i
NCARNA
NCAR TECHNICAL NOTE
h
AN INTR ODUCTION GENETIC ALGORITHMS
F OR NUMERICAL OPTIMIZA
P Charb onneau
HIGH AL TITUDE OBSER V A Y
NA CENTER F OR A TMOSPHERIC RESEAR
BOULDER COLORADO
CH TIONAL
TOR
aul
TION
TO
MarciiT
List of Figures v
List of T ables
Preface
In tro duction Optimization
Optimization and hill
The simplex metho d
Iterated simplex
A set of test problems
P erformance of the a iterated simplex m etho ds
Ev olution optimization and genetic algorithms
Biological ev olution
The po w er of cum ulativ e election s
A basic genetic algorithm
Information transfer in genetic algorithms
PIKAIA A genetic algorithm for n umerical optimization
Ov erview and problem deition
Minimal algorithmic comp onen
Additional comp onen ts
A case study GA P
Hamming w alls and creep m utation
P erformance on problems
A real application orbital elemen ts of binary
Binary stars
Radial v elo cities and Keplerian orbits
A genetic algorithm solution using PIKAIA
Final though ts and further readings
T o cross o v er or to o v er
Hybrid metho ds
cross not
stars
test
on
ts
nd simplex
bing clim
ix
vii
CONTENTS OF ABLE
iii When should y ou use genetic algorithms
F urther readings
Bibliograph y
ivv
LIST FIGURES
Op eration of a generic hill metho d
A hard maximization problem
An iterated hill clim bing sc heme
Absolute p erformance of the simplex metho d
T est problem P
T est problem P
Accelerated learning b y means of cum ulativ e selection
Con v ergence curv es for tence learning h problem
Breeding in genetic algorithms
Con v ergence curv es for GA on P
Ev olution of the p pulation o in parameter space
Global con v ergence probabilit y
Radial v elo cit yv ariations in Bo
Ev olution of a t ypical solution to the binary ting problem
iso con tours in four h yp erplanes of parameter space
orbit
otis
searc sen the
Norsk
bing clim
OFviLIST OF T ABLES
I Simplex p erformance measures on test problems
II P erformance on test problems vs iterated simplex
PIKAIA
viiviiiA CE
In I w in vited to presen t a lecture on genetic algorithms at a Mini
W orkshop on Numerical Metho ds Astroph ysics held June at Institute
for Theoretical Astroph ysics Oslo Norw a y I subsequen prepared a written
v ersion of the lecture n i the form of a tutorial n i tro uction d to genetic algorithms for
n umerical optimization Ho w ev er for reasons bey ond the organizers con trol the
planned Pro ceedings of the W orkshop w ere nev er published the written
v ersion a v ailable through the W eb P age since septem ber con tin ues
pro v e p opular with users of the PIKAIA soft w I decided to ublish the
pap er in the form of the presen t NCAR T hnical
The pap er is organized as follo ws Section establishes the distinction be
t w lo cal and global optimization and the meaning of p erformance
in con text of global optimization Section tro uces d the general idea of a
genetic algorithm as nspired i from the biological pro cess of ev olution b y means of
natural selection Section pro vides a detailed comparison of the p erformance of
three genetic algorithmased optimization sc hemes against iterated clim
using the simplex metho d Section describ es in full detail the use of a genetic
algorithm t o olv s e a real data mo deling problem namely the determination of or
bital elemen ts a binary star system from observ ed radial v elo cities The
closes in section rections on matters of a somewhat more philosophical
nature and includes a list of suggested further readings
I ended up making v ery few mo diations to the text originally prepared in
ev en though if I w ere to rewrite it w things undoubtedly w ould
turn out diren t The of test functions I no w use to test mo diations
PIKAIA has ev olv ed signian from that presen ted x herein V ersion of
PIKAIA publicly released in April w compare en v orably
the iterated simplex metho d against whic h x herein I
up dated expanded the of further reading x to b etter rect curren t
topic and trends the genetic algorithm literature addition to some minor
rew ording here and there throughout the text I restored a Figure x and
a al subsection to x b oth originally eliminated to within the age limit
of the ab o v een tioned illated W orkshop Pro ceedings
Bac k in I c hose e pap er the v or of a tutorial h section
ends with a ummary s of imp ortan t p oin ts to remem b r e rom f that section Y ou are
Eac this giv to
to also
In in
list and
in pitted is PIKAIA
to fa more ev ould
in tly
to suite
some no
with
er pap of
bing hill
in the
measures een
Note ec
are to
PIKAIA
Because
tly in
the in
as
PREF
ixx
of course encouraged to ber more than whatev er listed there Y ou
also d at the end of eac h section a s eries of Exercises are
some require programming on y our part These designed to be done
using PIKAIA a public omain d selfon tained genetic algorithmased optimization
subroutine The source co de PIKAIA s w ell as answ ers most
a v ailable on the tutorial W eb P age from whic hy ou can also access he t PIKAIA
W eb P age
httpwwaocar du publ ic r es earc h i ikai a to u ria lt ml
The T utorial P age also includes v arious animations for of solutions dis
cussed in the text The PIKAIA W eb P age con tains links to HA O ftp arc hiv e
from whic h y ou can obtain in addition to the source co de PIKAIA a User
Guide as w ell as source for the v arious examples discussed therein The
idea b ehind all this is that b y the y ou are reading through this
and doing the Exercises y should be go o d shap e to solv e global n
optimization problems y ou t encoun y our o wn researc h
The writing of this preface ors a e opp ortunit y to thank m y f riends and
colleagues Viggo Hansteen and ats M Carlsson for their in vitation and ancial sup
potr to attend their Mini orkshop on Numerical Metho ds in Astroph ysics
as w ell as for their y during m y extended y Norw a y The
CrB ata d and some source co des or f the orbital elemen t tting problem of x ere
pro vided b y Tim Bro wn who w as also generous with his ime t n i explaining to me
some of the subtleties of orbital elemen t determinations Thorough readings of
the draft of his t p ap er b y S andy and Gene Arnn Tim Bro wn Sarah Gibson
Barry K napp Hardi P eter are also gratefully ac kno wledged
Throughout m yt w e y ears w orking at NCAR High Altitude O bserv atory
it been m y privilege in teract with a large n um ber of brigh t en th usi
astic studen and p ostdo s c fora ys to algorithms ha v e particularly
b eneed from suc h collab orators Since Iha v e had k eep in turn with
T ed Kennelly arah S Gibson Hardi P eter Scott McIn tosh T ra vis Metcalfe I
thank them all for k eeping me on m ytoes all this time
P aul Charb onneau
h Boulder
Marc
and
up to
genetic in My ts
and to has
elv
and
in sta hospitalit kind
in ter migh
umerical in ou
er pap done time
des co
for
the
the some
are
exercises to for
are and so
less others easy Some
will is remem INTR ODUCTION OPTIMIZA TION
Optimization and hill clim
Optimization is something that readers of this tutorial ha v e
faced a long time ago in their st c alculus class one is giv en an analytic function
f x and presen ted of ding the v alue of x at whic h the function
reac its maxim um v alue pro cedure taugh tto w this end i s diren
tiate the function with resp ect to x set the resulting expression to zero
solv e for x call the result x and there y ou v e Ev en though most of
max
w ould no longer think t wice ab out it this is actually a prett y tric k
F or the eader r trained in ph ysics the limitation of this analytical metho d w as
encoun tered p erhaps st in optics hen w studying the diaction pattern of a sin
gle v ertical slit Jenkins White c Y ou migh t recall that the
in tensit y of the diaction pattern v aries as sin x ehre x is directly prop or
tional to the distance along the direction p erp endicular to the slit on the screen on
whic h the diaction pattern pro ected j lo of the in y minima
readily found be x n with n n is tric kier Ho w
min
ev er calculating the lo cations of the n i tensit y maxima b y t he analytical p ro cedure
describ ed ab o v e leads a nast y nonlinear transcenden tal equation whic h
be solv ed algebraically for x turn to iterativ e or graphical means
n the course of whic h the kier n of the minima is also resolv
This diult y with the diaction problem symptomatic of the fact that it is
usually harder often much harder d the zeros of functions than
extrema the more so the higher the dimensionalit y of f
et al x for a concise y et lucid discussion of this matter The nescapable i
conclusion is that once one mo v b ey ond high sc ho ol calculus minax problems
optimization is b est carried out n umerically
Up on op ening a t ypical in tro ductory textb o ok on n analysis one is
almost guaran teed to d therein a optimization metho ds describ ed in some
In fact y ou also v e to diren tiate the result of step once again and
v erify that the resulting expression is negativ e when ev aluated at x but
max
subtlet y migh tha v e b een elab orated up on in the lecture
next only
this
ha
few
umerical
es
Press see unctions said the
their to ery
is
ed case tric
to has One
cannot to
to are
tensit cation The is
hap
neat
us it ha
ard The hes
task the with
st will most
bing
detail In nearly all cases those m etho ds will fall under the b road category of hil l
climbing schemes op eration of a generic clim sc heme is illustrated
on Figure in the con text of maximizing a f unction of t w ov ariables i ding
the maxim um ltitude in a D andscap e bing b e gins b y c ho
a starting lo cation in parameter space anels One then determines the
lo cal steep est uphill direction mo v es a certain distance in direction anel
rev aluates the lo cal uphill direction and so on un til a lo cation in parameter
space is arriv ed at where all surrounding directions do wnhill This marks
the successful completion of the maximization task anel Most textb o ok
optimization metho ds basically op erate i n this w a y and simply dir in ho wthye
go ab out etermining d the steep est uphill direction c ho osing ho w a a is to
be akt en in that direction nd a whether or not in doing so use is made of gradien t
information accum ulated the course of previous steps
Hill clim bing metho ds w ork great if faced unimo al d landscap es suc h as
the one w ards whic h the rabid paratro op er of Fig A ab out dep osit
lo w er bac kside Unfortunately is not alw a ys that simple Consider instead
