GALAHAD BGO package#
purpose#
The bgo package uses a multi-start trust-region method to find an
approximation to the global minimizer of a differentiable objective
function \(f(x)\) of n variables \(x\), subject to simple
bounds \(x^l <= x <= x^u\) on the variables. Here, any of the
components of the vectors of bounds \(x^l\) and \(x^u\)
may be infinite. The method offers the choice of direct and
iterative solution of the key trust-region subproblems, and
is suitable for large problems. First derivatives are required,
and if second derivatives can be calculated, they will be exploited -
if the product of second derivatives with a vector may be found but
not the derivatives themselves, that may also be exploited.
The package offers both random multi-start and local-minimize-and-probe methods to try to locate the global minimizer. There are no theoretical guarantees unless the sampling is huge, and realistically the success of the methods decreases as the dimension and nonconvexity increase.
See Section 4 of $GALAHAD/doc/bgo.pdf for additional details.
method#
A choice of two methods is available.
In the first, local-minimization-and-probe, approach, local minimization
and univariate global minimization are intermixed. Given a current
champion \(x^S_k\), a local minimizer \(x_k\) of \(f(x)\) within the
feasible box \(x^l \leq x \leq x^u\) is found using TRB.
Thereafter \(m\) random directions \(p\) are generated, and univariate
local minimizer of \(f(x_k + \alpha p)\) as a function of the scalar
\(\alpha\) along each \(p\) within the interval \([\alpha^L,\alpha^u]\),
where \(\alpha^L\) and \(\alpha^u\) are the smallest and largest \(\alpha\)
for which \(x^l \leq x_k + \alpha p \leq x^u\),
is performed using UGO. The point \(x_k + \alpha p\)
that gives the smallest value of \(f\) is then selected as the new champion
\(x^S_{k+1}\).
The random directions \(p\) are chosen in one of three ways. The simplest is to select the components as
LHS.
Each components of \(p\) is then selected in its sub-box, either uniformly
or pseudo randomly.
The other, random-multi-start, method provided selects \(m\) starting points
at random, either componentwise pseudo randomly in the feasible box,
or by partitioning each component into \(m\) equal segments, assigning each to
a sub-box using Latin hypercube sampling, and finally choosing the
values either uniformly or pseudo randomly. Local minimizers within the
feasible box are then computed by TRB, and
the best is assigned as the current champion. This process is then
repeated until evaluation limits are achieved.
If \(n=1\), UGO is called directly.
We reiterate that there are no theoretical guarantees unless the sampling is huge, and realistically the success of the methods decreases as the dimension and nonconvexity increase. Thus the methods used should best be viewed as heuristics.
references#
The generic bound-constrained trust-region method is described in detail in
A. R. Conn, N. I. M. Gould and Ph. L. Toint, Trust-region methods. SIAM/MPS Series on Optimization (2000),
the univariate global minimization method employed is an extension of that due to
D. Lera and Ya. D. Sergeyev, ``Acceleration of univariate global optimization algorithms working with Lipschitz functions and Lipschitz first derivatives’’ SIAM J. Optimization 23(1) (2013) 508–529,
while the Latin-hypercube sampling method employed is that of
B. Beachkofski and R. Grandhi, ``Improved Distributed Hypercube Sampling’’, 43rd AIAA structures, structural dynamics, and materials conference, (2002) 2002-1274.
matrix storage#
The symmetric \(n\) by \(n\) matrix \(H = \nabla^2_{xx}f\) 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.
introduction to function calls#
To solve a given problem, functions from the bgo package must be called in the following order:
bgo_initialize - provide default control parameters and set up initial data structures
bgo_read_specfile (optional) - override control values by reading replacement values from a file
bgo_import - set up problem data structures and fixed values
bgo_reset_control (optional) - possibly change control parameters if a sequence of problems are being solved
solve the problem by calling one of
bgo_solve_with_mat - solve using function calls to evaluate function, gradient and Hessian values
bgo_solve_without_mat - solve using function calls to evaluate function and gradient values and Hessian-vector products
bgo_solve_reverse_with_mat - solve returning to the calling program to obtain function, gradient and Hessian values, or
bgo_solve_reverse_without_mat - solve returning to the calling prorgram to obtain function and gradient values and Hessian-vector products
bgo_information (optional) - recover information about the solution and solution process
bgo_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 bgo_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 bgo_control_type) |
status |
is a scalar variable of type Int32 that gives the exit status from the package. Possible values are (currently):
|
function bgo_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/bgo/BGO.template. See also Table 2.1 in the Fortran documentation provided in $GALAHAD/doc/bgo.pdf for a list of how these keywords relate to the components of the control structure. .. rubric:: Parameters:
control |
is a structure containing control information (see bgo_control_type) |
specfile |
is a one-dimensional array of type Vararg{Cchar} that must give the name of the specification file |
function bgo_import(T, control, data, status, n, x_l, x_u, 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 bgo_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. |
x_l |
is a one-dimensional array of size n and type T that holds the values \(x^l\) of the lower bounds on the optimization variables \(x\). The j-th component of |
x_u |
is a one-dimensional array of size n and type T that holds the values \(x^u\) of the upper bounds on the optimization variables \(x\). The j-th component of |
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 bgo_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 bgo_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 bgo_solve_with_mat(T, data, userdata, status, n, x, g, ne, eval_f, eval_g, eval_h, eval_hprod, eval_prec)
Find an approximation to the global minimizer of a given function subject to simple bounds on the variables using a multistart trust-region method.
