Maranas and Floudas, JOGO, 1995

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Journal o f Global O ptimization 7: 143-182, 1995. 1439 1995 Kluw erAc ade micP ublishe rs. Printed in theNetherlands.

Finding All Solutions o f N onlinearly C onstrained

Sys tems of Equ ations

C O S T A S D . M A R A N A S and C H R I S T O D O U L O S A . F L O U D A S *Department o f Chem ical Engineering, Princeton University, Princeton, N J 08544-5263, U.S.A.

(Received: 15 D ece m ber 1994; accepted: 15 M ay 1995)

Abstract . A new approach is proposed for f inding al l e-feasible solut ions for cer tain classes of

nonlinearly constrained system s of equat ions. B y introducing s lack variables , the init ia l pro blem is

t rans formed in to a g loba l op t imiza t ion prob lem (P) w hose mul t ip le g loba l min im um so lu tions wi th a

zero ob ject ive value (ff any) correspond to al l solutions of the ini tial constrained system of equali ties.

A ll e-global ly opt imal points o f (P) are then local ized w ithin a set of arbit rar ily small dis joint

rectangles . This is based on a branch and b ound ty pe global opt imizat ion algo ri thm wh ich atta ins

f ini te e-co nverge nce to each o f the mult iple global m inim a of (P) throug h the successive ref inement of

a conve x relaxat ion of the feasible region and the subsequent solut ion of a ser ies o f nonl inear co nve x

optim izat ion problems. Based o n the form of the partic ipating funct ions, a num ber of techniques forconstruct ing this co nve x relaxat ion are proposed. B y taking advan tage of the properties of products of

un ivariate func t ions , cus tomized con vex low er bounding func t ions a re in t roduced for a l a rge num berof expressions that are or can be t ransformed into products o f univariate funct ions. Alternat ive con vex

relaxa tion procedures invo lve e ither the d i f ference of two convex func t ions employ ed in aB B [23] o r

the exponential var iable t ransforma tion based underestimators em ploy ed for general ized geom etr ic

prog ram m ing problems [24]. The propo sed app roach is i llustrated with several tes t problems. Fo r

som e o f these prob lem s additional solutions are identified that existing meth ods failed to locate.

K e y w o r d s : Glo bal optimization, non linear system s of equations, all solutions.

1 . I n t r o d u c t i o n

A f u n d a m e n t a l t a sk i n a p p l i e d m a t h e m a t ic s , e n g i n e e r in g a n d s c i e n c e s i s f i n d in g

a l l s o lu t io ns o f a s e t o f equa t io ns . Th i s t a s k i s s o m et im es f ur ther co mp l i ca t ed by

requ ir ing t he s im ul t a neo u s s a t is f a c t io n o f a nu m ber o f inequa l i t y a nd /o r v a r ia b le

bo u nd co nst ra in ts . N o t o n ly t he pro b lem o f co mp ut ing a l l so lu t io ns o f no n l inea r ly

co ns t ra ined s y s t ems o f equa t io ns i s NP - ha rd , but i t i s a l s o po s s ib l e t ha t t here

e x i s t s e x p o n e n t i a l l y m a n y s u c h s o l u t i o n s [ 1 ] . I n a d d i t i o n , s i m p l y c h e c k i n g i f a

s o lu t io n ex i s t s i s N P - ha rd [ 2 ]. There ex i s t s a la rge bo d y o f l it era ture o n m et ho ds f o r

s o l v i n g s y s t e m s o f e q u a t i o n s. T h e s e m e t h o d s f a ll w i t h i n th e f o l l o w i n g th r ee b r o ad

c l as s es : ( i) N e w t o n a n d q u a s i - N e w t o n t y p e m e t h o d s ; ( ii ) h o m o t o p y c o n t i n u a t io n

t y pe met ho ds ; a nd ( i i i ) in t erv a l - Newt o n met ho ds .

N e w t o n a n d q u a s i - N e w t o n t y p e m e t h o d s a n d t h ei r m o d i f ic a t i o n s a c h i e v e s u p er -

l in e a r c o n v e r g e n c e o n l y w h e n t h e y a re w e l l w i t h i n th e n e i g h b o r h o o d o f t h e s o lu t io n .

H o w e v e r , t h e s e m e t h o d s a r e l i k e l y t o f a i l i f t h e i n i ti a l g u e s s i s p o o r , o r i f s in g u -

* Au thor fo r cor respondence .



144 C.D. MARANAS AND C. A. FLOUDAS

l a r po in ts a re encou n te red . Modi f i ca t ions in an a t t empt to avo id s ingu la r i ti e s m ay

incorpora t e trus t - reg ion t echn iques such as Pow el l ' s "do g leg " metho d [31 ] , s t eep -

es t de sce nt d i rec t ion inform at ion [7], [10] , [26] and a l tera tions on the quasi -N ew tonJacob ian es tima tes [30 ]. Th i s typ e o f me thod s , a l though ve ry co m puta t iona l ly ef fi-

c i en t , cann o t p rov ide guaran tees fo r conv ergenc e . Th i s i s m an i fes t ed in p rac t i ce

wi th the i r poor con verg enc e charac te ri s ti c s .

O n e o f t h e m o s t w i d e l y u s e d m e t h o d fo r l o c a ti n g s o l u ti o n s o f n o n l i n e a r s y s te m s

of equa t ions be longs to the b road c l ass of em bedding methods. Th i s c l a s s o f m e t h o d s

a re a ls o k n o w n a s continuation, homotopy continuation, or incremental loading,

and a re based on the p ioneer ing work o f [19 , 20 , 8 , 18] . Th e bas i c idea o f hom otop y

con t inua t ion methods i s to c rea t e a fam i ly o f a s ing le pa rame te r func t ions so tha t

t h e s o lu t io n fo r ( t = 0 ) i s k n o w n a n d th e n s o l v e a s e q u e n c e o f p ro b l e m s w i t h ts t ead i ly inc reas ing f rom ( t = 0 ) t o ( t = 1 ) u s ing the so lu t ion o f one p rob lem

as an es t ima te fo r t he nex t . A po pu la r va r i a t ion i s to use a sys t em var i ab le a s the

con t inua t ion pa ram ete r and in t eg ra t e the re su lt ing sys t em o f o rd ina ry d i f fe ren t ia l

equa t ions towards s t eady-s t a te by u t i li z ing AU TO [9] . A p rob lem co m m on to

a l l hom otopy va r i an t s i s t ha t va r i ab le bounds and inequ a l i ty cons t ra in t s cann o t be

hand led d i rec t ly . A com prehe ns ive rev iew o f the ex tens ive l i te ra tu re in th is a rea c an

be foun d in [12] . Wh i l e in p rac t i ce hom otopy con t inua t ion m ethod s a re f requen t ly

used in an a t t em pt to loca te a l l so lu t ions o f a rb it ra ry n on l inea r sys t ems o f equa t ions ,

m a them at i ca l guaran tees tha t a l l so lu tions wi l l be foun d ex i s t on ly in spec ia l cases