the D landscap e sho wn on Figure the um is the narro w cen tral spik e
indicated b y the arro w and is surrounded b y concen tric rings of secondary maxima
The only w a y that hill clim bing can d the true maxim um in case is if
paratro op r e happ ens to somewhere on the slop es of the cen maxim
hill clim bing rom f an y other landing site will lead to of the rings The cen tral
p eak co v ers a fractional surface area of ab out of the full parameter space
x y Unlik e on the landscap e of Fig A here the starting p oin t is
critical if hill clim bing is to w Hill bing is a lo c al optimization strategy
Figure ors a glob optimization problem
Of course if the sp eci optimization problem y ou are w orking on happ ens
be h that y ou can alw a ys come up with a go o d enough starting guess then
all y ou need is lo cal hilllim bing and y ou can pro ceed merrily ev after But
what if y ou are in the situation most p eople d themselv es in when dealing with
a ard h global optimization problem namely not bneiginapsiontoti o llag pu ood
starting guess out of y our hat
I kno wwhaty oue thinking If the cen tral p e ak co v ers ab out of parameter
space it means that y ou v e ab out one c hance in a h undred a random drop
land close enough for hill to w ork So the question y ou ha v e to
y ourself is do I f eel luc ky Y our answ er to this q uestion is em b o died in the First
W ell do y punk
ou
ask bing clim to
for ha
er
suc to
al
clim ork
one
um tral land
our this
maxim
life
his to is to
with
in
step big
are
that
osing clim Hill
bing hill The
Figure Op eration a generic hill clim bing d llegory F rom a ran
domly c hosen starting p oin t panel the direction of maxim um slop e is f ollo w ed
anel un til one reac hes a p oin t where all s urrounding directions are do wnhill
anel Landing anel not problematic from the computational p oin t
of view
is
metho of
Figure Tw o dimensional surface f x y x y deing a hard
maximization problem The global maxim um is f x y at x y
and is indicated b y the arro w
Rule of Global Optimization also kno wn as
THE HARR Y R ULE
ou should nev er feel luc ky
F aced with the landscap e of Figure the straigh ard solution lies with
a hnique called iter d l climbing This is a fancy name for something v
simple as illustrated on Figure Y ou run y our fa v orite lo cal hill clim
metho d rep eatedly eac h time from a diren t randomly c hosen starting p oin t
While doing so y ou k eep trac k of the v arious maxima lo cated and once y ou
satisd that all maxima v e b een found y ou pic k the tallest y ou
done with y our global optimization roblem p As y ou migh t imagine deciding
when to stop is the crux of otherwise straigh tforw pro cedure
ard this
are
and one ha are
so
bing just
ery hil ate tec
tforw most
TY DIR
with
Figure An iterated hilllim sc After landing eac h trial pro eeds c as
on Fig
heme bing
With a fractional co v erage of for the cen tral p eak of Figure y ou migh t
exp ect to ha v e to on a v erage something of the order of iterated
clim bing trials b fore e ding the cen tral p e ak As is faced with
problems of increasing parameter space dimensionalit y andr situations where
the global maxim um spans a tin y fraction of parameter space iterated
clim bing can add up to a lot of w This leads us naturally to the Rule
of Global Optimization also kno wn as
THE NO FREE R ULE
f y ou really w an t the global optim um y ou h a v e to w ork it
These considerations also lead us to distinguish bet w een three distinct asp ects of
p erformance when dealing a global optimization problem
Absolute p erformance Ho wn umerically accurate is the solution returned
b y m y adopted d
Global p erformance Ho w certain can I be that solution returned b y
m y metho d is the true global maxim um in parameter space
Relativ e p erformance Ho w m uc h computational w ork is b y m y
metho d to return a solution
Most fancy optimization m etho ds y ou migh t read ab out in textb o oks are designed
to do as w ell as p ossible on and s im ultaneously Suc h metho ds will do w ell
on only if pro vided with a uitable s starting guess If suc h a guess i s consisten tly
a v ailable for the roblems p y w orking on y ou need not ead r an y further But
assured that Dirt y h up y ou one of da ys
The simplex metho d
The distinction bet w lo cal and global optimization as w ell as related
p erformance issues are p erhaps best appreciated b y lo oking in some detail at the
beah vior of a l o al c hill clim bing metho d on a global optimization problem T o w ard
this end w e retain D landscap e of Figure as a b ed and attempt
maximize it using the Simplex Metho d
A d iren t terminology ma yw be used in optimization textb o oks but y ou
can be assured that they do discuss something equiv alen t
ell
to test the
the een
these with catc will Harry rest
are ou
required
the
metho
with
for will
LUNCH
Second ork
hill only
optimization one
hill run
The Simplex d of Nelder actually a v robust
clim bing sc heme A brief y et clear in tro duction to the metho d can be found in
Press et al x A simplex is a geometrical ure with n v ertices that
liv es in a parameter space of dimension n In D pace s a simplex is a triangle in
D space a t etrahedron and so on Giv en the function v alue ere he t ltitude
f x y at eac h of t he simplex v ertices ere an x y p t t he w orst v ertex is
displaced b y ha ving the simplex undergo one of p ossible t yp es of o v es
namely con traction expansion or rection ee Fig Press al The
mo v e is executed n i a manner suc h that the f unction v alue of the d isplaced v
is increased b y the mo v e in the con text of a aximization m problem
undergo es successiv e suc h mo v es un no mo v e can b e found that leads to further
impro v emen t bey ond some preset tolerance W atc the simplex con tract and
expand and squirt around the l andscap e of ig F is go o d visual fun and justis
w ell name giv en b y Press al to simplex subroutine amoeba This is
the implemen tation u sed here
By the standards of lo cal optimization metho ds simplex passes for a
lo w metho d The accuracy the solution increases ximately
linearly with the n um b r e of simplex mo v es Ho w ev er the simplex can pull itself out
of situations t hat w ould defeat or seriously mp i ede faster marter gradien tased
lo cal metho ds it can eien tly cra t v alleys and squeeze through
saddle p oin ts In this sense it can be said to exhibit pseudolobal capabilities
Eviden tly the simplex metho d requires that one pro vide initial co
x y for the simplex three v ertices Despite he t simplex metho d pseudolobal
abilities on a m ultimo dal global problem the c hoice of initial lo for the
simplex often determines whether the global maxim um is ultimately found Figure
ho ws a series of con v ergence curv es for the test problem of Figure Eac h curv e
corresp onds to a iren d t random initial simplex conuration
ds the cen tral p eak it do es so rather quic kly requiring ab out mo v es for
accuracy The p roblem of course is hat t the simplex often d o es not con v erge to the
cen tral p eak Rep eated rev eal that the metho d ac es global con v ergence
for only or so of trials
Iterated Simplex
The prosp ects of the simplex metho d for global p erformance are greatly en
hanced if one of starting v ertices lies high enough on slop es of cen tral
p eak This suggests that iterated hill clim bing using the simplex metho d ere
after iterated simplex should ac hiev e global p erformance within a few h undred
An animation of the simplex at w ork on the landscap e of Figure can b e
view ed on the tutorial W eb P age Chec k it
out
the the
hiev trials
simplex the When
cation
ordinates
long up wl
appro of absolute
the
their et the
hing
til
simplex The
ertex
et in
three
oin
hill ery is Mead Metho
Figure Absolute p erformance of the simplex d on the test problem of
Figure Eac h curv e corresp onds to a diren t starting simplex F ailure of the
simplex to lo cate the p eak to the con v ergence curv es eling o at
relativ ely high v alues of f x y With con v erged runs out of trials his t plot
is not represen tativ e of the simplex m etho d global p rformance e on this problem
whic h is in fact signian tly p o namely ab out
iterations And indeed it do es rep eatedly running the simplex times on the
problem of Figure to the cen tral p eak b e ing lo cated in of tri
als price to pa y of course is in n b er of function ev aluations required
ac hiev e this el of global p erformance nearly function ev aluations per
iterated simplex run on a v erage W elcome bac k to the No F ree Lunc h Rule
It is recommended practice when the simplex in singleun mo de
carry a r andom r estart once the simplex has v erged this en tails reinitial
izing randomly one of con v simplex v ertices letting the
simplex recon v erge again What is describ ed here as iterated simplex
in reinitializing al l v ertices randomly so as to mak e eac h successiv e trial
indep enden t from all others
A single simplex mo v e y than one function ev aluation F or
more tail en ma
fully
consists
and erged the but all
con out
to using
lev to
um the The
leads test
orer
lev leads tral cen
metho
A set of test problems
One should righ tfully susp ect that the simplex metho d p e rformance on the
problem Figure migh t not be represen e of its p erformance on
problems This v ery legitimate concern will eviden o v er the v arious
genetic algorithmased optimization sc hemes discussed further belo w It
therefore pro v e useful to ha v ea v ailable not just one but a set of est t roblems p The
four test problems describ ed b elo w a very hard global optimization problems
on whic h most con v tional lo cal optimization algorithms w fail miserably
Also k eep in mind that it is always p ossible design a test problem
defeat any global optimization metho d
P maximizing a function of t w o v ariables parameters
Our st test problem ereafter lab eled is our no w familiar D e
of Figure Mathematically t is deed as
f x y cos n r xp r a
r x y x y b
where n and are constan ts The global maxim is lo at
x y with f x y This global maxim um is surrounded b y
concen tric rings of secondary maxima cen tered on the maxim um at
distances
r f g
Bet w een these a re lo cated another series of concen tric rings c orresp onding to min
ima
m
r m
n
The error asso ciated with a en solution x y can be deed
f x y
example if the trial mo v e do es not lead an increase in f the mo v e migh t be