This call is for the case where \(H = \nabla_{xx}f(x)\) is provided specifically, and all function/derivative information is available by function calls.
Parameters:
data |
holds private internal data |
userdata |
is a structure that allows data to be passed into the function and derivative evaluation programs. |
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 |
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 |
g |
is a one-dimensional array of size n and type T that holds the gradient \(g = \nabla_xf(x)\) of the objective function. The j-th component of |
ne |
is a scalar variable of type Int32 that holds the number of entries in the lower triangular part of the Hessian matrix \(H\). |
eval_f |
is a user-supplied function that must have the following signature: function eval_f(n, x, f, userdata) The value of the objective function \(f(x)\) evaluated
at x=\(x\) must be assigned to f, and the function
return value set to 0. If the evaluation is impossible
at x, return should be set to a nonzero value. Data
may be passed into |
eval_g |
is a user-supplied function that must have the following signature: function eval_g(n, x, g, userdata) The components of the gradient \(g = \nabla_x f(x\)) of
the objective function evaluated at x=\(x\) must be
assigned to g, and the function return value set
to 0. If the evaluation is impossible at x, return
should be set to a nonzero value. Data may be passed
into |
eval_h |
is a user-supplied function that must have the following signature: function eval_h(n, ne, x, h, userdata) The nonzeros of the Hessian \(H = \nabla_{xx}f(x)\) of
the objective function evaluated at x=\(x\) must be
assigned to h in the same order as presented to
bgo_import, and the function return value set to 0. If
the evaluation is impossible at x, return should be
set to a nonzero value. Data may be passed into
|
eval_prec |
is an optional user-supplied function that may be C_NULL. If non-NULL, it must have the following signature: function eval_prec(n, x, u, v, userdata) The product \(u = P(x) v\) of the user’s preconditioner
\(P(x)\) evaluated at \(x\) with the vector v=\(v\), the
result \(u\) must be retured in u, and the function
return value set to 0. If the evaluation is impossible
at x, return should be set to a nonzero value. Data
may be passed into |
function bgo_solve_without_mat(T, data, userdata, status, n, x, g, eval_f, eval_g, eval_hprod, eval_shprod, eval_prec)
Find an approximation to the global minimizer of a given function subject to simple bounds on the variables using a multistart trust-region method.
This call is for the case where access to \(H = \nabla_{xx}f(x)\) is provided by Hessian-vector products, and all function/derivative information is available by function calls.
