G
DOCUMENTATION AND
USER INSTRUCTIONS OF
PROGRAM `OPTIMIZE'
Appendix G contains the documentation for the program
OPTIMIZE dis-
cussed in Chapters 3. It gives some information about the general features
and use of the programs.
G-1
G-2
G. Documentation and user instructions of program OPTIMIZE
G. Documentation and User Instructions of
Program OPTIMIZE
G.1 Program Documentation
G.1.1 Overview of the Program
Program OPTIMIZE deals with minimization of a non-linear function subject
to non-linear constraints. It uses the sequential quadratic programming (SQP)
based on the algorithm developed by Powell [1] OPTIMIZE program is already
implemented in SANOPT-S and SANOPT-F programs. It is an independent
and separate computer program which can be used for solving user dened
optimization problems.
G.1.2 Block Structure and Main Routines
A modular approach is adopted, in that separate subroutines are employed to
perform the various operation required in optimization. Each module in turn
is composed of one or more subroutines relevant only to its own needs and, in
some cases, of subroutines which are common to several modules. Control of
the modules is held by the main or master segment. The main segment of the
program structure for OPTIMIZE is given in Figure G.1.
FIMA
DESVAL
OPTIMIZE
SQPSET
FDFCDC
FIMA
Fig. G.1. Main block diagram of program OPTIMIZE.
1. FIMA handles the le management. This subroutine open and close the
necessary data les.
2. DESVAL is a user supplied subroutine which includes the optimizaiton
control parameters. These parameters are the number of design variable,
number of constraints and bounds of the design variables and constraints.
3. SQPSET is carry out sequential quadratic programming based on the algorithm developed by Powell [21]
G.2 Input instructions
G-3
4. FDFCDC is a user supplied subroutine. It computes the objective function,
constraints and their derivatives.
G.1.3 File Structures
Program OPTIMIZE uses the following channel for output:
unit 1 contains the results of the optimization. These are values of the optimum design variables, objective function and constraints at the optimum
solution.
G.2 Input Instructions
There is no input data le for program OPTIMIZE. However, the user has to
supply two subroutines which are DESVAL and FDFCDC.
Subroutine DESVAL: contains the control parameter for optimization problem
to be solved. The following parameters must be dened in the subroutine:
ndvab
ncons
necon
xdvab(idvab)
xlowb(idvab)
xuppb(idvab)
number of design variable
number of inequality constraint
number of equality constraint
initial values of the design variables, idvab = 1; : : : ; ndvab
lower limits of the design variables, idvab = 1; : : : ; ndvab
upper limits of the design variables, idvab = 1; : : : ; ndvab
Subroutine FDFCDC: computes the objective function and constraints for optimization problem to be solved. The following parameters must be dened in
the subroutine:
f(1)
f(icons)
d(idvab)
d(idvab)
objective function
constraint, icons = 2; : : : ; ncons
derivative of the objective function, idvab = 1; : : : ; ndvab
derivative of the constraint, idvab = ndvab+1; : : : ;ndvabncons
G-4
G. Documentation and user instructions of program OPTIMIZE
G.3 Specimen User Supplied Subroutines and Output
Data File
In this section user supplied subroutines and output data le are provided for
a selection of the numerical examples presented in the text. This information
will be of assistance to readers who wish to run the program contained in the
book on their own computer.
G.3.1 Example
Problem denition: The objective function is to minimize
nonlinear function of the design variables x
( ) = sin(x1 x2 ) + 2 exp (x1 + x2 ) +
F x
( ) which is a
F x
1
exp (x1 + x2 )
(G:1)
subject to inequality constraints
2
x1
+10
+ x2 0
x1 + 2x2 0
2
x2
3
x1
(G:2)
and bounds on the design variables
1
1
x1
x2
+1
+1
(G:3)
A typical user supplied subroutines for OPTIMIZE is displaced in Table G.1
and G.2. The output le of the program for the problem dened above is given
in Table G.3.
Table G.1.
c
c
c
c
c
c
c
The FORTRAN computer program listing of the subroutine DESVAL.
subroutine desval (ndvab,ncons,necon,xdvab,xlowb,xuppb)
Argument variables
integer
ndvab,
ncons,
necon
real*8
xdvab(*),
xlowb(*),
xuppb(*)
Local variables
real*8
bigbn
G.3 Specimen User Supplied Subroutines and Output Data File
G-5
c
bigbn = 1.0d20
c
c*** problem setup: control variables
c
ndvab = 2
ncons = 3
necon = 0
c
c*** problem setup: design variables with bounds
c
xdvab(1) = 1.d0
xdvab(2) = 2.d0
c
xlowb(1) = -bigbn
xlowb(2) = -bigbn
c
xuppb(1) = bigbn
xuppb(2) = bigbn
c
return
end
Table G.2.
c
c
c
c
c
c
The FORTRAN computer program listing of the subroutine FDFCDC.
subroutine fdfcdc(x,f,d)
Argument variables
real*8
real*8
f(*),
x(*)
d(*)
Local variables
real*8
ccos,
cexp
c
c*** function
c
ccos = dcos(x(1)*x(2))
cexp = dexp(x(1)+x(2))
f(1) = dsin(x(1)*x(2)) + 2*cexp + 1/cexp
c
c*** function derivatives
c
d(1) = x(2)*ccos + 2*cexp - 1/cexp
G-6
G. Documentation and user instructions of program OPTIMIZE
d(2) = x(1)*ccos + 2*cexp - 1/cexp
c
c*** constraints
c
f(2) = -x(1)**2-x(2)**2 + 1d0
f(3) = -x(1)**3+x(2)
f(4) = x(1)+2d0*x(2)
c
c*** constraints derivatives
c
d(3) = -2.d0*x(1)
d(4) = -2.d0*x(2)
d(5) = -3.d0*x(1)**2
d(6) = 1.d0
d(7) = 1.d0
d(8) = 2.d0
c
return
end
Table G.3.
The output of the OPTIMIZE for example problem.
_____ _____ _______ _____ _______ _____ ______ _______
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Optimization Program
Copyright (c) E. Hinton, J. Sienz and M. Ozakca
Swansea, 1988 - 2003
file created: Fri Sep 27 15:37:41 2002
solution:
number of calls of fdfcdc:
function value at the solution:
-0.8264473889
0.5630139549
11
0.2389516015D+01
constraint values at the solution: -0.2664535259D-14
0.1127490155D+01
0.2995805209D+00
References
G-7
References
[1]
Powell, M.J.D., Algorithms for nonlinear constraints that use Lagrangian functions, Math. Progr., 14, 224-248, 1978.
G-8
G. Documentation and user instructions of program OPTIMIZE