rep eated with a halv ed or doubled displacemen t ength l r a d iren tt yp e of mo v e
migh t be attempted dep ending on implemen tation On the maximization prob
lem of Figure one simplex mo v e r equires function ev aluations on a v erage
The high n high D v ersion of the fractal function discussed in x of B ac k
is a prett y go o d candidate for the ultimate killer test problem
to
as giv
min
max
radial global
cated um
landscap
will that to
ould en
all re
will
to carry tly
other tativ of test
Note that the eak corresp onding to the global maxim um co v ers a surface
n in parameter space If a illh clim bing sc heme w ere used the robabilit p yof
a andomly r c hosen s tarting p oin t landing close enough to this p e ak for the metho d
lo cate the true global um is only for n
P maximizing a function of t w o v ariables parameters
T est function P sho on Figure is again a D landscap e to be
It is deed b y
f x y exp r exp r a
r x y b
r x y c
The maxim um f x y is at x y and corresp onds to the p eak of
the s econd narro w er Gaussian P is ab out as hard a global optimization problem
as P he simplex succeeds times out of trials but f or a iren d t reason
There are no w only t w o lo cal maxima with the global maxim um again co v
ab out of arameter p space Unlik e P where m o ving to w ard successiv
secondary extrema actually brings one closer to the true m axim um ith w P mo v
ing to secondary maxim um pulls solutions away from the global maxim
Problems exhibiting this c haracteristics sometimes eceptiv e in the
optimization literature
P maximizing a function of v ariables parameters
T est problem P a direct generalization of P to indep e nden t v ariables
w x y z
f w x y z cos n r xp r a
r w x y z w x y z b
again with n and Comparing p erformance on P P
pro vide a measure of sc alability of the metho d under consideration ho w
p erformance degrades as parameter space dimensionalit y is increased ev
else b eing equal P a d global optimization problem the
metho d manages to d the global maxim um of trials
P Minimizing a least squares residual parameters
Our fourth and al test problem is deed as a eal nonlinear least
ting problem Consider a function of v ariable x eed d as sum of t w o
Gaussians
X
x x
j
y x A
j
j
j
exp
the one
squares
out times only
simplex har very is
erything
namely
will and
four is
four
called are
um the
higher ely
ering
maximized wn
again
maxim to
area
Figure T est problem P The problem consist maximizing a function of
t w o v ariables deed b y t w o Gaussians ee The global maxim um is
f x y at x y and is indicated b y the arro w
Dee no w a ataset b y ev aluating function a of K equidistan t
v alues of x in the in terv al i y y x x x x
k k k k
k
v alues of A etc Giv en dataset and the functional form used to generate
it eq the optimization problem then to v er the parameter v
for A originally used to pro uce d the dataset This is b y minimizing
the square residual
K
X
R A y y x A
k
k
with resp ect to the parameters deing the t w o Gaussians If one a
priori that t w o aussians G re a to b e to the data then this esidual r minimization
problem is ob viously equiv alen t a D function maximization problem for
a y whic h simply dees a function D space Figure ws the
generated using the parameter set
A
dataset sho in
to
told is
done etc
alues reco is
that
set some for
set for this
eqs
in
Figure T est problem P This is a parameter problem whic h consists in
ting t w o Gaussians a ataset of p oin ts Note ho w the second Gaussian
is p o orly sampled b y the discretization in x The thin line is
Gaussian function deed b y eq
and K discretization poin x Once the resulting
problem is not an easy one giv en the discretization in x the minimization is
largely dominated b y the need accurately the broader
Gaussian the second Gaussian s i not only of m uc hlo w er it is p o
sampled in x Fitting only the st Gaussian leads to a reasonably lo w residual
R global accuracy requires the second Gaussian be
and in whic h case only R
The simplex succeeds in prop erly ting b oth Gaussians out of
What are the econdary minima on whic h the simplex remains k They
can be divided in to t w o broad classes of mo Gaussians s the
broad higher amplitude comp onen t and the other is en to zero either b y
ha ving A or the metho d a t w o Gaussians solution where
x x and A A The D parameter space
con tains long t alleys and lains of w but sub optimal r esidual v alues in
whic h the simplex grinds to a halt
lo
returns
driv
del the one
stuc
trials
es do
etected also to
orly also amplitude
st amplitude high to
minimization again in ts
underlying the solid
to
T I
Simplex p erformance measures on problems
T est Problem P erformance Simplex Iterated simplex
P h f i
p
G
h N i
f
N
t
P h f i
p
G
h N i
f
N
t
P h f i
p
G
h N i
f
N
t
P h R i
p
G
h N i
f
N
t
P erformance of the simplex and iterated metho ds
T able I ummarizes s the p erformance of the implex s and iterated simplex meth
o d s on he t four test problems hen try represen ts an a v erage o v er at least
indep enden t runs p to for P and so should b e fairly r epresen e of the
metho d s beah vior on eac h test problem F or eac h problem the T able giv es the
absolute p erformance deed here as the a v erage o v er all runs of either f x
for P P and P or the r esidual R f eq f or P The global p erformance is
deed in terms of a p robabilit y easure m p as the fraction of all runs for whic h
G
the true global extrem um has been lo cated f for P P and P or the
second smaller Gaussian been prop erly R for P As amuasree of
relativ e p erformance the table simply the a v erage n um ber of functiono del
ev aluations h N i required b yeac h metho d The astl en try for eac h roblem p is the
f
n ber of trials N executed b y iterated simplex his n ber b y deition
t
for the basic simplex metho d without restart
is um um
lists
has
tativ
Eac
simplex
test
able
A t this stage only a commen ts need be made on the basis of T I
The st is that as adv ertised all four test problems are hard global optimization
problems as can b e judged from the p o or global p erformance of the basic simplex
metho d on eac h T urning to iterated simplex leads sp ectacular impro v t
in global p e rformance cases but of course the n um ber of required function
ev aluations go es up b ya orders of magnitude In fact the global p e rformance
of iterated simplex can be predicted on the of the singleun simplex The
global p e rformance on the later can b e view ed as a p robabilit y p of l o ating c the
global maxim the omplemen tary probabilit y of a giv en run to do so is
p the probabilit yof al l iterated simplex runs not ding the global maxim um
N
t
is then p that the probabilit y of any one of N iterations lo the
t
global maxim um is
N
t
p p terated hill clim bing
G
On the basis of eq one w ould predict global p erformances
on P through P giv en the n b er of h c bing iterations listed i n he t
righ tmost column of T able whic h compares quite w ell w ith he t actual measured
global p erformance One can also rewrite eq as
log p
G
N terated hill clim bing
t
log p
predict the exp ected n ber of hill clim bing iterations required to ac hiev e a
global p erformance lev el p p for P requiring p w ould
G G
demand n a v erage N hill clim bing trials adding up to a grand total of
t
ab out function ev aluations since a single simplex run on P carries out
on a v erage function ev aluations f T able Iterated hill clim
w orks there really is no h thing as a free lunc h
It is easy to predict the global p erformance of iterated simplex be
cause eac h trial pro ceeds completely indep enden tly impro v emen t in
p erformance simply rects the b etter initial sampling of parameter space asso ci
ated with the initial distribution of simplex v ertices Ev erything else b e
as problem dimensionalit y n increases the n um ber of trials N required can be
t
n
exp ected to scale as N a where a is n um ber c haracterizing in this case
t
the fraction of parameter space co v ered b y the global maxim um
is not only demanding in terms of function ev aluations but addition it
not scale w ell on a giv en problem dimensionalit y is increased in
fact is the cen tral problem facing iterated hill clim bing general not just its
simplexased incarnation
in
This as all at
es do in
simplex Iterated
some
equal ing
global The
ected exp
suc but
certainly bing
with
um to
lim ill um
cating so
not um
basis
few
all in
emen to
able few
The poor scalabilit y iterated stems from e
trial pr o c e e ds indep endently The c hallenge dev eloping global metho ds t hat are
outp erform iterated hill clim bing consists in tro ucing d a transfer of informa
tion bet w een trial solutions in a manner that con tin uously roadcasts eac h
paratro op r e in the squadron top ographical information garnered b y h in
dividual paratro p o e r in the course of hiser lo cal clim b The c hallenge of
course is to ac hiev e this without o v biasing the ensem ble of trials
A relativ w ellno metho d that often ac hiev es reliably is simulate d
anne aling etrop olis et see also Press et al x ulated
annealing is inspired b y he t global transfer of energynformation ac hiev ed b y col
liding constituen t p articles of a co oling liquid metal hic w hallo ws the ubstance s to
ac hiev e the crystallineetallic conuration that minimizes the total energy of
the hole w system The a lgorithmic implemen tation of the ec t hnique for n
optimization requires the sp eciation of a c o oling sche dule chi h i s ar f from triv
ial fast co oling is computationally eien otl w h N i but can lead to con v ergence
f
on a secondary extrem um o w p while slo w co oling impro v es global con v ergence
G
igh p but at the exp ense of a h N i No F h remem ber
G f
Genetic A lgorithms ac hiev e the are inspired b y the exc hange
of genetic information o c curring a breeding p opulation to
selection They can b e used to form the core of v ery robust global n umerical opti
mization metho ds as detailed Section beol w The follo wing Section pro
a brief in tro duction to genetic algorithms in a general sense
The least y ou should ber Section
Global optimization is a totally diren t game from lo cal optimization
Y ou should nev er feel luc ky
There is no suc h thing as a free h
Y ou can alw a ys design a problem that defeat an y global
metho d
Exercises for Section
Lo ok bac k at Figure Whenev er the simplex to ac hiev e global con v er
gence f seems remain stuc k at a discrete set of f