Parameters:
data |
holds private internal data |
userdata |
is a structure that allows data to be passed into the function and derivative evaluation programs. |
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 |
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 |
g |
is a one-dimensional array of size n and type T that holds the gradient \(g = \nabla_xf(x)\) of the objective function. The j-th component of |
eval_f |
is a user-supplied function that must have the following signature: function eval_f(n, x, f, userdata) The value of the objective function \(f(x)\) evaluated
at x=\(x\) must be assigned to f, and the function
return value set to 0. If the evaluation is impossible
at x, return should be set to a nonzero value. Data
may be passed into |
eval_g |
is a user-supplied function that must have the following signature: function eval_g(n, x, g, userdata) The components of the gradient \(g = \nabla_x f(x\)) of
the objective function evaluated at x=\(x\) must be
assigned to g, and the function return value set
to 0. If the evaluation is impossible at x, return
should be set to a nonzero value. Data may be passed
into |
eval_hprod |
is a user-supplied function that must have the following signature: function eval_hprod(n, x, u, v, got_h, userdata) The sum \(u + \nabla_{xx}f(x) v\) of the product of the
Hessian \(\nabla_{xx}f(x)\) of the objective function
evaluated at x=\(x\) with the vector v=\(v\) and the
vector $ \(u\) must be returned in u, and the function
return value set to 0. If the evaluation is impossible
at x, return should be set to a nonzero value. The
Hessian has already been evaluated or used at x if the Bool
got_h is true. Data may be passed into |
eval_shprod |
is a user-supplied function that must have the following signature: function eval_shprod(n, x, nnz_v, index_nz_v, v, nnz_u, index_nz_u, u, got_h, userdata) The product \(u = \nabla_{xx}f(x) v\) of the Hessian
\(\nabla_{xx}f(x)\) of the objective function evaluated
at \(x\) with the sparse vector v=\(v\) must be returned
in u, and the function return value set to 0. Only the
components index_nz_v[0:nnz_v-1] of v are nonzero, and
the remaining components may not have been be set. On
exit, the user must indicate the nnz_u indices of u
that are nonzero in index_nz_u[0:nnz_u-1], and only
these components of u need be set. If the evaluation
is impossible at x, return should be set to a nonzero
value. The Hessian has already been evaluated or used
at x if the Bool got_h is true. Data may be passed into
|
eval_prec |
is an optional user-supplied function that may be C_NULL. If non-NULL, it must have the following signature: function eval_prec(n, x, u, v, userdata) The product \(u = P(x) v\) of the user’s preconditioner
\(P(x)\) evaluated at \(x\) with the vector v=\(v\), the
result \(u\) must be retured in u, and the function
return value set to 0. If the evaluation is impossible
at x, return should be set to a nonzero value. Data
may be passed into |
function bgo_solve_reverse_with_mat(T, data, status, eval_status, n, x, f, g, ne, H_val, u, v)
Find an approximation to the global minimizer of a given function subject to simple bounds on the variables using a multistart trust-region method.
This call is for the case where \(H = \nabla_{xx}f(x)\) is provided specifically, but function/derivative information is only available by returning to the calling procedure
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:
|
eval_status |
is a scalar variable of type Int32 that is used to indicate if objective function/gradient/Hessian values can be provided (see above) |
n |
is a scalar variable of type Int32 that holds the number of variables |
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 |
f |
is a scalar variable pointer of type T that holds the value of the objective function. |
g |
is a one-dimensional array of size n and type T that holds the gradient \(g = \nabla_xf(x)\) of the objective function. The j-th component of |
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 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. |
u |
is a one-dimensional array of size n and type T that is used for reverse communication (see above for details) |
v |
is a one-dimensional array of size n and type T that is used for reverse communication (see above for details) |
function bgo_solve_reverse_without_mat(T, data, status, eval_status, n, x, f, g, u, v, index_nz_v, nnz_v, index_nz_u, nnz_u)
Find an approximation to the global minimizer of a given function subject to simple bounds on the variables using a multistart trust-region method.
This call is for the case where access to \(H = \nabla_{xx}f(x)\) is provided by Hessian-vector products, but function/derivative information is only available by returning to the calling procedure.
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:
|
eval_status |
is a scalar variable of type Int32 that is used to indicate if objective function/gradient/Hessian values can be provided (see above) |
n |
is a scalar variable of type Int32 that holds the number of variables |
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 |
f |
is a scalar variable pointer of type T that holds the value of the objective function. |
g |
is a one-dimensional array of size n and type T that holds the gradient \(g = \nabla_xf(x)\) of the objective function. The j-th component of |
u |
is a one-dimensional array of size n and type T that is used for reverse communication (see status=5,6,7 above for details) |
v |