(e .g . po lyn om ia l sys tems wi th no cons t ra in ts ) . Fo r po lyno m ia l sys tems o f equa t ions ,

how ever , M organ [27 ] p ropose d a d i f fe ren t ia l a rc l eng th con t inua t ion us ing a spec ia l

h o m o t o p y t h a t e s ta b l is h e s a n u m b e r o f c o n t in u a t i o n p a t hs g u a ra n t e e d to c o n v e rg e

t o a l l p o s s ib l e r e a l a n d c o m p l e x ro o t s. Tw o p o p u l a r s o f t w a re p a c k a g e s , C O N S O L

[2 7] a n d P O LS Y S [34 ] h a v e i m p l e m e n t e d t hi s m e t h o d .

In t e rva l -N ew ton m ethods c an f ind rec t ang les con ta in ing a ll so lu tions o f non-

l inea r sys t ems o f equa t ions w i th in ce r t a in va r iab le bounds w i th ma them at i ca l ce r -

ta in ty . T hey do so by app ly ing the c l a ss ica l New ton- l ike it e ra tive method s onin te rva l va r i ab les ra the r than va r i ab les co up led w i th a g enera l i zed b i sec t ion s t ra t e-

gy [29 , 13 ]. A ve rs ion o f t he bas ic In t e rva l -Ne wton m ethod has been im plem en ted

i n to t h e p u b li c d o m a i n s o f t w a re p ro g ra m IN TB IS [1 7] w h i c h i s c o u p l e d w i t h a

por t ab le in t e rva l s t andard func t ion l i b ra ry IN TL IB [16]. T he m ain a t t rac t ive fea -

t u r e o f In t e rv a l -N e w t o n m e t h o d s i s t h a t t h e y p ro v i d e m a t h e m a t i c a l g u a ra n t e e s

fo r co nve rgen ce to a l l so lu t ions o f fa i r ly a rb it ra ry no n l inea r sys t em s o f equ a t ions

wi th in ce r t a in va r iab le bound s . Ho we ver , t h is w ide app l i cab i l it y to a lm os t a rb i tra ry

non l inea r func t ions com es a t an expen se . Bec ause no spec i fi c st ruc tu re o f i nd iv id -

ua l express ions i s ana lyzed the ob ta ined in te rva l boun ds can som et imes be fa i r lyloose .

Th e p ro p o s e d a p p ro a c h is b a s e d o n c o n v e x l o w e r b o u n d i n g c o u p l e d w i t h a p a r-

t it ion ing s t ra t egy and l ike In t e rva l -N ew ton me thods , i t can p rov ide gu aran tees fo r

c o n v e rg e n c e t o a ll e - so lu t io n s . Th e fu n d a m e n t a l d if f e r e n c e, h o w e v e r , b e t w e e n o u r

p roc edure and In t e rva l -New ton m ethods i s t ha t wh i l e the fo rm er u ti li zes a s ing le



SOLUTIONSOF NONLINEAREQUATIONS 145

v a l u e t o l o w er b o u n d f u n c t i o n s w i t h i n r ec t an g u l a r d o m a i n s , w e l o w er b o u n d n o n -

con vex func t ions wi th con vex func t ions . By exp lo i t ing the ma them at ica l st ruc tu re

of the p rob lem , th is typ ica l ly r esu l t s in mu ch t igh te r bounds . In the ne x t sec t ion , adescr ip t ion o f the p rob lem i s p resen ted .

2 . P r o b l e m D e s c r i p t io n

This pape r addresses the p rob lem of iden t i fy ing a l l so lu t ions o f a non l inear sys tem

of equa t ions sub jec t to inequa l i ty cons t r a in t s and var i ab le boun ds a nd i s fo rmula ted

as:

h j ( x ) = O , jEA/ 'E (S )

gk (x) < o , k ~ X z

XL < X < xU~

w he re AlE is the set o f equal i t ies , .Mz the se t of inequ al i ty const ra ints , and x the

vec to r o f var i ab les. N ote tha t in fo rmula t ion (S ) the to ta l nu m ber o f var i ab les i s

a l low ed to be d i f f e ren t than the to ta l num ber o f equa l i ti es so as ne i ther the ex i s t ence

nor the un iqueness o f a so lu t ion o f (S ) is pos tu la ted . Therefore , bo th over spec i f i ed

and u nder spec i f i ed sys tems a re inc luded in the p resen t inves t iga tions . No te tha t anu m ber o f imp or tan t p rob lem s na tu ra l ly a r ise as spec ia l ins tances o f fo rm ula t ion

(S) . O n o ne hand , by o mi t t ing a l l inequa l i ty cons tr a in ts , (S ) cor responds to a sys tem

of non l inear equa t ions . On the o ther hand , by e l imina t ing a l l equa l i ty cons t ra in t s

(S) che cks the ex i s t ence o f f eas ib le po in t s fo r the g iven inequa l i ty cons t r a in t se t

( feasibi l ity problem ) .

F o r m u l a t io n ( S ) can b e t ran s f o rm ed i n to t h e f o l l o w i n g m i n - m ax o p t im i za t io n

prob lem [15]

m i n m a x I h j ( x ) lx j 6 A ; E

s u b j e c t t o g k ( x ) _< O , k 6 A / )

x L ~ x < x U.