v alues What do these v alues corresp ond to
Consider again the use iterated simplex on the test problem of Figure
calculate the fractional surface of the of the cen p eak lies
higher the innermost of secondary maxima On this basis what
ring that
that tral part area
of
to it
fails
optimization will
lunc
from remem
more
vides in
natural jected sub in
but goal same
Lunc ree high
umerical
Sim al
this wn ely
erly
hill
eac the
to
in to
in
ach that fact the bing clim hill of
w ould y ou predict the required n um b er of s to b e on a v erage or f
iterated simplex to lo cate cen tral p
Rep eat the same analysis as for Exercise ab o v e but in the con text of the
P test p roblem Are y our r esults in basic agreemen twtih T able Ho w can
y ou explain the dirences f an y
eak the
trials implex
EV OLUTION OPTIMIZA TION AND GENETIC ALGORITHMS
Biological ev olution
The general ideas of evolution and adaptation predate Charles Darwin On the
Origin of Sp ecies b y Means of Natural Selection but it is Darwin nd
more or less sim ultaneously Alfred Russell W allace who rst den i w is
considered to be primary d riving mec hanism of ev olution natur al sele ction
Nature is v m uc ho v ersubscrib ed A t a an ytiem in an y cosystem e far
more individuals are born than can p o surviv e en ecosystem a v ail
able resources This implies that man y mem besr of a en sp will die from
attrition or predation b e fore ha v e a c hance to d The of
natural selection states that individuals b e adapted to their en vironmen t i
for whatev er reason b etter at obtaining lunc h a v oiding b ecoming lunc h and nd
ingttractingomp eting for mates will on a v erage lea v e b more opring
than their less apt colleagues
F or natural selection to lead evolution t w o more essen tial ingredien ts are
required inheritanc e opring m ust retain at least some of the features that
made their paren ter than a v erage otherwise ev olution ectiv ely reset at
ev ery generation variability at an ygiv en time individuals of v arying nesses
m ust co exist in the p opulation otherwise natural selection h as nothing to op erate
on
Both these additional requiremen ts w ere plainly ob to Darwin and
con temp oraries but their underlying mec hanisms remained unexplained in
lifetime Ho w er this situation c hanged rapidly i n the early decades of the t w en
tieth tury and the p through whic h heredit y mediated and
v ariation main tained are n o w basically understo o d Inan the information
determining the gro wth and dev elopmen t of individuals is enco ded as linear se
quences of genes that can eac h assume a ite set of alues In sexual sp ecies
when t w o ndividuals i breed complemen tary p ortions of their genetic material are
passed on to their opring and com bined to dee that opring full genetic
mak That the inheritance part In the course of repro cessing the genetic
material to b e later passed on to opring cop y m istak es and t ruly random alter
ation of some gene v alues also o ccur o c casionally m utation ev en ts coupled
These
eup
utshell
is cesses ro primary cen
ev
their
his vious
is ts
to
ehind
tter
principle uce repro they
ecies giv
the giv ssibly
lmost ery
the
still hat tid
to the fact that an opring receiv es complemen tary genes from t w o arenp ts hic h
is true of most animals pro vides the needed source of v ariabilit y
The individual that mo v feeds and mates in real space can b e lo ok ed at as
an outer manifestation of its deing genes Think t hen of an i ndividual ness
as a function of the v alues assumed b y its genes What ev olution do es s i to riv d ea
gradual increase n i a v erage ness v alues o v er the course of an m y generations
is what Darwin called adaptation No w that b eginning to sound lik e illh limc bing
do esn it In fact ev olution do es not optimize at least not in the mathematical
sense of the w ord Ev olution blind Ev olution do es not e a ab out
globally maximal ness n d eplaise a T eilhard de Chardin Ev en if it did
ev olution m ust accommo date ysical constrain ts asso ciated with dev elopmen t
and gro wth so that not all paths p ossible genetic arameter space
ev olution do es is pro duce individuals of ab o v e v erage ness Nonetheless the
basic ideas of natural selection and inheritance with v ariation can be used
construct v ery robust algorithms or f global n umerical optimization
The po w ulativ e selection
The idea that natural selection lead to a of hill clim in ness
space y b ecome in tuitiv ely ob vious after thinking ab out it for a while
remains less ob vious is the degree to h cum ulativ e selection i selection
op erating on successiv e generations can accelerate w ould in its absence be
a random searc h of genetic parameter space The follo wing p opularized
in Ric hard Da wkins Blind W atc hmak er a book w ell w orth reading
inciden tally mak es for suc h a nice demonstration of this v ery poin t it
b y no w found its w a y in to at least one textb o ok on ev olutionary a y
nard Smith an excellen t d the topic Consider the
sen tence
JEG SNAKKER BARE LITT NORSK
This sen tence is c haracters including blank spaces and is up of
an alphab et of letters a blank c haracter included lease note that I am
taking in to accoun t the famous Scandina vian letters A and Consider no w
This is said without at all den ying that a large part of what mak es us who w e
are arises rom f learning and other in teractions with the en vironmen t in the course
of elopmen t and gro wth what genes de is some sort of b eha
Bauplan from whic h these higher lev el pro cesses tak e o
The original sen tence b y Da METHINK IT IS A WEASEL
whic h of course is tak en from Shak e Hamlet
are esp
LIKE is wkins used
vioral basic enco dev
is if
made long
wing follo to uction tro in
genetics
has that
The
example
what
whic
What ma
bing form can
cum of er
to
All in are
ph
damn giv is
This
es
the ro p cess of ro p ducing haracterong sen tences b y r
from the a v ailable c haracters of the alphab et Here an example
GE YT AUMNBGH JH QMWCXNES
Do esn lo ok m h lik e the original sen although careful comparison
sho wtath t w o letters actually coincide n um b er of d istinct haracter
long sen tences that can be made of a haracter alphab et
This is a very large n um b er ev en b y astronomical standards The corre
sp onding probabilit y of generating our st target sen tence b y this random pro cess
on the st trial is then suc h a n um ber that in
v oking the Dirt y Harry Rule at this poni t w be mo ot Instead consider the
follo wing pro cedure
Generate sen tences of randomly c hosen c haracters
Select the sen tence that has the most correct letters
Duplicate this b est sen tence ten times
F or eac h suc h duplicate randomly eplace r a few letters
Rep eat steps through un til the target sen tence been matc
This searc h algorithm incorp orates the three ingredien ts men tioned previously as
essen tial to the ev olutionary pro cess Step is natural selection in a
deterministic and rather extreme form the b est and only the best acts as
progenitor to the next eneration Step inheritance again of a
extreme form as opring start o as exact replicas of the ingle progenitor
Step is a sto c hastic pro cess whic h pro vides the v ariabilit y Note
that the algorithm op erates with minimal ness i nformation all it has a v ailable
w man y correct etters l a sen tence con tains but not which letters are correct or
incorrect What is still missing is exc hange of information b et w een trial solutions
but be patien t this will in due time
Figure illustrates the v olution of the b estf sen tence starting from
an initial ten random en s tences as describ ed ab o v e The m utation ate r w as set at
p meaning that an y giv letter a probabilit y of bengi sub jected
to random replacemen t Iteration coun t s i l isted i n t he leftmost column and error
in righ tmost column is deed here simply as n um ber of
letters in the bets sen tence generated the course of the curren t iteration Note
ho w the error decreases rather rapidly at st m uc h more wly later on it
es ab out as man y iterations to get the st letters righ t as it tak es to get
the last one The target sen tence is found after only iterations in the course
More precisely dee a mutation r as probabilit y p that a
giv en constituen t letter be replaced
randomly
the ate
tak
slo but
in
incorrect the Error the
has en
come
ho is
also required
rather is
since
fact in
hed has
ould
small is This
is out
total The
will tence uc
letters selecting andomly
Figure Accelerated Norsk l earning b y means of cum ulativ e selection Iteration
coun t is listed in the left column and the error deed as the n um b e r of ncorrect i
letters in he t righ tmost column The target sen tence is found after iterations
of whic h trial sen tences w ere generated and v aluated against the
This is almost initely than the of en umerativ e or random
searc h
Figure sho ws con v ergence curv es for three runs starting w ith the same initial
random sen tence ev under diren t m utation rates The solid is
the solution of Figure Note ho w the solution with the highest m utation rate
con v erges more rapidly at but ev en tually lev o at a ite nonzero
lev el What is happ ening here is that m utations are pro ducing the n eeded
letters as fast as they are destro ying curren tly correct letters en an et
size and sen tence length will alw a exist a critical m utation rate ab o v e
whic h this will happ en
There t w o imp ortan t things to remem ber at this poin t First m utation
This is in fact a notion cen to our understanding of the emergence of
life Among a v ariet y of selfeplicating olecules m of diren t l engths omp eting
for c hemical constituen ts limited supply in the primaev al soup those
closest to the critical m utation rate can adapt the fastest to an ev olving c
hemical
lying in
tral
are
ys there
alphab Giv
correct
error els st
line olving but
purely less
target
Figure Con v ergence curv es for he t sen tence searc h roblem p or f three diren t
m utation rates The curv es sho w the error c the best sen tence
pro d uced at h iteration The solid line corresp onds to the solution sho wn on
Figure
is a mixed blessing It is clearly needed as a source of v ariabilit y to o m uc h
of it is deitely deleterious Second the general s hap e of the con v ergence curv es
in Figure is w orth noting Con v ergence is rather swift at st but lev els
o This is a b eha vior w e again again in what follo ws Time no w
