DIRECT
Version 2.0
User Guide
J. M. Gablonsky
North Carolina State University
Department of Mathematis
Center for Researh in Sienti Computation
Box 8205
Raleigh, NC 27695 - 8205
April 18, 2001
DIRECT
v2.0 User Guide
i
Contents
1 Intro dution to
DIRECT
1
1.1 Problem desription . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Using
DIRECT
2
2.1 What is inluded in the pakage . . . . . . . . . . . . . . . . . 2
2.2 Calling
DIRECT
. . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.3 Sample main program . . . . . . . . . . . . . . . . . . . . . . 7
3 A short overview of the
DIRECT
algorithm and our mo dia-
tions 11
3.1 Dividing the domain . . . . . . . . . . . . . . . . . . . . . . . 12
3.1.1 Dividing of a hyperub e . . . . . . . . . . . . . . . . 12
3.1.2 Dividing of a hyperretangle . . . . . . . . . . . . . . 12
3.2 Potentially optimal hyp erretangles . . . . . . . . . . . . . . . 14
3.3 The
DIRECT
algorithm . . . . . . . . . . . . . . . . . . . . . . 14
3.4 Our mo diation to the
DIRECT
algorithm . . . . . . . . . . . 16
3.5 Extensions to the
DIRECT
algorithm . . . . . . . . . . . . . . . 16
4 The Test Problems 20
4.1 Elementary funtions . . . . . . . . . . . . . . . . . . . . . . . 20
4.1.1 Constant funtion . . . . . . . . . . . . . . . . . . . . . 20
4.1.2 Linear funtion . . . . . . . . . . . . . . . . . . . . . . 20
4.1.3 Quadrati funtion . . . . . . . . . . . . . . . . . . . . 20
4.2 Example for hidden onstraints . . . . . . . . . . . . . . . . . 20
4.2.1 Gomez #3 [11℄ . . . . . . . . . . . . . . . . . . . . . . 21
4.3 Test funtions desrib ed in Jones et.al. [15℄ . . . . . . . . . . . 21
4.3.1 Shekel's family (S5,S7,S10) [7℄ . . . . . . . . . . . . . . 22
4.3.2 Hartman's family (H3, H6) [7℄ . . . . . . . . . . . . . . 22
4.3.3 Branin funtion (BR) [7℄ . . . . . . . . . . . . . . . . . 22
4.3.4 Goldstein and Prie funtion (GP) [7℄ . . . . . . . . . . 24
4.3.5 Six-hump amelbak funtion (C6) [16℄ . . . . . . . . . 24
4.3.6 Two-dimensional Shub ert funtion (SH) [16℄ . . . . . . 25
4.4 Main features of the Test Funtions . . . . . . . . . . . . . . . 25
4.5 Numerial results . . . . . . . . . . . . . . . . . . . . . . . . . 27
DIRECT
v2.0 User Guide
ii
Prefae
This do ument is a revision of the 1998 guide [8℄ to Version 1 of our imple-
mentation of
DIRECT
. The ma jor hanges in the o de are
inlusion of the original
DIRECT
algorithm as desrib ed by Jones [15℄,
extension of the algorithm to handle hidden onstraints,
and a parallel version b oth with PVM and MPI alls.
The o de and do umentation an b e found at the following WWW-
address :
http://www4.nsu.edu/eos/users//tkelley/www/optimization
o des.html
The primary ontat for
DIRECT
is
J. M. Gablonsky
Department of Mathematis
Center for Researh in Sienti Computations
North Carolina State University
Raleigh, NC 27695-8205
jmgablonunity.nsu.edu
Eletroni mail is preferred.
This pro jet was supp orted by National Siene Foundation grants #DMS-
9700569, #DMS-9714811, and #DMS-0070641, and an allo ation from the
North Carolina Sup eromputing Center.
Owen Esslinger and Alton Patrik ontributed to the parallel version of
the o de and the test programs.
DIRECT
v2.0 User Guide
1
1 Intro dution to
DIRECT
This user guide overs an implementation of b oth the original
DIRECT
algo-
rithm and our mo diation, whih we alled
DIRECT-l
.
DIRECT
is a metho d
to solve global bound onstraint optimization problems and was originally
develop ed by Jones et.al. [15℄. It has b een used in many industrial applia-
tions [1, 2, 4, 5, 6, 14℄. We will only briey desrib e our mo diations to
DIRECT
and how we extended it to handle problems with hidden onstraints
and p oint to [9, 10℄ for further information.
After a short introdution in Setion 1 we desrib e in Setion 2 what is
inluded in the pakage and how to use our implementation. Setion 3 then
gives a short explanation of the algorithm and our mo diations. Finally
Setion 4 desrib es several test funtions and rep orts some numerial results.
1.1 Problem desription
DIRECT
was develop ed to solve problems of the following form:
Problem 1 (P
0
)
Let
a; b
2
R
N
;
=
f
x
2
R
N
:
a
i
x
i
b
i
g
, and
f
:
!
R
N
be Lipshitz ontinuous with onstant
. Find
x
opt
2
suh that
f
opt
=
f
(
x
opt
)
f
+
;
(1)
where
is a given smal l positive onstant.
Our extension also solve more diÆult problems of the following form:
Problem 2 (P
00
)
Let
B
and
f
:
B
!
R
be Lipshitz ontinuous with
onstant
. Let
f
be
f
= min
x
2
B
f
(
x
)
:
Find
x
opt
2
B
suh that
f
opt
=
f
(
x
opt
)
f
+
;
(2)
where
is a given smal l positive onstant.
If
B
is not given analytially, we say that the problem has hidden on-
straints. Problems with hidden onstraints often o ur in so alled \blak-
b ox" optimization problems, where the ob jetive funtion is given by a om-
puter program. Note that problems of kind P
00
and P
0
are the same if
B
= .
DIRECT
v2.0 User Guide
2
2 Using
DIRECT
2.1 What is inluded in the pakage
The o de and do umentation an b e found at the following WWW-address :
http://www4.nsu.edu/eos/users//tkelley/www/optimization
o des.html
One you have the le
DIRECTv2.0.3.tar.gz
, do the following in an UNIX
environment:
unix
>
gunzip DIRECTv2.0.3.tar.gz
unix
>
tar -xf DIRECTv2.0.3.tar
If you use a omputing environment other than UNIX, you may need to use
other programs to unompress the les. One you have unompressed the
le, you will have a sub diretory alled
diret
ontaining the following les:
DIRet.f
The main routine.
DIRserial.f
Sp eial routines for serial version of
DIRECT
.
DIRparallel.f
Sp eial routines for parallel version of
DIRECT
.
DIRsubrout.f
Subroutines used in
DIRECT
.
main.f
Sample program for the serial version of
DIRECT
. This sample pro-
gram optimizes the test funtions desrib ed in Setion 4.
mainparallel.f
Sample program for parallel version of
DIRECT
.
myfun.f
Sample test funtions.
DIRmpi.f
Routines for parallel version of
DIRECT
using MPI.
DIRpvm.f
Routines for parallel version of
DIRECT
using PVM.
makele
Makele for sample programs (b oth serial and parallel).
mpi
test.md
File to run MPI version of parallel o de on the IBM SP/2
sup er omputer.
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