By in t roduc ing a s ing le s l ack var iab le s , t he min-m ax prob lem can be w r i tt en as

the fo l lowing op t imiza t ion p rob lem (P0) .

m in s (PO)x,s_>O

subject to hj ( x ) - s < O, j 6 ACE

- h j ( x ) - s < O, j e . M E

gk (x) < o , k e N 1

xL_< X _< XU



146 C . D . M A R A N A S A N D C . A . F L O U D A S

C l ear ly , t h e re i s a o n e t o o n e co r r e sp o n d en ce b e t w een m u l ti p le g l o b a l m i n i m a

(x* , s* ) o f (P0) fo r wh ich s* = 0 and so lu tions o f (S ). Th i s m eans tha t i f t he

g loba l min imum of (P0) invo lves a nonzero s l ack var i ab le s* then the o r ig ina lp rob lem (S) has no so lu t ions . Note tha t , un less the func t ions h ( x ) a n d g k ( x )

are l i near and convex r espec t ive ly , fo rmula t ion (P ) cor responds to a n o n c o n v e x

opt imiza t ion p rob lem. Th i s im pl ies tha t i f a loca l op t imiza t ion approach i s us ed to

so l v e (P 0 ) , o n e m i g h t m i s s so m e o f t h e m u l t ip l e g l o b a l m i n i m a o f ( P 0 ) o r ev en

er roneo us ly ded uce tha t there a re no so lu t ions fo r (S) . Therefore , an ap proach tha t

i s guaran teed to a lways loca te a ll m u l t i p le g l o b a l m i n i m a o f ( P 0 ) ap p ea r s t o b e

nec ess ary for solving (S) so that ( i) the cor rect solut ion vec tor (x* , s*) i s ident i f ied

and ( ii ) a l l solut ions (x*) o f (S ) w i th s* = 0 are fou nd in a l l instances. In th is w ork ,

a de te rmin i s t i c g loba l op t imiza t ion i s p roposed w hich i s guaran teed to loca te a l le - gl o b al m i n i m a o f ( P 0 ) t h r o u g h th e su cces s iv e r e f in em en t o f co n v e r g i n g l o w er

an d u p p e r b o u n d s o n t h e so l u ti o n b a sed o n t h e so l u ti o n o f co n v ex o p t im i za t io n

p r o b l em s d e f i n ed b y a b r an ch an d b o u n d ap p r o ach . A l o w er b o u n d o n t h e so l u t io n

of (P0) i s found by fir st r ep lac ing each non con vex cons t r a in t in (P0) w i th a conv ex

unde res t imat ion o f i t and then f ind ing the so lu t ion o f the con vex r e laxa t ion OR) o f

(P0) w i th com m erc ia l ly ava i l ab le so lver such as MIN OS 5.4 [28] as sh ow n in [22]

and [23] . This approach natural ly par t i t ions the const ra ints of formulat ion (P0)

in to conv ex ( fo r whic h no r e laxa t ion i s r equ i red) and n onc onv ex cons t ra in t s . Th i s

par t it ion ing y ie lds the fo l lowing a l t e rna t ive fo rmula t ion (P ) :

subjec t to

m i n sx,s>O

n o n e _ \hj ( x ) - s <_ O, j E .A/noncE

n o n c- h i (x) - s < O, j E .A/noncE

g~~ _< 0 , kEA/ ' noncI

h~m (x) = O , j e J~nnE

g~ ~ < O , j E A / 'c o n v l

x L _ _ _ x _ < x v

(P )

H e r e J ~ f n o n c E , J ~ f l i n E a r e t he se t s o f nonconvex and l inear equa l i ty cons t r a in t s

resp ectiv ely, a nd Afnonc~,A/'convZ are the sets o f no nc on ve x and co nv ex ineq ual i ty

const raints ,

= A; .onc u mE, X 1 = Y.o .c u X onv .

A con vex r e laxa t ion OR) o f (P ) o f the fo rm,

m i n s ( R )X,s>_O

sub jec t t o ~ , n o nc { _ \ _, ~ + , j ~ x ) - - 8 < O , j E . A / n o n c E




h n o n c / x

_ , j ( x ) - s < 0 , j E A / n o n c E

t F ~ < o , k ~ N n o n a

h ~ ( x ) = O , j e A /n aB

g~on v(x) < 0 , j e A/convz

xL _< x _< x v

c a n b e o b t a i n e d b y r e p l a c i n g t h e o r i g in a l n o n c o n v e x f u n c t io n s h ff~ - h y~ ( x ) ,

hnonc (X~ ]znonc X ~,~noncr-..~ w i t h s o m e c o n v e x t i g h t l o w e r b o u n d i n g f u n c t i o n s + , J ~ J ' - o , / 'Yk k~J

~nonc/. ~ + , j ~ / , _ , j ,x ) , g k ( x ) m u s t b e( x ) . T h e s e l o w e r b o u n d i n g f u n c ti o n s h n ~ ~ h n ~ ~ -none .

( i) c o n v e x i n I x L , x U j ; ( ii ) v a l i d u n d e r e s t i m a t o r s o f t h e o r i g i n a l f u n c t i o n s hnonc

_/,nonc.~3 ~k"n~ a n d (iii) fo r e v e ry po in t x e .IxL , x U ] t h e m a x i m u m s e p a r a t i o n

b e t w e e n t h e o ri g i n a l f u n c t i o n s a n d t h e c o n v e x u n d e r e s t i m a t o r s m u s t b e c o m e a rb i -

t r a ri ly e s m a l l b y a p p r o p r i a t e l y r e d u c i n g th e s i z e o f t h e r e c t a n g u l a r d o m a i n [ x l , x u ]

a r o u n d t h e p o i n t x i n s i d e w h i c h t h e c o n v e x u n d e r e s t i m a t o r s a r e d e f i n e d 9 T h e s e

r e q u i r e m e n t s a re e x p r e s s e d m a t h e m a t i c a l ly a s fo l lo w s :

9 h n~ ~ h n~ a n d z no nc (v a b e c o n v e x V X E [ x L , x U ]R O P E R T Y 1 + , j ~ / , - , j ~ / , v k ~ ' /

P R O P E R T Y 2 . h~~ ">/ Izn~ ~ none _ iZn~ an d .noncr. .~ >_ + , j ~ j , - h i ( x ) > - o ~ J ' v k ~ -'-J -

o n o n c r v ~ V x ~ [ x L , x U ]k \""1 ~

an ,>

...l II 1/2IIx ~ ' - 7 , , 2 > 0 s u c h th a t

~ x m a X ( h~ ~ h n~e[xZ,x~] - + 's ~ /J < e ,

m a x ( - h ~ ~ h n ~ < e,x e [x l ,x , ,] - - J ' ' /

m a x ( g ~ ~ ~ ~ < e.x~[x l ,x , , ]