mo v e on ally o genetic algorithms
en vironmen t without selfestructing and so rapidly tak eo v er the soup see Eigen
for a comprehensiv e though somewhat dated review This is conjectured
be the explanation b ehind the ersalit yof the genetic co de among v
all living organisms
nearly ery univ
to
to
and meet will
then
but
eac
with iated asso
A basic genetic algorithm
F undamen tally genetic are a class of searc h tec hniques that use sim
plid forms of the b iological pro cesses of selectionnheritance ariation
sp eaking they are not optimization ds p but can be used to form the
core of a class of robust and xible ds kno wn as genetic algorithm ase d
optimizers
Let go ac b k to a generic optimization problem One s i giv en a mo del that
dep ends on a of parameters u and a functional relation f u that returns
a measure of qualit y or ness asso ciated with the corresp onding mo his
could be a yp e goodesns of measure the mo del is compared to data
example The optimization task usually consists ding the oin t u in
parameter space corresp onding to the mo that maximizes the ness function
f u Dee no w a p opulation as a set of N realizations of the parameters u A
p
topev el view of a basic genetic algorithm is then as follo ws
Randomly initialize p opulation and ev aluate ness of besr
Breed selected mem b ers of curren t p opulation to pro duce opring p opulation
election based on ness
Replace curren t p opulation b y opring p opulation
Ev aluate ness of new p opulation b
Rep eat steps through un til the test mem b r e of the curren t p opulation
is deemed enough
W ere not that what it b eing cycled through the iteration is a p opulation of
solutions rather than a s ingle trial olution s this w ould v ery m uc h smell of iterated
hill clim bing It should also giv e y ou that uncann y feeling of d ej a unless y our
memory is really shot or on y ou y ou ha v e ed o v er the
section The crucial v elt y lies with Br e e ding is in the course of
breeding that information is passed and hanged across p opulation mem b
Ho w this information transfer t ak es place s i r ather p eculiar and merits discussion
in some detail and only b ecause this genetic algorithms justify the
enetic in their name
Figure illustrates the breeding pro cess the con of a simple D max
imization problem suc h as the P or P test problems In an individual
is a x y p oin t and so is eed b yt w o oating p oin tn um b e rs The st step
is o de the t w o ating p oin t n um bers deing h individual selected
breeding Here this is done b y dropping the decimal p oin t and concate
nating the resulting set of simple decimal n i tegers i n a c hromosomelik e
ines on Figure Breeding prop er a t w o step pro cess The st step
is
string to
simply
for eac enc to
case this
text in
where is not
ers exc
It step no
preceding skipp shame
vu
it
ers mem
mem its
del
in
for if
del
set
metho
se er metho
Strictly
algorithms
is ossover t w o strings generated b y the enco ding pro cess are laid side b y
side and a cutting p oin t is randomly selected along the length of the
strings string fragmen lo cated righ t of the cutting p oin t are then in ter
c hanged and spliced on to the fragmen ts originally lo cated left of the cutting p oin t
ines for a cutting poin t lo cated bet w een the third fourth
digit The second breeding step is mutation F or eac h string pro ucedd b y the
crosso v pro cess a few randomly selected digits r enes are b y a
new randomly selected digit v alue ines for am utation hitting the ten th
digit of the second opring string The resulting fragmen ts then deco ded
in t w o x y pairs whose ness then ev aluated here simply b y
the function v alue f x y
Some additional commen ts are in order First n ote that opring incorp orate
in tact h unks of genetic material coming from b oth paren ts that the needed
inheritance as w ell as the promised exc hange of information bet w trial solu
tions Ho w ev er both the crosso v er m utation op erations also in v olv e purely
sto c hastic comp onen suc h as the c hoice of cutting poni t of m utation and
new v alue of m utated digit This is w e get the v ariabilit y to sus
tain the ev olutionary p ro cess as discussed earlier Se c ond d
pro cess illustrated on Figure is just of man y possible suc h sc hemes T ra
ditionally genetic algorithms ha v e use of binary enco ding this often
not particularly adv an tageous for n umerical optimization use of a
genetic lphab et is no more artiial than a inary b represen tation ev en so
giv en that v ery nearly all kno wn living organisms nco e de their genetic information
in a base alphab et fact in terms of enco ding ating oin t n um b e rs b oth
binary and decimal alphab ets sur from signian t shortcomings can
the p erformance of the resulting optimization algorithms Thir d the crosso v
and m utation op erators op erating conjunction the d
pro cesses as illustrated on Figure preserv e total range parameter space
That is if the ating oin t parameters deing paren t solutions are restricted to
the range then the opring solution parameters will be restricted
to This is a v ery imp ortan tpropter y through whic h one can rtlessly e
hardwire constrain suc h p ositivit y F ourth ving the m utation op erator
act on the enco ded form of paren t solution in teresting consequence
that opring can dir v m uc horv ery little from their p aren ts dep ending on
whether digits acted b y m utation eco d de in to one of the leading or
digits of the corresp onding ating oin tn um ber This means that rom f the p oin t
of view of parameter space exploration a genetic algorithm can carry out b oth
wide exploration and e tuning in parallel Fifth it es t w o paren ts to pro
duce im ultaneously t w o opring One can of course devise orgiastic
hemes that in v olv e more t w o paren ts yield an y n um ber of opring
and than sc
breeding
tak
trailing the
ery
the has the
ha as ts
also
in the
ding ingeco enco with in
er
act that
In
more
decimal The
is but made
one
ding ingeco enco the
needed where
site ts
and
een
computing is to
are
replaced er
decimal and
ts The
deing
The cr
Figure Breeding in genetic algorithms Here the pro cess illustrated in the
con text of a D maximization problem uc h asPrPf x An individual is
an x y p oin t t w oscu h t individuals are needed for breeding enoted
P and P here one oint cr ossover and one oint mutation op erators
act on string represen tations of the paren ts and S to pro duce opring
strings S and S whic h are ally deco ded i n t w o opring x y p oin
P and P
Exp erience sho that this rarely impro v es the p erformance of the resulting algo
rithms Sixth f u m ob viously be computable for all u but necessarily
not ust
ws
ts to
The
paren and
is
diren tiable since deriv ativ es of the ness function with resp ect to its i nput pa
rameters not required for the algorithm to op erate F rom a practical p oin t of
view this can be a great adv an tage
Information transfer in genetic algorithms
Time to step bac k and revisit the issue of information pro cessing Genetic
algorithms ac hiev e transfer of information through the breeding of trial solutions
selected on the basis of their ness whic his wh y he t crosso v er op erator is usually
deemed to be the deing feature of algorithms as compared to
classes of ev olutionary algorithms ee e B ac k
The join t action of crosso v er and nessased selection on a p opulation of
strings enco ding trial solutions is to increase the o currence c frequency of sub
strings that con v ey their trial solution ab o v e v erage ness at a rate
prop ortional to dirence bet w een the a v erage ness of trial solutions includ
ing that substring in their enot yp e the stringnco ded v ersion of
deing parameter set and a v erage ness of the whole p opulation The
mathematical expression of the preceding mouthful adequately expanded tak e
in accoun t the p o ssibilit y of substring disruption b y crosso v er or m utation is
kno wn as the Schema or and is originally due Holland see
Goldb erg As the p opulation ev olv es resp onse to breeding ness
based selection adv an tageous substrings are con uously sorted and om c bined b y
crosso v in to individuals leading to an inexorable ness increase in the
p opulation as a whole Because this in v olv es the concurren t c of a great
man y distinct substrings Holland ed this prop ert y intrinsic p ar al lelism and
argues that therein fundamen tally ies l he t exploratory p o w er of genetic algorithms
The least y ou should ber Section
Natural selection alone cannot lead to ev olution inheritance and v ariation are
also needed
Cum ulativ e s election can accelerate an otherwise random searc h pro cess b ya
factor that astronomically enormous
Genetic Algorithms searc h tec hniques that e use of simplid forms
of the biological selectionnheritance ariation triad
Exercises for Section
All exercises for hist part of the tutorial aim at etting l y ou explore quan titativ
the probabilistic asp ects of sen tence searc h example of x
the
ely
mak are
is
from remem
dubb
essing pro
single er
tin
and in
also to em The
to
to
the
their
all
ded deco
other genetic
are
First some asic b probabilit y calculations to w arm up what is the proba
bilit y of getting l of the l etters w rong on an initial r andom trial getting
at least one etter l n y etter l righ t getting exactly one l etter n y letter
righ t
In the run of Figure it to ok iterations to get the poni t of ha
correct letters What is no w the probabilit y of obtaining a
fully correct tence in one of the ten m utated copies after the subsequen t
iteration What is the probabilit y of al l m utated copies ha ving regressed
only correct letters
Giv the sen tence S alphab et size A and a m utation
rate p obtain an estimate not a formal calculation the n ber of
iterations required on a v erage to reac h zero error Ho w do es y estimate
compare Figure Do y ou think that Figure a t ypical solution
Giv again a sen tence length S an alphab et size A and a m utation rate p
calculate the error lev at whic h the sen tence searc h algorithm saturate
ik eteh dotted line on Figure Use this result o t estimate an optimal m u
tation rate as a function of S and A that will on a v erage lead to con v ergence
in the smallest p ossible n um ber of iterations
In terms of an analogy biological ev olution what do y ou think are the
most signian t failings of the sen tence searc h example
for
will el
en
is to
our
um for