is a one-dimensional array of size n and type T that is used for reverse communication (see status=5,6,7 above for details) |
index_nz_v |
is a one-dimensional array of size n and type Int32 that is used for reverse communication (see status=7 above for details) |
nnz_v |
is a scalar variable of type Int32 that is used for reverse communication (see status=7 above for details) |
index_nz_u |
is a one-dimensional array of size n and type Int32 that is used for reverse communication (see status=7 above for details) |
nnz_u |
is a scalar variable of type Int32 that is used for reverse communication (see status=7 above for details). On initial (status=1) entry, nnz_u should be set to an (arbitrary) nonzero value, and nnz_u=0 is recommended |
function bgo_information(T, data, inform, status)
Provides output information
Parameters:
data |
holds private internal data |
inform |
is a structure containing output information (see bgo_inform_type) |
status |
is a scalar variable of type Int32 that gives the exit status from the package. Possible values are (currently):
|
function bgo_terminate(T, data, control, inform)
Deallocate all internal private storage
Parameters:
data |
holds private internal data |
control |
is a structure containing control information (see bgo_control_type) |
inform |
is a structure containing output information (see bgo_inform_type) |
available structures#
bgo_control_type structure#
struct bgo_control_type{T} f_indexing::Bool error::Int32 out::Int32 print_level::Int32 attempts_max::Int32 max_evals::Int32 sampling_strategy::Int32 hypercube_discretization::Int32 alive_unit::Int32 alive_file::NTuple{31,Cchar} infinity::T obj_unbounded::T cpu_time_limit::T clock_time_limit::T random_multistart::Bool hessian_available::Bool space_critical::Bool deallocate_error_fatal::Bool prefix::NTuple{31,Cchar} ugo_control::ugo_control_type{T} lhs_control::lhs_control_type trb_control::trb_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. Possible values are:
\(\leq\) 0 no output,
1 a one-line summary for every improvement
2 a summary of each iteration
\(\geq\) 3 increasingly verbose (debugging) output
Int32 attempts_max
the maximum number of random searches from the best point found so far
Int32 max_evals
the maximum number of function evaluations made
Int32 sampling_strategy
sampling strategy used. Possible values are
1 uniformly spread
2 Latin hypercube sampling
3 niformly spread within a Latin hypercube
Int32 hypercube_discretization
hyper-cube discretization (for sampling stategies 2 and 3)
Int32 alive_unit
removal of the file alive_file from unit alive_unit terminates execution
char alive_file[31]
see alive_unit
T infinity
any bound larger than infinity in modulus will be regarded as infinite
T obj_unbounded
the smallest value the objective function may take before the problem is marked as unbounded
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 random_multistart
perform random-multistart as opposed to local minimize and probe
Bool hessian_available
is the Hessian matrix of second derivatives available or is access only via matrix-vector products?
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
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 ugo_control_type ugo_control
control parameters for UGO
struct lhs_control_type lhs_control
control parameters for LHS
struct trb_control_type trb_control
control parameters for TRB
bgo_time_type structure#
struct bgo_time_type{T} total::Float32 univariate_global::Float32 multivariate_local::Float32 clock_total::T clock_univariate_global::T clock_multivariate_local::T
detailed documentation#
time derived type as a Julia structure
components#
Float32 total
the total CPU time spent in the package
Float32 univariate_global
the CPU time spent performing univariate global optimization
Float32 multivariate_local
the CPU time spent performing multivariate local optimization
T clock_total
the total clock time spent in the package
T clock_univariate_global
the clock time spent performing univariate global optimization
T clock_multivariate_local
the clock time spent performing multivariate local optimization
bgo_inform_type structure#
#include <galahad_bgo.h> struct bgo_inform_type{T} status::Int32 alloc_status::Int32 bad_alloc::NTuple{81,Cchar} f_eval::Int32 g_eval::Int32 h_eval::Int32 obj::T norm_pg::T time::bgo_time_type{T} ugo_inform::ugo_inform_type{T} lhs_inform::lhs_inform_type trb_inform::trb_inform_type{T}
detailed documentation#
inform derived type as a Julia structure
components#
Int32 status
return status. See BGO_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 f_eval
the total number of evaluations of the objective function
Int32 g_eval
the total number of evaluations of the gradient of the objective function
Int32 h_eval
the total number of evaluations of the Hessian of the objective function
T obj
the value of the objective function at the best estimate of the solution determined by BGO_solve
T norm_pg
the norm of the projected gradient of the objective function at the best estimate of the solution determined by BGO_solve
struct bgo_time_type time
timings (see above)
struct ugo_inform_type ugo_inform
inform parameters for UGO
struct lhs_inform_type lhs_inform
inform parameters for LHS
struct trb_inform_type trb_inform
inform parameters for TRB
example calls#
This is an example of how to use the package to minimize a multi-dimensional objective within a box; the code is available in $GALAHAD/src/bgo/Julia/test_bgo.jl . A variety of supported Hessian and constraint matrix storage formats are shown.