P r o p e rt y 3 r e q u i re s t h a t t h e m a x i m u m s e p a ra t io n b e t w e e n t h e n o n c o n v e x f u n c t io n

a n d i t s t ig h t c o n v e x l o w e r b o u n d i n g f u n c t io n , d e f i n e d in s i d e s o m e r e c t a n g u l a r

r e g i o n , m u s t g o t o z e r o

l i m e = O .~---~0+



1 4 8 C.D. MARANAS AND C. A. FLOUDAS

a s t h e s i ze o f t h e r e c t a n g u l a r d o m a i n a p p r o a c h e s z e r o ( 6 = 0 ) . T h i s is i m p o r t a n t f o r

p r o v i n g f in i te e - c o n v e r g e n c e . T h e o r d e r O ( e ) = 0 ( 6 n ) w i t h w h i c h e a p p r o a c h e s

z e r o a s 6 g o e s t o z e ro i s i m p o r t a n t b e c a u s e i t d e t e r m i n e s t h e s p e e d o f c o n v e r g e n c e .C l e a rl y , th e l a rg e s t p o s s i b l e v a l u e f o r n i s d e s i r a b le s o a s th e m a x i m u m t o l e r a n c e

r e a c h e s a n a r b it ra r y v a l u e e f o r a n o t t o o s m a l l v a r i a b le r a n g e 6 . F o r e x a m p l e , i f

t h e m a x i m u m s e p a r a t io n e g o e s a s 6 2 t h e n a v a l u e o f j u s t 6 = 0 .0 1 s u ff ic e s t o m e e t

a c o n v e r g e n c e t o l e r a n c e o f e = 0 . 0 0 0 1 .

A n e f fi c ie n t c o n v e x l o w e r b o u n d i n g o f n o n c o n v e x f u n c t io n a l s a p p e a r in g i n

f o r m u l a t i o n ( P ) i s c le a r ly c e n t r a l t o th e d e s i g n o f th e p r o p o s e d g l o b a l o p t i m i z a -

t i o n a p p r o a c h f o r l o c a t i n g a l l s o lu t i o n s . U n d o u b t e d l y , t h e ti g h t e r th e c o n v e x l o w e r

b o u n d i n g i s t h e b e t t e r t h e q u a l it y o f th e o b t a i n e d l o w e r b o u n d s w i l l b e , a n d c o n s e -

q u e n t l y t h e f as t e r t h e a l g o r i t h m w i l l c o n v e r g e . T h e t i g h te s t p o s s i b l e c o n v e x l o w e r

b o u n d i n g f u n c t io n fo r a n y a r b i tr ar y n o n c o n v e x f u n c t i o n f ( x ) i n s id e s o m e re c t an -

g u l a r r e g i o n P i s c a l l e d t h e convex envelope ~ b(x ) o f f ( x ) , a n d i t m u s t c o n f o r m t o

t h e f o l l o w i n g p r o p e r t i e s [ 15 ]:

( i) ~ b ( x ) c o n v e x f o r a l l x E P .

( i i) f ( x ) _> ~ b ( x ) f o r a l l x E P .

( ii i) F o r a ll f u n c t i o n s 9 ( x ) t h a t s a t i s f y (i ) a n d ( i i ), ~ b (x ) _> g ( x ) f o r a l l x E P .

U n f o r t u n a t e ly , i n a ll b u t t h e s i m p l e s t c a s e s th e r e e x i s ts n o m e t h o d f o r d e r i v in gt h e c o n v e x e n v e l o p e f o r a rb i tr a ry f u n c t i o n s d e f i n e d i n s id e a r b it ra r y d o m a i n s . A s a

r e su l t, t h e f o c u s i n t h i s w o r k i s t o id e n t i fy t h e m a x i m u m p o s s i b l e f u n c t i o n w h i c h

s a t is f ie s p r o p e r t i e s ( i) a n d ( i i) . T h e r e e x i s t s a n u m b e r o f t e c h n i q u e s f o r o b t a i n i n g

f u n c t i o n s t h a t s a t i s f y p r o p e r t i e s ( i ) , ( i i ) . I n t h e f o l l o w i n g s e c t i o n s , a n u m b e r o f

c o n v e x l o w e r b o u n d i n g p ro c e d u r e s a r e d i s c u s s e d w h i c h c a n b e o f u s e n o t o n l y f o r

t h e p r o b l e m o f l o c a t i n g a ll m u l t i p l e s o l u t i o n s b u t a l s o f o r a n y d e t e r m i n i s t i c b r a n c h

a n d b o u n d g l o b a l o p ti m i z a t i o n a lg o r i th m b a s e d o n c o n v e x l o w e r b o u n d i n g . T h e

f ir st c o n v e x l o w e r b o u n d i n g t e c h n i q u e i s m o t i v a t e d b y t h e f a c t th a t a la r g e n u m b e r

o f n o n c o n v e x t e r m s a p p e a r in g i n d i f fe r e n t m o d e l s a r e o r c a n b e t r a n s fo r m e d i n tot h e p r o d u c t o f f u n c t i o n s o f a s i n g l e v a ri a b l e ( u n i v a r ia t e f u n c t i o n s ) . B y e x p l o i t i n g

t h e p r o p e r ti e s o f p r o d u c t s o f u n i v a r i a te f u n c t i o n s , t i g h t c o n v e x l o w e r b o u n d i n g

f u n c t i o n s a r e d e r i v e d i n t h e n e x t s e c t io n .

3 . P r o d u c t s o f U n i v a r i a t e F u n c t i o n s

A f u n c t i o n f : T ~ ~ R o f a s i n g l e v a r i a b l e z i s c a l l e d u n i v a r i a t e f u n c t i o n . P r o d u c t s

o f u n i v a r i a t e f u n c t i o n s f i ,

N

: ( x ) = I - [i=1

a r e i n g e n e r a l n o n c o n v e x f u n c t i o n s e v e n i f t h e c o r r e s p o n d i n g u n i v a r ia t e f u n c -

t i o n s a r e c o n v e x . B y u t i l i z i n g a p p r o p r i a t e l i n e a r t r a n s f o r m a t i o n s , i f n e c e s s a r y , a



SOLUTIONS OF NONLINEAR EQUATIONS 149

l a rg e n u m b e r o f n o n li n e a r it ie s a p p e a r i n g i n a p p l i e d m a t h e m a t i c s a n d e n g i n e e r i n g

p r o b l e m s c a n b e d e s c r i b e d a s p r o d u c t s o f u n i v a r ia t e fu n c t i o n s .