length en
to
sen
of out
ving to
al
PIKAIA A GENETIC ALGORITHM
F NUMERICAL OPTIMIZA TION
Ov erview problem deition
In this section w e will be primarily concerned with the comparison of genetic
algorithmased optimizers with other global optimization sc hemes sp ecially
iterated hill clim bing using simplex metho d x T o w e st need
settle on a sp eci implemen tation a genetic algorithm
PIKAIA is a p ublic omain d general purp ose genetic algorithmased optimiza
tion subroutine is written in F TRAN is completely selfon tained and
is designed to be as easy to use as the optimization subroutines found in
et al Numeric R e cip or example their simplex routine amoeba It comes
with limited I capabilities and no fancy graphics The w is describ ed in
great detail in the User Guide to PIKAIA harb onneau Knapp
hereafter PUG whic h n umerous references made in what follo ws The
soft w are and User can both be obtained from the PIKAIA W eb P age
httpwwaocar du publ ic r es earc h i ikai a to u ria lt ml
This section op ens a brief o v erview of the op erators and tec hniques
included in PIKAIA In ternally seeks to maximize a usereed function
f x in a b ounded n imensional space i
x x x x k
n k
The restriction of parameter v alues in the range allo xibilit y
and p ortabilit y across problem domains This ho w ev er mplies i that the u ser m ust
adequately normalize the input parameters of the function to be maximized with
to those b ounds
The maximization s i carried out on a p opulation up of N individuals
p
rial solutions This p opulation size remains ed throughout ev
Rather than olving p opulation un some tolerance criterion is satisd
PIKAIA carries the ev olution o v er a u sereed preset n um b r e o f generations N
g
til the ev
olution the
made
ect resp
greater ws
PIKAIA
with
Guide
are to
are soft
es al
Press
OR It
of
to so do the
and
OR
PIKAIA ors the ser u the xibilit y to sp ecify a n um b r e of other input param
eters that con trol the b eha vior of the underlying genetic algorithm The subroutine
do es include builtn default settings that ha v e pro v en across problem do
mains All suc h input parameters passed to in the imensional
con trol v ector Section of the PUG for allo w ed default v
of those con parameters
The topev el structure of PIKAIA the same as the sequence of algorithmic
steps listed in x an outer lo op trolling generational iteration and an
inner lo op con trolling breeding Since breeding in v olv the d of
opring the inner lo op executes N times p er generational iteration where N
p p
is the p pulation o size N is the default v alue
p
All parameter v alues deing the individual mem b ers of the initial p opulation
assigned a random n ber the range extracted from a uniform
distribution of random deviates ee x of the PUG This ensur es no
bias whatso ever is intr o duc e d the initialization
Minimal algorithmic comp onen ts
Selection UG x
PIKAIA uses astoc hastic selection pro cess to assign to eac h individual n i the p op
ulation a pr ob ability of being selected for breeding Sp ecially probabilit y
is made linearly prop ortional the nessased r ank of eac h individual within
the curren t p o pulation This is carried using a heme kno wn as the R oulette
Whe el A lgorithm as detailed x of the PUG ee also Da vis c hap
Note that general it is a go o d idea to e selection probabilit y
prop ortional to ness value as often leads to a of selection pressure
late in the ev olutionary once p opulation b ers v e ound the
global optim um some it can also lead early on to a up e
b eing selected so frequen tly that the p opulation b ecomes degenerate through the
computational equiv alen t of breeding The prop ortionalit y constan t bet w een
nessased rank and selection probabilit y is sp ecid as an input parameter
PIKAIA The default v alue is
Breeding UG xx and
Once t w o individuals ha v e b een selected breeding pro ceeds exactly as in Figure
The enco ding pro cess requires one to sp ecify the n ber of digits to be retained
in the enco ing d pro ess c a userp ecid quan tit y whic h is set to in
calculations rep orted up on here his also the default v alue in PIKAIA Tw o
additional quan tities need to be sp ecid the crosso v er rate whic h sets the
is
all is this
um
to
in
rindividual cases In
ha mem most run
loss this
directly mak not in
in
sc out
to
that
by
initial that
in um are
two uction pro es
the con
is
trol
alues and the See ctrl
PIKAIA are
robust
probabilit y that the crosso v er op eration actually tak es place efault
the m utation rate whic h s ets the probabilit y for e m up
string of an opring that a m utation tak es place at that lo cation
is
P opulation replacemen t UG x
Under PIKAIA default settings t he opring p opulation is accum ulated in to tem
poaryr storage and once n um ber of suc h opring equals of the curren t
breeding p pulation o the latter is deleted and replaced b y the opring p opulation
This is the default strategy used b y PIKAIA although it is p ossible for the
sp ecify other p opulation replacemen t tec hniques ee PUG xx
Additional comp onen ts
The comp onen ts listed ab o v e a minimal genetic algorithm Suc h an
algorithm can be used for n umerical optimization but as w e so on see
out to be far from optimal what follo ws w e refer to this algorithm as GA
The ollo f wing t w o imple s additions o t A G lead to an algorithm to b e referred to
as GA that ac hiev es far b etter p erformance on n umerical optimization problems
So m uc h b etter in fact that use of these t w o additional comp onen ts is the
default c hoice in PIKAIA
Elitism UG x
This simply consists in storing a w a y the parameters deing the test mem ber
of curren t p opulation and later cop ying tact the opring p
This represen ts a s afeguard gainst a the p ossibilit y that crosso v er andr m utation
destro y he t curren t b est solution hic w hw ould a h v e a go o d c hance of unnecessarily
wing do wn the optimization pro cess Elitism in b comes e essen up on
in tro ucing d our second impro v emen t to GA
V ariable m utation rate UG x
This one is p erhaps the single most i mp ortan t mproi v emen t t hat can nd should
be made to GA As discussed x m utation is v ery m uc h a mixed blessing
F or those of y ou t w an t to run PIKAIA to repro duce the
b elo w GA is pro d uced b y explicitly setting the follo elemen of PIKAIA
con trol v ector ctrl ctrl ctrl and all other e lemen ts of ctrl
negativ e v alues to activ ate default options
This means initializing al l elemen of the trol v ector negativ e
v alues Note that this sets a p opulation size equal to ia ctrl and a
n ber of generations equal to ia ctrl
um
to ctrl con ts
to
ts wing
results migh who
in
vital
tial fact slo
opulation in in it the
the
In
turns will
dee
to user
that the
efault digit
deing the aking digit ach
is
it vides the m uc h needed source of v ariabilit y through h no v el parameter
v alues are injected i n to the p opulation Ho w ev er it also leads to the destruction of
go o d solutions This w as precisely the p oin t of igure F dotted line Finding the
exact v alue for the m utation ate r t hat ac es optimal balance b t e w een those t w o
ects to maximize t he former while minimizing the latter is of course p ossible
Ho w ev er in doing ds that the optimal parameter settings often end up
b eing highly problem dep enden t
One po w erful solution to this problem to dynamically adjust the m utation
rate The k ey to this strategy lies with recognizing that as long as the p opulation
is broadly distributed in parameter space the crosso v er op erator leads t o a prett y
eien t earc h as it recom bines fragmen ts of existing solutions Ho w er once
the p opulation has con v hether on a secondary or absolute optim um
crosso v no longer ac hiev es m uc h as it leads to hange of fragmen ts that
are early n iden tical since all paren ts ha v e n early iden tical parameter v alues
ob viously is where a high m utation rate is needed to reinject v ariabilit y in to the
p opulation
Consider then the follo wing pro cedure A t an y giv en time k eep trac k of the
ness v alue of t he test p opulation mem b er and of t he median rank ed mem ber
The ness dirence f bet w een those t w o individuals is clearly a measure of
p opulation con v ergence if f large the p opulation is presumably distributed
more broadly in parameter space than if f is v Therefore if f
b ecomes to o small increase the m utation rate if it besomec to o large decrease
the m utation rate again This ho w PIKAIA dynamically adjusts m utation
rate during runime This strategy represen a s imple of selfdaptation of
a arameter p con trolling the b eha vior of the underlying genetic algorithm F
details and implemen tation issues are discussed x of the PUG
A case study GA on P
It will pro v e u seful to st tak e a etailed d lo ok at the beha vior of the genetic
algorithm in the con text of a s imple problem Figure sho ws ten con v ergence
curv es for GA w orking on P with N What plotted is min us the
p
In fact this often done b y letting the m utation rate nd other con
parameters of the algorithm ev e under trol of a second higher lev
genetic algorithm with ness b eing t hen deed as the p erformance of the genetic
algorithm deed b y t hose parameters on the p roblem under consideration Prett y
cute but as y ou migh t i magine ather r time consuming
Just as the optimal m utation y ou op efully w ork out in Problem
of x is rather sensitiv ely d ep enden t on the sen tence length and alphab e t size
ed rate
el con the olv
trolling is
one is
in
urther
form ts
its is
small ery
is
This
exc the er
erged
ev
is
one so
hiev
whic pro
ness v alue of the test individual v ersus generation coun t for separate
of GA Figure should be compared to Figure the con v ergence
of the simplex on the same problem Early on the curv es ha v e qualitativ
similar shap es either con v ergence o ccurs relativ ely uicq kly h more quic
for simplex when it do es con v erge or solutions remain k on one of the
rings of secondary extrema f Fig whic h leads the error lev o at a
ed v alue Unlike simplex however GA is able to pul l itself o the se c ondary
extr ema rings It do es so primarily through m utation although crosso v er b et w een
t w o paren ts prop erly p ositioned in parameter space can ac e the ect
Mutation b eing a fundamen sto c hastic pro cess is then surprising
see diren t runs requiring diren t generation b e needed
fa v orable m utation tak place
