DIRECT SEARCH FOR OPTIMAL PARAMETERS WITHIN ...

DIRECT SEARCH FOR OPTIMAL P A R A M E T E R S WITHIN S I M U L A T I O N MODELS

Hans-Paul Schwefel Nuclear Research Centre Juelich, F.R.G. (KFA) Programme Group of Systems Analysis and T e c h n o l o g i c a l Development

(STE)

Abstract. The tool of systems simulation can be improved by s u p e r p o s i n g optimization techniques onto the computer model of the object or system investigated. Direct search methods for p a r a m e t e r o p t i m i z a t i o n are applicable in a much wider field than any other technique. O u t s t a n d i n g among such partly heuristic methods is one that is based on p r i n c i p l e s of organic evolution. The results of a comp r e h e n s i v e test p r o g r a m in which all important algorithms have been compared reveal the superiority of this e v o l u t i o n strategy. Reference is made to examples of actual application.

INTRODUCTION A model of a system or an object is devised where it is desirable its p e r f o r m a n c e

or b e h a v i o u r in future or under certain conditions

p e r i m e n t i n g on the actual object is precluded

mulated in terms of direct m u l t i t u d e of details.

and where ex-

for cost or other reasons.

the case e.g. in biology, medical science, and socioeconomy. always involve highly complex r e l a t i o n s h i p s ;

to know

hence,

This is

Here the problems

they usually cannot be for-

c a u s e - a n d - e f f e c t relations r e q u i r i n g attendance to a

Contrary to p h y s i c a l or e n g i n e e r i n g models, b e h a v i o u r a l

and statistical relations b e t w e e n system quantities have to be taken into account. This explains why any model is n e c e s s a r i l y

confined to certain aspects of the

overall system and greatly depends on the q u e s t i o n that has to be answered. Computerized models have preference over imaginary models. consistent

They call for a

image of reality on any level of a g g r e g a t i o n and for an even quanti-

tative disclosure of all assumptions.

The results are always reproducible.

When finally all important relations b e t w e e n system quantities have been formulated,

a model is a v a i l a b l e for equivalent experimenting.

as "what happens if

..." can be answered in the format

Simple questions

"if ... , then

...".

S y s t e m quantities a p p e a r i n g at the "if" end are the independent variables x = { Xl, x 2 ... x n } : { xi; i = i, 2 ... n } of the system that are sometimes variable w i t h i n limits only.

91

92

SCHWEFEL

The other end contains the dependent variable(s). F(x) or { Fi(x) , F2(x) The number

n

... ).

of the p a r a m e t e r s

xi

d e t e r m i n e s the m u l t i p l i c i t y

of p o s s i b l e

v a r i a t i o n s that r a p i d l y becomes u n m a n a g e a b l e with i n c r e a s i n g ~ n. In such situations, no progress

can be a c h i e v e d unless the original q u e s t i o n

is r e v e r s e d into "what should be done to achieve a d e s i r a b l e result?".

THE SEARCH FOR D E S I G N A T E D STATES OF THE S I M U L A T E D SYSTEM This r e v e r s a l of the d i r e c t i o n of i n f o r m a t i o n nearly always leads to arithmetic problems, the simplest terms,

e s p e c i a l l y where

several p a r a m e t e r values are searched for. In

case, a q u a n t i t a t i v e l y known result is desirable.

In m a t h e m a t i c a l

the s p e c i f i c a t i o n is !

F(x)

~

Fse t

Now the p r o b l e m assumes the form of an e q u a t i o n or a set of (simultaneous) equations;

however,

exact a p p r o a c h e s to solution,

e.g,

the G a u s s i a n elimination,

exist only in the case of purely linear or f i r s t - d e g r e e r e l a t i o n s b e t w e e n and

F. In other cases, an iterative or s t e p - b y - s t e p

the p a r a m e t e r s

x

is r e q u i r e d

I! F(x) - Fse t is g r a d u a l l y reduced. deviation, problem. tained.

x

search with v a r i a t i o n of

in which the d i f f e r e n c e

il

D e p e n d i n g on the n o r m

standard methods are a v a i l a b l e

II" II

for the w e i g h t i n g of the

for the a p p r o x i m a t i v e

After a finite n u m b e r of arithmetic

operations,

solution of the

the s o l u t i o n is ob-

This solution is not in all cases exact, however,

but rather a suffici-

ently accurate a p p r o x i m a t i o n as can be a c h i e v e d with the computer e m p l o y e d and its limited c o m p u t i n g accuracy. The q u e s t i o n for the best of all p o s s i b l e solutions without s p e c i f i c a t i o n of the latter is also permissible, to indicate an objective or maximum) value.

function

F(x)

but only where

quantitative it is p o s s i b l e

that is to assume an extreme

(minimum

This s p e c i f i c a t i o n for an u n d e t e r m i n e d value is sometimes

e x p r e s s e d as F(x)

+

extr.

(min. or max.)

.

This is now a genuine o p t i m i z a t i o n p r o b l e m where only an iterative a p p r o a c h to the s o l u t i o n can be applied unless the r e l a t i o n

F(x)

is so simple that the

n e c e s s a r y o p t i m a l i t y conditions Fx(X )

:

{ ~F(x) ~x.