A 1 - K h a y y a l a n d F a l k [3 ] s h o w e d t h a t th e n o n c o n v e x b i li n e a r p r o d u c t o f x yi n s id e t h e r e c t a n g u l a r d o m a i n ] x L , x U I x [yL , yU [ c a n b e t ig h t ly c o n v e x l o w e r

r ~ P q

I . . I I . . I

b o u n d e d b y t h e f o l l o w i n g l i n e a r c u t:

m a x ( x L y q - x y L - - x L y L, x U y + x y U -- x U y U )

F i rs t, t h e c o n d i t i o n s u n d e r w h i c h a s i m i la r r e s u lt h o l d s f o r t h e p r o d u c t o f t w o

a r b i t ra r y u n i v a r i a t e f u n c t i o n s f ( x ) a n d g ( y ) a r e i n v e s t i g a t e d .

T H E O R E M 1. I f f , g a r e t w i ce d if fe r en t ia b l e u n i va r i a te f u n c t i o n s f ( x ) , g ( y ) 6 C2d e f i n e d i n s i d e a r e c t a n g l e [ ( x L , x U ) , ( y L y U ) ] a n d

l ( x , y ) = m a x { r + r f ( x ) ) _ f L g L ,

r + r I ( x ) ) _ f U g U }

w h e r e

f L = i n f f ( x ) ,x L < m < X U

f u = su p f ( x ) ,x L < x < Z U

g L = i n f g ( x ) ,x L < x < x U

g U = sup g ( x )x L < x < x U

a n d r 1 6 2 1 6 2 r a re th e c on ve x

e n v e l o p e s o f t h e u n iv a r ia t e f u n c t i o n s f Lg ( y ) , g Z f ( x ) ' f i g ( y ) , a n d g V f ( x ) r e s p ec -t i ve ly then:

( i) l ( x , y ) i s c o n v e x , V ( x , y ) 6 [ x L , x U ] • [ yL , y U ] .

( g ~ f ( x ) ) - S ~ g ~ a r e c o n v e x as t he s u m o f t he c o n v e x e n v e lo p e s o f u n i v ~ i a t e

f u n c ti o n s. S i n c e t h e m a x i m u m o f t w o c o n v e x f u n c t io n s is a c o n v e x f u n c t io n a s w e l l ,

s t a t e m e n t ( i ) i s t ru e a n d l ( x , y ) i s c o n v e x f o r a ll ( x , y ) i n [ x L , xU ] • [ yL , yU ]

B e c a u s e ,



1 5 0

w e h a v e

( f ( x ) - f ~ ) ( ~ ( ~ ) - ~ ' ) _ > 0 , ~ ( ~ , ~ ) ~ [ x ~ , x ~ ] x [ ~ , ~ ]A f t e r r e a r r a n g i n g t e r m s w e o b t a i n ,

f ( x ) g ( y ) ) _ f L g ( y ) + f ( x ) g L - - f L g L , V ( x , v ) E [ x L , x U ] X [ y L y U ] .

B y t h e d e f i n i t io n o f t h e c o n v e x e n v e l o p e w e k n o w t ha t ,

f L g ( y ) > _ r E [ y L , y U ] ,

g L f ( x ) )_ r [ x L , x U ] .

T h e r e f o r e ,

f ( x ) g ( y )

F u r t h e r m o r e , b y f o l l o w i n g t h e s a m e l i n e o f r e a s o n i n g o n t h e r e l a ti o n ,

( f ( x ) - f ) ( ~ ( ~ ) - ~ ) _ > 0 , ~ ( ~ , ~ ) ~ [ ~ , , x ~ ] x [ ~ , , ~ ] ,

w e o b t a in :

f ( x ) g ( y ) >

R e l a t i o n s ( 1 ), (2 ) i m p l y t h a t s t a t e m e n t (i i) i s t r u e .

C. D. MARANAS AND C. A. FLOUDAS

(1 )

T H E O R E M 2 . I f ( i ) t h e u n i v a r ia t e u n c t i o n s ,

g L f ( x ) , g U f ( x ) a n d f L g ( y ) , l U g ( y )

a r e c o n c a v e in [ x L , x U ] a n d [ y L y U ] r e s p e c t i v e l y a n d ( i i ) t h e fu n c t io n s f ( x ) , 9 ( Y )

a r e m o n o t o n i c , t h e n l ( x , y ) i s t h e c o n v e x e n v e l o p e o f f (z)g(v).

B a s e d o n T h e o r e m ( 1 ) f u n c t i o n l ( z , y ) c a n b e u t i l i z e d a s a t ig h t c o n v e x l o w e r

b o u n d i n g f u n c t i o n o f t h e p r o d u c t o f t w o c o n t i n u o u s a n d t w i c e d i f f e re n t ia b l e f u n c -

t io n s . It c a n a l s o b e s h o w n t h a t u n d e r c e r t a in c o n d i t i o n s l ( z , y ) c o r r e s p o n d s t o th e

a c t ua l c o n v e x e n v e l o p e o f t h e p r o d u c t f ( x ) g ( y ) . T h e s e c o n d i t i o n s a r e s t a t ed i n t h e

f o l lo w i n g t h e o r e m :

( 2 )

[]



SOLUTIONS OF NONLINEAREQUATIONS 15 1

Proof . T h e o r e m ( 1) p r o v e s t h a t fu n c t i o n l ( x , y ) c o n f o r m s w i t h P r o p e r t i e s ( i )a n d ( ii ) o f S e c t i o n 3 . T h e r e f o r e , i t r e m a i n s t o s h o w t h a t i t s a t is f ie s P r o p e r t y ( i ii ) o f

S e c t i o n 3 . B e c a u s e t h e c o n v e x e n v e l o p e o f a u n iv a r ia t e c o n c a v e f u n c t i o n d e f i n e d

i n a n i n te r v a l is th e l i n e s e g m e n t c o n n e c t i n g t h e th e t w o e n d p o i n t s w e h a v e :

r ) g U I f (x ; )-- ( x L ) x U I ( x L ) - x LI ( xU ) }= ~L x + - Z - 7 J '

i y~ 7 + y~ y~ ] '

+ ( f g ( ,,)) : s < ,g ( y u ) _ g ( y L ) x +

y V _ y L

y ~ g ( y L ) _ ~ ,~g ( d . . l ) ]

Y ' - 7 J "