Clearly m utation pla ys a c ritical role here Figure sho ws the nesses of
the b est olid line and medianank ed ashed ine l individuals in the p opulation
as a function of generational t for the GA run plotted a thic k er line
on panel The otted d ine l sho ws the v ariation of the m utation rate Figure
sho ws the distribution of the p opulation in D parameter space at the ep o c hs
indicated b y solid dots Fig
T o start with note on that no individual in the initial random
p opulation has landed an ywhere close enough to cen tral p eak for hill clim
w ork The rst few generational iterations see the p opulation cluster itself closer
and loser c to en c ter ig but the ness dirence b t e w een b est and median
is still quite large The m utation decreases sligh tly rom f i ts initial lo w v
th
then remains constan t By the generation Fig ost m of the p opulation
has con v erged somewhere inner ring of secondary extrema f
that the nesses of the bets and median are no w comparable This leads to a
th
sharp increase of the m utation rate et w een the and generations The
high m utation rate results in opring begin kno c k ed all o v er parameter space
in the course of breeding ig some m utan t individuals do land
regularly on the slop e of cen p it is only b y the generation that
one suc h m utan t is catapulted high enough b ecome the test of the curren t
p opulation ig F urther breeding during subsequen t generations
more and more individuals to the cen tral p eak and further increases ness of
Y ou migh t notice that GA already starts o doing signian tly b e than
the simplex metho d this merely results from the initial random p opulation of
GA ha ving ampled poni ts parameter space compared only for the
simplex
An animation of the ev olving p opulation for this solution can be ed on
the T utorial W eb P age
view
to in
tter
in
brings
to
eak tral the
th
While
th
so
the on
but alue
to
bing the
Fig
on
with coun
es
the fore ts coun GA
to not it tally
same hiev
eling to
tuc
kly uc
ely
wing sho
runs
Figure P anel sho ws con v ergence es for distinct of GA on
P As b efore the error is deed as f x y P anel B sho ws for the single run
plotted with a thic k er line on panel t he v ariations with generation coun tof hte
bets individual of the p opulation olid line medianank ed individual ashed
line m utation rate otted line
and
runs curv
Figure Ev olution of the p opulation of trial s olutions parameter space
the GA run sho wn as a thic k er line on Fig The concen tric circles indicate the
rings of secondary maxima and the l arger solid blac k d ot is the test solution of
the curren t generation
for in
the curren t b est via b oth crosso v er and m utation ig Note ho w litism e is
essen tial here otherwise the utan t ha ving landed on the slop es of the cen tral
p eak w ould ha v e a w lik ielhood of replicating itself tact in to the subsequen t
generation in view of the high m utation rate
GA basically b eha v es in exactly the same w a y w ith the imp ortan t exception
that man y more generations needed for the v orable m utation to sho w up
this is b e cause GA op rates e w ith a ed lo wm utation rate G A
rate v ary dep ending on the degree of con v ergence of the p opulation f x
Hamming w alls creep m utation
W e are doing prett yw ell ith w GA but w e still need to correct a fundamen tal
shortcoming of the one oin tm utation op erator arising from the ecimal d enco ding
heme of Fig Consider a problem where sough tfter optimal solution
requires the follo wing substring to be pro duced b y the ev olutionary pro cess
deco ding in to the ating p oin tn b er no w early n i the ev olutionary run
an individual ha ving sa y
will lik ely b e ter than a v erage and so this genetic material ill w spread through
out the p opulation After a while follo wing v orable m utations or crosso v
recom binations the substring migh t lo ok lik e sa y
whic h is admittedly quite close to Ho w ev er t w o v w ell co ordinated
m utations are needed o t p ush t his to w ards the target ust m utate
a the st o a Note hat t either m utation o ccurring in isolation
andr m utating to a diren t v alue tak es us farther from the t arget ating
p oin t n b er Mutation benig a slo w pro cess probabilit y of the needed pair
of m utations o ccurring sim ultaneously will in general b e quite s mall meaning that
the ev olution w ould ha v e t o b e pushed o v er y generations for it to app h en The
p opulation is getting iled at in ternal b oundaries of the enco ding system
These b oundaries are called Hamming wal ls They can b e b ypassed b yc ho os
ing n a enco ding sc heme suc h that successiv esiglne m utations can alw a ys to a
con uous v ariation in the deco ded p arameter This is wh y the soalled Gra y bi
nary co ding Press et al x is no w used almost univ ersally in genetic
algorithms based on binary enco ding Another p ossibilit y is to devise m utation
op erators that can jump o v er Hamming w alls
tin
lead
up
man
the um
digit
and to
the
ery
er fa
um
the sc
and
this lets while
fa are
in lo
Cr e ep mutation do es precisely this Once a digit on the enco ding string
b een targeted for m utation instead of replacing the existing digit b y a
randomly c hosen one just add either or ith equal probabilit y and if
the resulting digit ecause a has b een with or bauecse
a has b een hit carry the one o v er to the next digit on the
Just lik e in grade sc ho ol So example creep m utation hitting the middle
with in the last substring ab o v e w ould to
whic h ac hiev es the desired ect of umping the w all
The one thing creep m utation do es not wis to akt e l arge jumps in p arame
ter pace s As argued b fore e j umping is actually a n eeded apabilit c y consequen
in practice for eac h opring individual a probabilit y est t will decide whether one
p oin t or creep m utation i s to be used ith equal probabilities
Creep m utation i s included the original release of PIKAIA o w kno wn
as PIKAIA although it in v ersion h b een released in
ee the PIKAIA W eb P age and the Release PIKAIA NCAR
T ec hnical Note STR results describ ed follo ws w ere obtained
using a mo did v ersion of PIKAIA G A whic h includes m
is otherwise iden tical GA
P erformance on test problems
Time w turn lo ose our algorithms on the suite of problems x
W e ha v e three v ersions of genetic algorithmased optimizers GA whic h rep
resen ts a minimal algorithm a nd GA whic h tical to GA includes in
addition elitism and dynamic adjustmen t of the m utation rate GA includ
ing creep m utation otherwise iden tical to GA As a comparison algorithm
w e r etain iterated hill clim bing using the simplex metho d as describ d e in x As
will b ecome eviden t GA is actually not a v ery go o d optimizer so that the
more in teresting comparison will b e among GA GA and iterated simplex
Before getting t o o carried a w a y l et ause p and rect on what w e to
ac hiev e here Ideally ne w an ts a metho d t hat ac es con v ergence to
optim um with high probabilit y p sa y while requiring smallest
G
p ossible n um ber of mo del r function ev aluations doing so This latter p oin t
can b ecome a dominan t constrain t dealing with a application where
ev aluating the tness of a en trial solution computationally in tensiv e
Consider the helioseismic in v ersions describ ed in Charb onneau et al
using a genetic algorithm giv en a set of parameters deing a trial solution ness
is giv
real when
in
the
global the hiev
trying are
on so
but
and
but iden is
of test to no
to
but utation creep
what in The
for Notes
April has whic is
in not
tly
allo
lead
for
left with
hit is
simply
has
Suc h considerations are easily quan tid Let N and N be the p opulation
p g
size and generation length a the required n ber of function ev
N is ob viously
f
N N N A GA GA a
f p g
while for iterated simplex N is the n um ber of hill clim bing trials N times the
f t
a v erage n ber fo function aluations r equired b y a single simplex run N
s
quan tit y is run and problemep enden t
N N N terated simplex b
f t s
So w e pla y the follo wing game w e run iterated simplex and GA increas
ing n um b ers of generationsterations and c hec k whether global con v ergence is
ac hiev ed to get statistically meaningful results w e times for eac h
metho d and eac h generationteration coun t allo ws to empirically estab
lish the probabilit y of global con v ergence p as a function of genera
G
tionteration coun t In doing so to decide whether or not a giv en h as
con v erged w e use again the f for P P P and R
P The results of his t p ro cedure applied to eac h t est problem is s ho wn in Figure
It should be easy to con vince y ourself of the follo wing on P and P
b oth iterated implex s and GA p erform equally w ell on all asp cts e of p e rformance
when pushed long enough to ha v e p P is a and
G
tec hnique p erforms satisfactorily on it Still GA outp erforms iterated
simplex on global p erformance P GA iterated simplex do
w ell up to p but then GAs p erformance starts to lag b ehind as the
G
solutions are pushed to p
G
An ob vious conclusion to be dra wn at this juncture is that iterated
clim bing using the simplex metho d es for a prett y decen t global optimiza
tion sc heme Not quite y ou w ere exp e cting as a sales pitc h for genetic
algorithmased optimization righ t This is in a consequence of the rel
ativ ely lo w dimensionalit y of our test problems Recall x that iterated
simplex leads impro v ed p erformance ith resp e ct to single run simplex pri
marily as a consequence of the b etter sampling of arameter p space asso ciated with
the initial andom distribution of simplex v ertices giv en enough trials one is
almost guaran teed v e one initial simplex v ertex landing close enough to the
ev aluation in v olv es the construction of a rotation curv e a l arge matrix
v ector m ultiplication the calculation of a against some data p oin
This adds up to ab out half a PUecond C on a Cra y J All test p roblems of x
require v little computation in comparison
ery
ts
ha to
to
from
part
what
mak
hill
equally and On
largely
neither problem hard
for and criteria
globally run
us This
this do
for
this ev um
aluations um run of
Figure Global con v ergence robabilit p y as a function of the n um b er of unction f
ev aluations N required b y iterated simplex iamonds and GA olid dots on
f
the four test problem dotted line P dashed line P solid line P dash