# test_bgo.jl
# Simple code to test the Julia interface to BGO
using GALAHAD
using Test
using Printf
using Accessors
using Quadmath
# Custom userdata struct
struct userdata_bgo{T}
p::T
freq::Int
mag::Int
end
function test_bgo(::Type{T}) where T
# Objective function
function fun(n::Int, x::Vector{T}, f::Ref{T}, userdata::userdata_bgo)
p = userdata.p
freq = userdata.freq
mag = userdata.mag
f[] = (x[1] + x[3] + p)^2 + (x[2] + x[3])^2 + mag * cos(freq * x[1]) + sum(x)
return 0
end
# Gradient of the objective
function grad(n::Int, x::Vector{T}, g::Vector{T}, userdata::userdata_bgo)
p = userdata.p
freq = userdata.freq
mag = userdata.mag
g[1] = 2.0 * (x[1] + x[3] + p) - mag * freq * sin(freq * x[1]) + 1.0
g[2] = 2.0 * (x[2] + x[3]) + 1.0
g[3] = 2.0 * (x[1] + x[3] + p) + 2.0 * (x[2] + x[3]) + 1.0
return 0
end
# Hessian of the objective
function hess(n::Int, ne::Int, x::Vector{T}, hval::Vector{T},
userdata::userdata_bgo)
p = userdata.p
freq = userdata.freq
mag = userdata.mag
hval[1] = 2.0 - mag * freq^2 * cos(freq * x[1])
hval[2] = 2.0
hval[3] = 2.0
hval[4] = 4.0
return 0
end
# Dense Hessian
function hess_dense(n::Int, ne::Int, x::Vector{T}, hval::Vector{T},
userdata::userdata_bgo)
p = userdata.p
freq = userdata.freq
mag = userdata.mag
hval[1] = 2.0 - mag * freq^2 * cos(freq * x[1])
hval[2] = 0.0
hval[3] = 2.0
hval[4] = 2.0
hval[5] = 4.0
return 0
end
# Hessian-vector product
function hessprod(n::Int, x::Vector{T}, u::Vector{T}, v::Vector{T},
got_h::Bool, userdata::userdata_bgo)
p = userdata.p
freq = userdata.freq
mag = userdata.mag
u[1] = u[1] + 2.0 * (v[1] + v[3]) - mag * freq^2 * cos(freq * x[1]) * v[1]
u[2] = u[2] + 2.0 * (v[2] + v[3])
u[3] = u[3] + 2.0 * (v[1] + v[2] + 2.0 * v[3])
return 0
end
# Sparse Hessian-vector product
function shessprod(n::Int, x::Vector{T}, nnz_v::Cint, index_nz_v::Vector{Cint},
v::Vector{T}, nnz_u::Ref{Cint}, index_nz_u::Vector{Cint},
u::Vector{T}, got_h::Bool, userdata::userdata_bgo)
p = userdata.p
freq = userdata.freq
mag = userdata.mag
p = zeros(T, 3)
used = falses(3)
for i in 1:nnz_v
j = index_nz_v[i]
if j == 1
p[1] = p[1] + 2.0 * v[1] - mag * freq^2 * cos(freq * x[1]) * v[1]
used[1] = true
p[3] = p[3] + 2.0 * v[1]
used[3] = true
elseif j == 2
p[2] = p[2] + 2.0 * v[2]
used[2] = true
p[3] = p[3] + 2.0 * v[2]
used[3] = true
elseif j == 3
p[1] = p[1] + 2.0 * v[3]
used[1] = true
p[2] = p[2] + 2.0 * v[3]
used[2] = true
p[3] = p[3] + 4.0 * v[3]
used[3] = true
end
end
nnz_u[] = 0
for j in 1:3
if used[j]
u[j] = p[j]
nnz_u[] += 1
index_nz_u[nnz_u[]] = j
end
end
return 0
end
# Apply preconditioner
function prec(n::Int, x::Vector{T}, u::Vector{T}, v::Vector{T},
userdata::userdata_bgo)
u[1] = 0.5 * v[1]
u[2] = 0.5 * v[2]
u[3] = 0.25 * v[3]
return 0
end
# Objective function
function fun_diag(n::Int, x::Vector{T}, f::Ref{T}, userdata::userdata_bgo)
p = userdata.p
freq = userdata.freq
mag = userdata.mag
f[] = (x[3] + p)^2 + x[2]^2 + mag * cos(freq * x[1]) + x[1] + x[2] + x[3]
return 0
end
# Gradient of the objective
function grad_diag(n::Int, x::Vector{T}, g::Vector{T}, userdata::userdata_bgo)
p = userdata.p
freq = userdata.freq
mag = userdata.mag
g[1] = -mag * freq * sin(freq * x[1]) + 1
g[2] = 2.0 * x[2] + 1
g[3] = 2.0 * (x[3] + p) + 1
return 0
end
# Hessian of the objective
function hess_diag(n::Int, ne::Int, x::Vector{T}, hval::Vector{T},
userdata::userdata_bgo)
freq = userdata.freq
mag = userdata.mag
hval[1] = -mag * freq^2 * cos(freq * x[1])
hval[2] = 2.0
hval[3] = 2.0
return 0
end