A f t e r s u b s t i tu t i n g t h e s e e x p r e s s i o n s i n t o t h e r e l a t io n f o r l ( x , y ) w e o b t a i n:

l ( x , y ) = m a x (/1 ( x , y ) , / 2 ( x , y ) )

w h e r e

l l ( X , y ) - ~

+

g L I ( X U ) _ f ( x L ) " [ I L U ( y U ) _ g ( y L ) ,- ~ x L X + L - ~ - ~ Y

g L X U f ( x L ) _ x L f ( x U ) y U g ( y L ) _ y L g ( y U )

" - ~ = " ~ ..[_ L y V _ y L _ f L g L ]

r , < . , ( , ' - - ' ) - , ( , ' - . ) 17 7 J ~ + [" 7e yL j Y

+ [ ffy%(yr')-yLn(yU)L + u U _ y L -- U o

B e c a u s e f ( x ) , g ( y ) a r e m o n o t o n i c o n e o f th e f o l l o w i n g a l t e rn a t iv e s i s tr u e:

( a ) I ( x L ) = I L , i ( x U ) = I v ' g ( x L ) = g L , g ( x U ) = g U ,

@ ) f(x L ) = . f L , i ( z u ) = i ~ , g(x L ) = g U , g(x ~ ) = g L ,

( c ) I ( x L ) = I U , i ( z U ) = i s , , g ( z L ) = g L , g ( x t ' ) = g U ,

(d ) f ( x L ) = i v , f ( x U ) = I L , g ( x L ) = g V , g ( x U ) = gL .



1 5 2 C . D . M A R A N A S A N D C . A . FL O U D A S

F i g . 1 .

( x ~ , # ) R ( x% y~

%

(x L # ) (x~,y )

Decom position of rectangleR into two triangles T~, T2.

A s s u m i n g t h a t (a ) is tr u e w e h a v e :

l l ( x L , y L ) = f ( x L ) g ( y L ) ,

l l ( x L , y U ) = f ( x L ) g ( y V ) , a n d

l l ( x U , y L ) -~ - f ( x U ) g ( y L ) .

T h i s i m p l i e s t h a t o n e c a n p a r t it i o n t h e o r i g i n a l r e c t a n g l e

i n t o t h e f o l l o w i n g t w o d i s j o i n t t ri a n g l e s ,

T 1 = [ ( x L , y L ) , ( x L , y U ) , ( x U , y L ) ] , a n d

1 2 ( x U , y U ) = f ( x U ) g ( y U ) ,

1 2 ( x L , y U ) = f ( x L ) g ( y U ) ,

1 2 ( x v , y L ) = f ( x V ) g ( y L ) .

a t w h o s e v e r t i c e s t h e l i n e a r f u n c t i o n s l l ( x , y ) , 1 2 ( x , y ) m a t c h t h e o r i g i n a l p r o d u c t

o f u n i v a r ia t e f u n c t i o n s f ( z ) g ( y ) r e s p e c t i v e l y ( S e e F i g u r e 1 ) .

I f l ( z , y ) w e r e n o t t h e c o n v e x e n v e l o p e o f f ( z ) g ( y ) o v e r th e r e c t a n g u l a r d o m a i n

T~ t h e n , t h e r e w o u l d b e a t h i r d c o n v e x f u n c t i o n 13 ( x , y ) u n d e r e s t i m a t i n g f ( x ) g ( y )

o v e r T~ a n d a p o in t ( 2 , if ) E R su c h th a t :

l ( : , ~ 7 ) < 1 3 ( 2 , ~ ).

S u p p o s e th a t ( 2 , y ) E ~ . T h e n ( 2 , ~ ) is a u n i q u e c o n v e x c o m b i n a t i o n o f t h e t h r e ee x t r e m e p o i n t s v 1 v 2 , v 3 o f T 1 . H e n c e , f o r t h e a f f in e f u n c t i o n l , t h e r e e x i s t s u n i q u e

~ i = l A i = 1 s u c h t h atos i t iv e A1, )~2 ,/~3 sa t i s fy in g 3

l ( 2 , f f ) = 1 / ~ i V = ) ~ i l ( v i ) .



S O L U T I O N S O F N O N L I N E A R E Q U A T I O N S 153

B e c a u s e 13 i s th e c o n v e x e n v e l o p e o f f ( x ) g ( y ) ins ide T~, ( i ) 13 i s co n ve x a nd ( i i ) i t

m a t c h e s f ( x ) g ( y ) a t a l l v e r t e x p o i n t s l i k e l ( x , y ) d o e s w h i c h i m p l i es :

t 3 ( s : , ~ ) = 1 3 : ~ < : ~ d 3 ( ~ ~ ) = ~ , x d ( r = l ( s : , ~ ) .

i = 1 i = 1

T h i s c o n t r a d i c t s t h e i n i t i a l h y p o th e s i s l (:~, ~7) < / 3 ( x , Y ) a n d t h e r e f o r e , l ( x , y ) is

i n d e e d th e c o n v e x e n v e l o p e o f f ( x ) g ( y ) i n ~ . N o t e t h a t a s im i l a r a r g u m e n t h o l d s

i f (: L 77) E T ~ . M o r e o v e r , d e p e n d i n g o n w h i c h m o n o t o n i c i t y c o m b i n a t i o n ( a ), (b )

, (c ) o r (d ) i s t r u e i t i s a lw a y s p o s s ib l e t o p a r t i t i o n ~ i n to tw o t r i a n g l e s T 1 , T 2

b y h a l v i n g a l o n g o n e o f t h e d i a g o n a l s . T h e r e f o r e , b y f o l l o w i n g t h e s a m e l i n e o f

t h o u g h t f o r c o m b i n a t i o n s ( b ) , ( c ) a n d ( d ) i t i s s t r a i g h t f o r w a r d t o e x t e n t t h i s p r o o f

f o r a l l m o n o t o n i c i t y c o m b i n a t i o n s . [ ]

T h e a n a l y s i s f o r t h e c o n v e x l o w e r b o u n d i n g o f p r o d u c t s o f t w o u n i v a r ia t e f u n c t io n s

c a n b e e x t e n d e d t o a c c o m m o d a t e th e p r o d u c t o f N u n i v a r i a te f u n c ti o n s . T h i s is

a c c o m p l i s h e d b y s u c c e s s iv e l y c o n v e x l o w e r b o u n d i n g p a i rs o f u n i v ar ia t e f u n c ti o n s

i n a r e c u r s i v e m a n n e r u n t i l n o p a i r s a r e l e f t . O n e o f t h e p o s s i b l e a l t e r n a t i v e s o f

c o m b i n i n g p a i r s i s t o s ta rt w i t h c o n v e x l o w e r b o u n d i n g t h e l a st tw o f u n c t i o n s o f th e

p r o d u c t a n d w o r k y o u r w a y t o t h e fr o n t o f t h e e x p r e s si o n . T h e o r e m ( 3) s t a te s t h att h is p r o c e d u r e y i e l d s a c o n v e x l o w e r b o u n d i n g f u n c t i o n f o r t h e i n it ia l p ro d u c t .

