H_ne, H_val, g, f,
x_l, x_u, x, z, x_stat );
break;
printf(" case %1" i_ipc_ " break\n",d);
case 2: // sparse by rows
st = 'R';
bqp_import( &control, &data, &status, n,
"sparse_by_rows", H_ne, NULL, H_col, H_ptr );
bqp_solve_given_h( &data, &status, n, H_ne, H_val, g, f,
x_l, x_u, x, z, x_stat );
break;
case 3: // dense
st = 'D';
bqp_import( &control, &data, &status, n,
"dense", H_dense_ne, NULL, NULL, NULL );
bqp_solve_given_h( &data, &status, n, H_dense_ne, H_dense,
g, f, x_l, x_u, x, z, x_stat );
break;
case 4: // diagonal
st = 'L';
bqp_import( &control, &data, &status, n,
"diagonal", H_ne, NULL, NULL, NULL );
bqp_solve_given_h( &data, &status, n, n, H_diag, g, f,
x_l, x_u, x, z, x_stat );
break;
}
bqp_information( &data, &inform, &status );
if(inform.status == 0){
#ifdef REAL_128
// interim replacement for quad output: $GALAHAD/include/galahad_pquad_f.h
#include "galahad_pquad_f.h"
#else
printf("%c:%6" i_ipc_ " iterations. Optimal objective "
"value = %.2f status = %1" i_ipc_ "\n",
st, inform.iter, inform.obj, inform.status);
#endif
}else{
printf("%c: BQP_solve exit status = %1" i_ipc_ "\n",
st, inform.status);
}
//printf("x: ");
//for( ipc_ i = 0; i < n; i++) printf("%f ", x[i]);
//printf("\n");
//printf("gradient: ");
//for( ipc_ i = 0; i < n; i++) printf("%f ", g[i]);
//printf("\n");
// Delete internal workspace
bqp_terminate( &data, &control, &inform );
}
printf("\n tests reverse-communication options\n\n");
// reverse-communication input/output
ipc_ nz_v_start, nz_v_end, nz_prod_end;
ipc_ nz_v[n], nz_prod[n], mask[n];
rpc_ v[n], prod[n];
nz_prod_end = 0;
// Initialize BQP
bqp_initialize( &data, &control, &status );
// control.print_level = 1;
// Set user-defined control options
control.f_indexing = false; // C sparse matrix indexing
// Start from 0
for( ipc_ i = 0; i < n; i++) x[i] = 0.0;
for( ipc_ i = 0; i < n; i++) z[i] = 0.0;
st = 'I';
for( ipc_ i = 0; i < n; i++) mask[i] = 0;
bqp_import_without_h( &control, &data, &status, n ) ;
while(true){ // reverse-communication loop
bqp_solve_reverse_h_prod( &data, &status, n, g, f, x_l, x_u,
x, z, x_stat, v, prod,
nz_v, &nz_v_start, &nz_v_end,
nz_prod, nz_prod_end );
if(status == 0){ // successful termination
break;
}else if(status < 0){ // error exit
break;
}else if(status == 2){ // evaluate Hv
prod[0] = 2.0 * v[0] + v[1];
for( ipc_ i = 1; i < n-1; i++) prod[i] = 2.0 * v[i] + v[i-1] + v[i+1];
prod[n-1] = 2.0 * v[n-1] + v[n-2];
}else if(status == 3){ // evaluate Hv for sparse v
for( ipc_ i = 0; i < n; i++) prod[i] = 0.0;
for( ipc_ l = nz_v_start - 1; l < nz_v_end; l++){
i = nz_v[l];
if (i > 0) prod[i-1] = prod[i-1] + v[i];
prod[i] = prod[i] + 2.0 * v[i];
if (i < n-1) prod[i+1] = prod[i+1] + v[i];
}
}else if(status == 4){ // evaluate sarse Hv for sparse v
nz_prod_end = 0;
for( ipc_ l = nz_v_start - 1; l < nz_v_end; l++){
i = nz_v[l];
if (i > 0){
if (mask[i-1] == 0){
mask[i-1] = 1;
nz_prod[nz_prod_end] = i - 1;
nz_prod_end = nz_prod_end + 1;
prod[i-1] = v[i];
}else{
prod[i-1] = prod[i-1] + v[i];
}
}
if (mask[i] == 0){
mask[i] = 1;
nz_prod[nz_prod_end] = i;
nz_prod_end = nz_prod_end + 1;
prod[i] = 2.0 * v[i];
}else{
prod[i] = prod[i] + 2.0 * v[i];
}
if (i < n-1){
if (mask[i+1] == 0){
mask[i+1] = 1;
nz_prod[nz_prod_end] = i + 1;
nz_prod_end = nz_prod_end + 1;
prod[i+1] = prod[i+1] + v[i];
}else{
prod[i+1] = prod[i+1] + v[i];
}
}
}
for( ipc_ l = 0; l < nz_prod_end; l++) mask[nz_prod[l]] = 0;
}else{
printf(" the value %1" i_ipc_ " of status should not occur\n",
status);
break;
}
}
// Record solution information
bqp_information( &data, &inform, &status );
// Print solution details
if(inform.status == 0){
#ifdef REAL_128
// interim replacement for quad output: $GALAHAD/include/galahad_pquad_f.h
#include "galahad_pquad_f.h"
#else
printf("%c:%6" i_ipc_ " iterations. Optimal objective "
"value = %.2f status = %1" i_ipc_ "\n",
st, inform.iter, inform.obj, inform.status);
#endif
}else{
printf("%c: BQP_solve exit status = %1" i_ipc_ "\n", st, inform.status);
}
//printf("x: ");
//for( ipc_ i = 0; i < n; i++) printf("%f ", x[i]);
//printf("\n");
//printf("gradient: ");
//for( ipc_ i = 0; i < n; i++) printf("%f ", g[i]);
//printf("\n");
// Delete internal workspace
bqp_terminate( &data, &control, &inform );
}