# Hessian-vector product
function hessprod_diag(n::Int, x::Vector{T}, u::Vector{T}, v::Vector{T},
got_h::Bool, userdata::userdata_bgo)
freq = userdata.freq
mag = userdata.mag
u[1] += -mag * freq^2 * cos(freq * x[1]) * v[1]
u[2] += 2.0 * v[2]
u[3] += 2.0 * v[3]
return 0
end
# Sparse Hessian-vector product
function shessprod_diag(n::Int, x::Vector{T}, nnz_v::Cint, index_nz_v::Vector{Cint},
v::Vector{T}, nnz_u::Ref{Cint}, index_nz_u::Vector{Cint},
u::Vector{T}, got_h::Bool, userdata::userdata_bgo)
freq = userdata.freq
mag = userdata.mag
p = zeros(3)
used = falses(3)
for i in 1:nnz_v
j = index_nz_v[i]
if j == 1
p[1] -= mag * freq^2 * cos(freq * x[1]) * v[1]
used[1] = true
elseif j == 2
p[2] += 2.0 * v[2]
used[2] = true
elseif j == 3
p[3] += 2.0 * v[3]
used[3] = true
end
end
nnz_u[] = 0
for j in 1:3
if used[j]
u[j] = p[j]
nnz_u[] += 1
index_nz_u[nnz_u[]] = j
end
end
return 0
end
# Derived types
data = Ref{Ptr{Cvoid}}()
control = Ref{bgo_control_type{T}}()
inform = Ref{bgo_inform_type{T}}()
# Set user data
userdata = userdata_bgo(4.0, 10, 1000)
# Set problem data
n = 3 # dimension
ne = 5 # Hesssian elements
x_l = T[-10, -10, -10]
x_u = T[0.5, 0.5, 0.5]
H_row = Cint[1, 2, 3, 3, 3] # Hessian H
H_col = Cint[1, 2, 1, 2, 3] # NB lower triangle
H_ptr = Cint[1, 2, 3, 6] # row pointers
# Set storage
g = zeros(T, n) # gradient
st = ' '
status = Ref{Cint}()
@printf(" Fortran sparse matrix indexing\n\n")
@printf(" tests reverse-communication options\n\n")
# reverse-communication input/output
eval_status = Ref{Cint}()
nnz_u = Ref{Cint}()
nnz_v = Ref{Cint}()
f = Ref{T}(0.0)
u = zeros(T, n)
v = zeros(T, n)
index_nz_u = zeros(Cint, n)
index_nz_v = zeros(Cint, n)
H_val = zeros(T, ne)
H_dense = zeros(T, div(n * (n + 1), 2))
H_diag = zeros(T, n)
for d in 1:5
# Initialize BGO
bgo_initialize(T, data, control, status)
# Set user-defined control options
@reset control[].f_indexing = true # Fortran sparse matrix indexing
@reset control[].attempts_max = Cint(10000)
@reset control[].max_evals = Cint(20000)
@reset control[].sampling_strategy = Cint(3)
@reset control[].trb_control.maxit = Cint(100)
# @reset control[].print_level = CInt(1)
# Start from 0
x = T[0.0, 0.0, 0.0]
# sparse co-ordinate storage
if d == 1
st = 'C'
bgo_import(T, control, data, status, n, x_l, x_u,
"coordinate", ne, H_row, H_col, C_NULL)
terminated = false
while !terminated # reverse-communication loop
bgo_solve_reverse_with_mat(T, data, status, eval_status,
n, x, f[], g, ne, H_val, u, v)
if status[] == 0 # successful termination
terminated = true
elseif status[] < 0 # error exit
terminated = true
elseif status[] == 2 # evaluate f
eval_status[] = fun(n, x, f, userdata)
elseif status[] == 3 # evaluate g
eval_status[] = grad(n, x, g, userdata)
elseif status[] == 4 # evaluate H
eval_status[] = hess(n, ne, x, H_val, userdata)
elseif status[] == 5 # evaluate Hv product
eval_status[] = hessprod(n, x, u, v, false, userdata)
elseif status[] == 6 # evaluate the product with P
eval_status[] = prec(n, x, u, v, userdata)
elseif status[] == 23 # evaluate f and g
eval_status[] = fun(n, x, f, userdata)
eval_status[] = grad(n, x, g, userdata)
elseif status[] == 25 # evaluate f and Hv product
eval_status[] = fun(n, x, f, userdata)
eval_status[] = hessprod(n, x, u, v, false, userdata)
elseif status[] == 35 # evaluate g and Hv product
eval_status[] = grad(n, x, g, userdata)
eval_status[] = hessprod(n, x, u, v, false, userdata)
elseif status[] == 235 # evaluate f, g and Hv product
eval_status[] = fun(n, x, f, userdata)