; i

=

I, 2 ... n }

=

0

1

(i.e. all first-order partial d e r i v a t i v e s of the o b j e c t i v e become

function should

zero) can be h a n d l e d w i t h one of the standard methods

sets of equations and certain sufficient

for the solution of

conditions are still met.

DIRECT SEARCH FOR OPTIMA THE PREREQUISITES

OF OPTIMIZING

Before the various seems appropriate one dependent Fj(x).

93

procedures

to find an optimum are discussed,

to the case where the objective

variable

F(x),

cannot be represented

but is described by several partial

In this case, either the partial objectives F(x)

=

Fl(X)

=

F(x)

~ Fj(x)

=

objectives

have to be interrelated,

e.g.

by factors wj, e.g.

wj

•

O or one of them has to be selected while the other objectives lated into constraints

by only

/ F2(x)

or they have to be weighted

F(x)

a reference

have to be reformu-

as, for instance,

FI(x) ;

Gj

=

Fj+ 1

cj

~

0

with cj as the upper or lower bounds. In doing so, compromises fails,no ables

optimization

(parameters)

pensable

are frequently

is possible.

The identification

and the selection

prerequisites o f

necessary

any optimization.

This may be illustrated

by an example

emissions

of pollutants.

might be advisable loading minimum, with different

to first determine

and finally

Apart dependent

from methods variables

such controversial

ALGORITHM

particularly

suitable

it

parameter

minimizing

of multiple

1976, ref.

the cost

criteria

i .

values

multitude

optimization

the linear optimization linearity

- these problems

of methods

are

problem.

or Linear Programming

of the relations

F(x)

and

to find the optimum in a finite number of steps and can be cost even for large problems.

solution may be quite unusable non-linear

for a region

to find minima and maxima for time-

here -, an almost unsurveyable

at reasonable

originally

More details

OPTIMIZATION

that makes use of the at least piece-wise

employed

is taken.

partial objectives,

solutions,

in Keeney and Raiffa,

for the continuous

It guarantees

function

the pure cost minimum, then the environment-

to the emissions.

Most widely used is probably

Gj(x).

energy

or those that may assume only discrete

will not be discussed already available

on which a decision

look for intermediate

decision making may be found e.g.

CHOOSING AN APPROPRIATE

are the indis-

the objective

from energy economy:

vari-

cost, but at the same time with the least possible

Considering

constraints

objective

In some cases,

at a family of solutions

at lowest

of the independent

of an unambiguous

can be varied to arrive

is to be provided

and if this procedure

in reality,

and was impermissibly

optimum of a linear p r o g r a m up of the restrictions,

is always

However,

the computed best

e.g. where the relation simplified

located

in the model.

F(x)

was

While the

in a corner of the polyhedron

made

the optimum of a non.-linear p r o b l e m may well be found in

the interior of the feasible

area.

94

SCHWEFEL The extension of linear programming to convex or concave, usually purely

quadratic problems yields only gradual improvements It presupposes unimodality, to be excluded; moreover, partial derivatives

of the area of application.

that is, the existence of several local optima has

such an extension calls for the specification of the

Fx(X),

mostly in analytical

form, and their continuity.

Frequently the simulation model can be specified in algorithmic derivatives are not available, discontinuities

form only;

nothing can be stated about the topology,

are known to exist. Then only direct search strategies

climbing methods)

can be applied.

is no theoretically

or

(hill-

These are partly of heuristic nature and there

founded guarantee for the convergence to the (absolute)

mum. But they have proven to yield practical

opti-

solutions even when other methods

fail. They include quite a number of different

strategy concepts.

A potential

user would therefore profit from their numerical comparison with respect to reliability and cost, based on many test models. (Schwefel,

1977, ref.

Such comparison has been made

2) and has comprised the strategies

listed in Table 1.

Table 1: List of Strategies Compared Code

Description of strategy or variant

RBO GOLD LAGR HOJE DSCG

Univariate strategy with Rbonacci search Univariate strategy with golden-section search Univariate strategy with Lagrange interpolation Strategy of Hooke and Jeeves (pattem search) Strategy of Davies, Swann, and Campey with Gram-Schmidt orthonormalization Strategy of Davies, Swann, and Campey with Palmer orthonormalization Strategy of Powell (conjugate directions) Strategy of Davidon, Retcher, and Powell (variable metdc] modified by Stewart Simplex strategy of Nelder and Mead Strategy of Rosenbrock (rotating coordinates) Complex strategy of M.J. Box (1+1) evolution strategy (10,100) evolution strategy without recombination (10,100) evolution strategy with recombination

DSCP POWE DFPS SIMP ROSE COMP EVOL GRUP REKO

D I R E C T SEARCH FOR OPTIMA

95

COMPARISON AMONG D E R I V A T I V E - F R E E O P T I M I Z A T I O N P R O C E D U R E S Fig.

1 shows the c o m p u t i n g time for a quadratic test p r o b l e m vs. number of

parameters. non-uniform. gations.

As long as the number of variables is small, the results are highly This explains the c o n t r o v e r s i a l evidence of many early investi-

The trends become very clear, however, where the number of parameters

is large enough; these trends are indicated by the e x t e n d e d lines plotted.

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