dotted line The probabilities w ere estimated from distinct trials and in he t
case of iterated simplex N is an a v erage o v er the trials
f
global maxim um to ensure subsequen t global con v ergence w dimensional
searc h spaces iterated simplex th us ends up bengi quite comp e titiv e Figure
already indicates that dge do es not carry o v er to higher dimensionalit y
ompare results for P and P
GAs p erformance on P is actually a delicate matter T ak e another lo ok
at Figure consider what happ ens the p opulation has v erged to the
broad secondary maxim s do es early in the run for nearly ev ery single
trial for m utation to prop el a olution s from x y to the n arro wpake
x y t w o v w co ordinated m utations m ust e place sim ul
taneously otherwise m utan t solutions up regions of rather lo w elev ations
and do not con tribute m uc h the next generation a w probabilit y
o ccurrence ev en at relativ ely high m utation rates the pro c ess tak es
GAs global p erformance on P then results from an in terpla ybet w een one oin t
Notice on Figure ho w f ew solutions sho w up n i the corners of he t
time so
lo is This to
in end
tak ell ery
it um
con once and
this
lo In
m utation the rather direct relationship that exists bet w een a solution
deing parameters and its string represen tation on hicw hm utation and c rosso v
op erate If the narro w Gaussian is cen tered on x y then a single
m utation can prop el a solution from the broad cen Gaussian to the narro w
one Not surprisingly on this mo problem GA outp erforms iterated sim
plex to a signian t degree p with only N i faster than
G f
iterated simplex b y a factor of Enco ding is a tric ky business with p
fareac hing consequences p erformance
Iterated simplex sup erior p erformance on P is certainly notew y y et
rects in part the p eculiar structure of parameter space deed b y the Gaussian
ting problem whic h is relativ w accommo dated b y the simplex metho d
pseudolobal capabilities Other lo optimization metho ds do not
as w ell F or a detailed comparison of genetic algorithms and other metho d s on
ting Gaussian proes to real thetic data see McIn tosh et al
It is really only with v ery hard problems suc h as P that GA starts sho
w orth By an y s tandards P s i a v ery hard global optimization roblem p While
on its D v ersion GA nd a iterated simplex do ab out as w ell as dimensionalit yis
increased the global p erformance of iterated simplex egrades d m uc h
than GA This is in fact the po w er of algorithmased optimizers
lies although for searc h spaces of high dimensionalit y n s a y the one oin t
crosso v m utation op erators describ ed in x usually sub optimal and
m be impro v ed up on
In some sense a fairer comparison of the resp ectiv e exploratory capabilities
of iterated can be carried b y setting n um ber of trials
in iterated simplex so that original distribution simplex v ertices samples
parameter space with same densit y as GAs initial random p opulation in
other w ords using the notation of w e
N N n
t p
where n is the dimensionalit yof parameter space and compare results of the
resulting iterated simplex to some tandard GA and GA runs Suc h a
comparison is presen in T able II in a format essen tially tical to T able I
P erformance measures also listed for a set of GA runs extending o v er the
domain
PIKAIA to be released in April includes a t w o oin t crosso v er op
erator whic h generally impro v es p erformance for problems in v olving man y pa
rameters See e Section fthe Release Notes for harb onneau
PIKAIA
are
iden ted
runs
the
set eqs
the
of the
the out simplex and GA
ust
are and er
genetic where
rapidly more
its
wing
syn and
nearly fare cal
ell ely
orth
on
tially oten
did
tral
er
here and
T able II
P erformance on test problems ith eq enforced
T est Problem P erformance Iter Simplex G A GA GA
P h f i
p
G
h N i
f
N N
t g
P h f i
p
G
h N i
f
N N
t g
P h f i
p
G
h N i
f
N N
t g
P h R i
p
G
h N i
f
N N
t g
same n um ber of generations as the GA and GA Once again p e rformance
measures are established on the basis of distinct runs for eac h metho d
Eviden tly GA and GA outp erform GA on all asp e cts of p rformance e to a
staggering degree GA is not m uc h of a global n umerical optimization algorithm
Comparison with T able I sho ws that its global p erformance exceeds somewhat that
of the s implex metho d n i singleun o m de but the n um b er of unction f v e aluations
required b y GA to ac hiev e orders magnitude l arger
What is also lainly p eviden ton T able II i s the degree to whic h GA GA
outp erform iterated simplex for a given level of initial sampling of p ar ameter sp ac e
Although the n ber of function ev aluations is t ypically an order of
magnitude larger b oth algorithms are far b etter than iterated simplex at activ
exploring parameter space This is plain evidenc efro the p ositive e cts of tr ansfer
of information b etwe en trial solutions in c ourse of the se ar ch pr o c ess
The w orth of creep m utation can be ascertained b y comparing
global the
the
ely
required um
and
of is this
runs
or
or
or
or
p erformance of the GA and GA solutions results are not clearut
do es b etter than GA on P alttewPil orse on P and signian w orse
on P The usefulness of creep m utation is con tingen t on there actually b
Hamming w alls in vicinit y of the global solution if there are creep m utation
helps sometimes uite q a bit Otherwise ectiv ely d ecreases probabilit yof
taking large jumps parameter space and can be deleterious cases
This is what is happ ening here P where mo ving a w a y the secondary
maxim um requires a large in parameter space to tak e place x y
to
A tan y rate the ab o v e iscussion d amply llustrates i the degree to whic h glob al
p erformanc e oblemep endent This cannot be o v eremphasized Y ou should
certainly b ew are of an y empirical comparisons b et w een v arious global optimization
metho d s that rely on a small of test problems esp cially e of w dimensional
it y Y ou should also k eep in mind that GA one sp eci instance of a genetic
algorithmased optimizer and that other incarnations ma y beha v e diren tly
ither betert or w orse the same test problems
The least y ou should ber Section
Through random initialization of the p opulation genetic algorithms in
no initial bias whatso ev er in the h pro cess
F or n umerical optimization elitism and an adjustable m utation rate are t w o
crucial additions a genetic algorithm
Iterated hill clim bing using t he simplex metho d m ak es a prett y ecend t
optimization t ec hnique esp ecially for lo wimensionalit y problems
P erformance measures of an y global optimization metho d are ighly h problem
dep enden t
Exercises for Section
Lo ok bac k at Figure The dynamically adjusting m utation rate lev els o
at a v alue of ab out One could ha v e predicted a v erage v alue b efore
running the co de Ho w in t reead x
Co de up a D D and D v ersion of P in its GA form
efault settings except for generation coun t in v estigate ho w global p erfor
mance degrades with problem dimensionalit y Keep the generation oun c etxd
at ctrl Ho w do es compare to iterated simplex
this
PIKAIA Using
this
global
basic to
searc
duce tro
from remem
on
is
lo set
pr is
from jump
from with
some in so in
the it
the
eing
tly and
GA The
A REAL APPLICA TION
ORBIT BINAR Y ST ARS
Binary stars
More than half all stars observ ed in the solar neigh b orho o d are comp onen
of binary systems This is presumed to be a consequence of angular momen tum
conserv ation leading to fragmen tation in the later tages s of collapse of protostellar
clouds Some binary stars can b e resolv ed optically with ev en a s mall telescop e the
st suc h visual binary w as disco v ered b y G Riccioli Binarit y
can also be established sp ectroscopically y measuring the small Doppler shift in
narro w sp ectral lines caused b y the comp onen t V along linefigh t of the
orbital v elo cit y ab out the system ter of mass F or nonelativistic
sp eeds the w a v elength shift is
V
c
where c is the sp eed of ligh t The st suc h e ctr w as disco v ered
in b y E Pic k ering Curren t hardw are and analysis tec hniques no w allo w
us to measure tellar s radial v c y w ith useful accuracy do wn to p r e
second This lev el of accuracy is what has made p ossible the t sp ectacular
disco v ery of extrasolar planets Figure sho ws radial v c it y measuremen ts of
the s tar Bo otis a lassical sp ectroscopic b inary star F rom these data one can
determine the orbital parameters of the system
Am usingly sp ectroscopic binary the brigh ter comp onen t of the
visual binary disco v ered b y Riccioli the star Mizar A in the constellation
Majoris Ev en b tter e w as later realized that Mizar B is a sp ectroscopic
binary
This ure is for bac k in this er w as originally
it w as giv en as m s Prett y remark able impro v emen t a little o v
three y ears
er just in
written pap when
also it
Ursa
st is this
elo
recen
meters it elo
binary opic osc sp
orbital cen
the
in system
ts of
OF ELEMENTS AL
Figure Radial v c y v ariation observ ed in the sp ectroscopically visible
comp onen t of the binary star Bo axis is giv units of Julian
Date ne JD one solar y are from Bertiau with one lone
datum at JD on this plot solid is the b estt solution
obtained later section The asymmetrical shap e of curv e is due to the
eccen tricit y of the orbit a circular orbit w to a purely sin usoidal
v elo cit yv ariation
Radial v elo cities and Keplerian orbits
If the shap e and size of an orbit are kno wn as w as orien tation of its semi
ma jor axis with resp ect to the l ine of s igh t the exp ected radial v cit yv ariations
can b e computed and compared to observ ations There are actually a few subtleties
in v olv Determining the radial v c y v ariation asso ciated the motion of
a b inary comp onen t in an arbitrarily p ositioned elliptical orbit ab out the common
cen of mass of the s ystem i s a straigh tforw ard b ut somewhat messy problem in
spherical trigonometry The pro cedure is in great gory details in man y
astronomical monographs ee e mart S W e shall simply write d o wn the
resulting expression here
V t V K os v t e cos
out laid
ter
with it elo ed
elo
the ell
radial lead ould
the this in
line The missing
Data da
in en time The otis
it elo
The quan tit y V is the radial v c y of the binary system cen of mass and
eq only holds once the Earths orbital motion ab the Sun as been sub
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