eval_status[] = grad(n, x, g, userdata)
eval_status[] = hessprod(n, x, u, v, false, userdata)
else
@printf(" the value %1i of status should not occur\n",
status)
end
end
end
# sparse by rows
if d == 2
st = 'R'
bgo_import(T, control, data, status, n, x_l, x_u,
"sparse_by_rows", ne, C_NULL, H_col, H_ptr)
terminated = false
while !terminated # reverse-communication loop
bgo_solve_reverse_with_mat(T, data, status, eval_status,
n, x, f[], g, ne, H_val, u, v)
if status[] == 0 # successful termination
terminated = true
elseif status[] < 0 # error exit
terminated = true
elseif status[] == 2 # evaluate f
eval_status[] = fun(n, x, f, userdata)
elseif status[] == 3 # evaluate g
eval_status[] = grad(n, x, g, userdata)
elseif status[] == 4 # evaluate H
eval_status[] = hess(n, ne, x, H_val, userdata)
elseif status[] == 5 # evaluate Hv product
eval_status[] = hessprod(n, x, u, v, false, userdata)
elseif status[] == 6 # evaluate the product with P
eval_status[] = prec(n, x, u, v, userdata)
elseif status[] == 23 # evaluate f and g
eval_status[] = fun(n, x, f, userdata)
eval_status[] = grad(n, x, g, userdata)
elseif status[] == 25 # evaluate f and Hv product
eval_status[] = fun(n, x, f, userdata)
eval_status[] = hessprod(n, x, u, v, false, userdata)
elseif status[] == 35 # evaluate g and Hv product
eval_status[] = grad(n, x, g, userdata)
eval_status[] = hessprod(n, x, u, v, false, userdata)
elseif status[] == 235 # evaluate f, g and Hv product
eval_status[] = fun(n, x, f, userdata)
eval_status[] = grad(n, x, g, userdata)
eval_status[] = hessprod(n, x, u, v, false, userdata)
else
@printf(" the value %1i of status should not occur\n", status)
end
end
end
# dense
if d == 3
st = 'D'
bgo_import(T, control, data, status, n, x_l, x_u,
"dense", ne, C_NULL, C_NULL, C_NULL)
terminated = false
while !terminated # reverse-communication loop
bgo_solve_reverse_with_mat(T, data, status, eval_status,
n, x, f[], g, div(n * (n + 1), 2),
H_dense, u, v)
if status[] == 0 # successful termination
terminated = true
elseif status[] < 0 # error exit
terminated = true
elseif status[] == 2 # evaluate f
eval_status[] = fun(n, x, f, userdata)
elseif status[] == 3 # evaluate g
eval_status[] = grad(n, x, g, userdata)
elseif status[] == 4 # evaluate H
eval_status[] = hess_dense(n, div(n * (n + 1), 2), x, H_dense, userdata)
elseif status[] == 5 # evaluate Hv product
eval_status[] = hessprod(n, x, u, v, false, userdata)
elseif status[] == 6 # evaluate the product with P
eval_status[] = prec(n, x, u, v, userdata)
elseif status[] == 23 # evaluate f and g
eval_status[] = fun(n, x, f, userdata)
eval_status[] = grad(n, x, g, userdata)
elseif status[] == 25 # evaluate f and Hv product
eval_status[] = fun(n, x, f, userdata)
eval_status[] = hessprod(n, x, u, v, false, userdata)
elseif status[] == 35 # evaluate g and Hv product
eval_status[] = grad(n, x, g, userdata)
eval_status[] = hessprod(n, x, u, v, false, userdata)
elseif status[] == 235 # evaluate f, g and Hv product
eval_status[] = fun(n, x, f, userdata)
eval_status[] = grad(n, x, g, userdata)
eval_status[] = hessprod(n, x, u, v, false, userdata)
else
@printf(" the value %1i of status should not occur\n", status)
end
end
end
# diagonal
if d == 4
st = 'I'
bgo_import(T, control, data, status, n, x_l, x_u,
"diagonal", ne, C_NULL, C_NULL, C_NULL)
terminated = false
while !terminated # reverse-communication loop
bgo_solve_reverse_with_mat(T, data, status, eval_status,
n, x, f[], g, n, H_diag, u, v)
if status[] == 0 # successful termination
terminated = true
elseif status[] < 0 # error exit
terminated = true
elseif status[] == 2 # evaluate f
eval_status[] = fun_diag(n, x, f, userdata)
elseif status[] == 3 # evaluate g
eval_status[] = grad_diag(n, x, g, userdata)
elseif status[] == 4 # evaluate H
eval_status[] = hess_diag(n, n, x, H_diag, userdata)
elseif status[] == 5 # evaluate Hv product
eval_status[] = hessprod_diag(n, x, u, v, false, userdata)
elseif status[] == 6 # evaluate the product with P
eval_status[] = prec(n, x, u, v, userdata)
elseif status[] == 23 # evaluate f and g
eval_status[] = fun_diag(n, x, f, userdata)
eval_status[] = grad_diag(n, x, g, userdata)
elseif status[] == 25 # evaluate f and Hv product
eval_status[] = fun_diag(n, x, f, userdata)
eval_status[] = hessprod_diag(n, x, u, v, false,
userdata)
elseif status[] == 35 # evaluate g and Hv product
eval_status[] = grad_diag(n, x, g, userdata)
eval_status[] = hessprod_diag(n, x, u, v, false,
userdata)
elseif status[] == 235 # evaluate f, g and Hv product
eval_status[] = fun_diag(n, x, f, userdata)
eval_status[] = grad_diag(n, x, g, userdata)
eval_status[] = hessprod_diag(n, x, u, v, false,
userdata)
else
@printf(" the value %1i of status should not occur\n", status)
end
end
end
# access by products
if d == 5
st = 'P'
bgo_import(T, control, data, status, n, x_l, x_u,
"absent", ne, C_NULL, C_NULL, C_NULL)
nnz_u = Ref{Cint}(0)
terminated = false
while !terminated # reverse-communication loop
bgo_solve_reverse_without_mat(T, data, status, eval_status,
n, x, f[], g, u, v, index_nz_v,
nnz_v, index_nz_u, nnz_u[])
if status[] == 0 # successful termination
terminated = true
elseif status[] < 0 # error exit
terminated = true
elseif status[] == 2 # evaluate f
eval_status[] = fun(n, x, f, userdata)
elseif status[] == 3 # evaluate g
eval_status[] = grad(n, x, g, userdata)
elseif status[] == 5 # evaluate Hv product
eval_status[] = hessprod(n, x, u, v, false, userdata)
elseif status[] == 6 # evaluate the product with P
eval_status[] = prec(n, x, u, v, userdata)
elseif status[] == 7 # evaluate sparse Hess-vect product
eval_status[] = shessprod(n, x, nnz_v[], index_nz_v, v,
nnz_u, index_nz_u, u,
false, userdata)
elseif status[] == 23 # evaluate f and g
eval_status[] = fun(n, x, f, userdata)
eval_status[] = grad(n, x, g, userdata)
elseif status[] == 25 # evaluate f and Hv product
eval_status[] = fun(n, x, f, userdata)
eval_status[] = hessprod(n, x, u, v, false, userdata)
elseif status[] == 35 # evaluate g and Hv product
eval_status[] = grad(n, x, g, userdata)
eval_status[] = hessprod(n, x, u, v, false, userdata)
elseif status[] == 235 # evaluate f, g and Hv product
eval_status[] = fun(n, x, f, userdata)
eval_status[] = grad(n, x, g, userdata)
eval_status[] = hessprod(n, x, u, v, false, userdata)
else
@printf(" the value %1i of status should not occur\n",
status)
end
end
end
# Record solution information
bgo_information(T, data, inform, status)
if inform[].status == 0
@printf("%c:%6i evaluations. Optimal objective value = %5.2f status = %1i\n", st,
inform[].f_eval, inform[].obj, inform[].status)
else
@printf("%c: BGO_solve exit status = %1i\n", st, inform[].status)
end
# @printf("x: ")
# for i in 1:n
# @printf("%f ", x[i])
# end
# @printf("\n")
# @printf("gradient: ")
# for i in 1:n
# @printf("%f ", g[i])
# end
# @printf("\n")
# Delete internal workspace
bgo_terminate(T, data, control, inform)
end
return 0
end
@testset "BGO" begin
@test test_bgo(Float32) == 0
@test test_bgo(Float64) == 0
@test test_bgo(Float128) == 0
end