T H E O R E M 3 . I f f i E c= [x ,xy I - - , r + , i = 1 , . . . , N a n d

L ( x ) = Y o

w h e r e

= L ( Y j + l f j + l ( X j + l ) ) L La x { + ( f j + l y j+ l ) _ l _ + L - - Y j + l f j + l ,

+ ( f Y - I - l Y J - i - 1 ) " i - + ( Y Y ' - I - l f j - t - I ( Z J " P l ) ) - - Y j ' - P - l f j + l ,U ,

j = 0 , . . . , N - 3

a n d

Y N - 2 m a x { + ( f D _ l f N ( X N ) ) - i - + ( f D f N - I ( X N - 1 ) ) L L

q ~ ( f N U _ l f N ( Z N ) ) " - ' l - ( f U N f N - I ( X N - 1 ) ) - - f N U _ l f N }

L- - Y j + l f j + l ,j+I ,Yj+I

+ + <' <'

- - Y j + I ~ - I - 1 , } ,

j = 0 , . . . , N - 3



154 C.D . MARANAS AND C. A. FLOUDAS

L/UN - 2

w e h a v e,

in f su p L X L LX N , X N _ I ( r N ) ) - } - ~ ) ( f L f N - I ( X N - 1 ) ) - f N - l f ~ r

t h e n( i ) L ( x ) i s c onv e x , V x E [ x L , x U ] 9

N

(ii) 11 f i ( x i ) ) _ L ( x ) , V x E [ x L , x U ] .i----1

P r o o f . S t a r t i n g f r o m t h e b e g i n n i n g o f t h e r e c u r s i v e d e f i n i t i o n o f L ( x ) ,

Y N - 2 i s a c o n v e x f u n c t i o n o f ( X N - I , X N ) a s th e m a x o f tw o c o n v e x f u n c -t i o n s . F o r t h e s a m e r e a s o n Y N - 3 i s a c o n v e x f u n c t i o n o f ( X N - 2 , Y N - 2 ) o r

o t h e r w i s e o f ( X N - 2 , X N - I , X N ) . B y r e c u r s i v e l y s u b s t i t u t i n g y j i n t o t h e e x p r e s -

s i o n f o r Y j - 1 w e d e d u c e t ha t f o r e v e r y j = 0 , . . . , N - 2 , y j is a c o n v e x f u n c t io n

o f ( x j + l , x j + 2 , . . . , X N ) . T h e r e f o r e , L ( x ) = Y0 is a c o n v e x f u n c t i o n o f

(Xl , x2 , 99 9 X N ) w h i c h p r o v e s p a r t ( ii) o f T h e o r e m ( 3) .

F r o m T h e o r e m ( 1 ) a n d t h e s t a te m e n t o f T h e o r e m ( 3) w e h a v e ,

f N _ t ( X N _ I ) f N ( X N ) ~ > - - f N - l f N 'a x ( r -} - ~ ) ( f L f N - I ( X N - 1 ) ) L L

Y N - 2

a n d

~ + I ( X j + I ) Y j + I > m a x ( r + r ( y L + l f j + I ( X j + I ) ) L L- - Y j + l f j + l ,

= y j ,

j = 0 , . . . , N - 3 .

B y c o m b i n i n g t h e s e l a s t t w o s e ts o f i n e q u a li ti e s w e h a v e ,

N

I I s , ( x , / > _ - -

i=1

w h i c h p r o v e s p a r t ( ii) o f T h e o r e m ( 3) . []

T h e o r e m ( 3 ) d e s c r i b e s o n e p o s s i b le w a y o f re c u r s i v e l y c o m b i n i n g p a i r s o f u n i v a r i-

a t e fu n c t io n s . T h e o r e m ( 4) s t a te s w h o m a n y o f th e s e a l t e rn a t iv e s e q u e n c e s e x i s t

f o r c o n v e x l o w e r b o u n d i n g t h e p r o d u c t o f N u n i v a r i a te f u n c t io n s .



SOLUTIONSOF N ONLINEAREQUATIONS 155

T H E O R E M 4 . T h e r e a r e ,

( N ! ) 2

N 2 N - 1 ,

w a y s o f c o m b i n in g p a i r s o f u n iv a r ia t e f u n c t i o n s i n a p r o d u c t o f N u n i v ar ia t e

f u n c t i o n s .

P r o o f . C l e a r l y , t h e r e e x i s t s ( g ) = g (g -1 )2 w a y s o f s e l e c t i n g t h e f i rs t p a i r o f

u n i v a r ia t e f u n c t i o n s t o b e c o n v e x l o w e r b o u n d . A f t e r th i s a c ti o n , w e a re l e f t w i t h

g - 1 f u n c t i o n s w h i c h im p l i e s t h a t t h e r e a r e ( Y 21 ) ~-- (N-1 )(N-2)2 a l t e r n a t i v e s

f o r p i c k i n g t h e n e x t p a i r o f f u n c t i o n s . T h i s r e c u r s i v e c o n v e x l o w e r b o u n d i n g

i s c o n t i n u e d u n t il w e a r e l e f t w i t h o n l y a p a i r o f f u n c t i o n s i n v o l v i n g a s i n g l e

c o n v e x l o w e r b o u n d i n g a l te rn a ti ve . B e c a u s e e v e r y c o n v e x l o w e r b o u n d i n g s t ag e

is i n d e p e n d e n t o f t h e p r e v i o u s o n e , th e t o ta l n u m b e r o f w a y s o f c o m b i n i n g p a i r s

t h e N u n i v a r i a t e f u n c t i o n s in p a i rs o f tw o i s:

I I 2 = I I i ( i - 1) N ! ( N - 1 )! ( N ! ) 2

i = 2 i = 2 2 - 2 g - 1 - N2N_------ . O

E x a m p l e s o f c o n v e x l o w e r b o u n d i n g o f p r o d u c t s o f N u n i v a r i a t e f u n c t i o n s ar e g iv e n

i n A p p e n d i x A . F u r t h e r m o r e , c o n d i t i o n s fo r c o n v e x i t y / c o n c a v i t y a re p r o v i d e d f o r

g e n e r a l i z e d p o l y n o m i a l te r m s , w h i c h a re a s p e c ia l ca s e o f p r o d u c t s o f u n i v a ri a te

f u n c t i o n s , i n a p p e n d i x B . F r o m t h e a n a l y s i s i n t h e p r e v i o u s s e c t i o n s it i s c l e a r th a t

i n o r d e r to c o n v e x l o w e r b o u n d t h e p r o d u c t o f u n i v a r ia t e f u n c t io n s i t i s n e c e s s a r y

t o b e a b l e to o b t a i n t h e c o n v e x e n v e l o p e , o r a t l e as t a ti g h t c o n v e x l o w e r b o u n d i n g

f u n c t i o n , o f ar b i tr a r y u n i v a r i a t e f u n c t i o n s . T o t h is e n d , g u i d e l i n e s f o r c o n s t r u c t i n g

t h e c o n v e x e n v e l o p e o f a r b it r ar y f u n c t i o n s o f a s in g l e v a r i a b le i n s i d e a c e r t a in

i n t e rv a l a r e p r e s e n t e d i n t h e f o l l o w i n g s e c t i o n .

3 .1 . C O N V E X E N V E L O PE S O F U N IV A R IA T EFUNCTIONS

C o m p u t i n g t h e c o n v e x e n v e l o p e s o f a r b it ra r y t w i c e d i f fe r e n ti a b le f u n c t i o n s i n

a s in g l e v a ri a b le a p p e a r s f r e q u e n t ly a s a ta s k i n m a n y c o m p l e x c o n v e x l o w e r

b o u n d i n g s i t u a t i o n s . I n s o m e c a s e s , t h i s i s a s t r a i g h t f o r w a r d t a s k , f o r e x a m p l e i f

f E C 2 " [ a , b] ---+ R i s c o n v e x t h e n i ts c o n v e x e n v e l o p e c o i n c i d e s w i t h t h e o r i g i n a l

f u n c t i o n :

r = f ( x ) , V x E [ a , b ] i f a n d o n l y i f f ( x ) i s c o n v e x i n [ a , b ] .

I f n o w f ( x ) i s c o n c a v e , th e n i ts c o n v e x e n v e l o p e i s a li n e s e g m e n t c o n n e c t i n g th e

e n d p o i n t s o f th e g r a p h o f th e f u n c t i o n :

r = f ( b ) - r ~jaJ + b f ( a ) - a f ( b ) , v x E [ a , b ] .b - a b - a





SOLUTIONS OF NONLINEAREQUATIONS 157

w i t h w h i c h t h e s e p o i n ts a p p e a r i n t h e g r a p h o f t h e u n i v a r i a te f u n c t i o n f ( x ) w h i c h

is:

9 . [ d l - l - d u - u ] i ' " .

T h i s n a t u r a l l y p r o v i d e s a p a r t i ti o n i n g o f t h e i n it ia l i n t e r v a l [ a , b ] i n t o c o n v e x

s u b i n t e r v a l s [d l i , d u i ] a n d c o n c a v e o n es [d u i , d l i ] .

T h e p r o c e d u r e f o r l o ca t in g t h e f i rs t p o i n t w h e r e t h e c o n v e x e n v e l o p e c h a n g e s

r e p r e s e n t a t i o n d e p e n d s o n w h e t h e r f ( x ) i s c o n v e x o r c o n c a v e a t x = a . I f f i s

c o n c a v e a t x = a t h e n t h e in i ti a l s e g m e n t o f th e c o n v e x e n v e l o p e is a l in e . T h e

n e x t s e g m e n t o f t h e c o n v e x e n v e l o p e is t h e f u n c t i o n it s e ff s ta r ti n g a t t h e p o i n t x

w h e r e th e s l o p e f ~ o f f e q u a l s t h e s l o p e o f th e l in e c o n n e c t i n g a w i t h x .

f ( x ) - f ( a ) = f ' ( x ) .

x - - a

N o t e t h a t x b e l o n g s t o o n e o f t h e c o n v e x s u b i n t e r v a l s [ d l i , u i ] s i n c e f m u s t b e

c o n v e x a t x . T h i s i m p l i e s t h a t t h e t a s k o f l o c a t i n g x c o r r e s p o n d s t o d r a w i n g a t a n g e n t

f r o m t h e f i x e d p o i n t ( a , f ( a ) ) t o e a c h o n e o f t h e c o n v e x f u n c t i o n r e p r e s e n t a t i o n s

d e f i n e d i n t h e s u b i n t e r v a l s [ d l i, u i ] . B e c a u s e t h e r e e x i s t s a s i n g l e t a n g e n t t o a

c o n v e x f u n c t i o n d r a w n f r o m a p o i n t o u t s i d e t h e a c o n v e x f u n c t i o n [ 2 1 ] a n y s t a n d a r d

b i s e c t i o n a l g o r i t h m c a n b e u t i l i z e d t o l o c a t e x . T h e c o r r e c t s u b i n t e r v a l [ d l i , u i ] i st h e n t h e o n e w h i c h p r o v i d e s a l i n e t h a t d o e s n o t c u t - o f f a n y p o r t i o n o f t h e c u r v e

f ( x ) .

I f f i s c o n v e x a t x = a , t h e n t h e i ni ti al s e g m e n t o f t h e c o n v e x e n v e l o p e

c a n b e e i t h e r a l i n e o r t h e f u n c t i o n i t se l f. I f t h e r e e x i s ts a c o n v e x s u b i n t e r v a l

[ l i u i ] , = 2 , . . . w h e r e t h e e q u a t i o n f ( ) f ( ) = f ' ( ) ( - a ) h a s a s o l u t i o n

x w h i c h d e f i n e s a l i n e t h a t d o e s n o t c u t - o f f a n y p o r t i o n o f t h e c u r v e f ( x ) t h e n t h e

i n it ia l s e g m e n t o f t h e c o n v e x e n v e l o p e is a l in e c o n n e c t i n g t h e p o i n t s ( a , f ( a ) ) a n d

( x , f ( x ) ) . O t h e r w i s e , t h e i n it ia l s e g m e n t o f t h e c o n v e x e n v e l o p e i s t h e f u n c t i o n

f i ts e lf . T h e l a st p o i n t o f th i s s e g m e n t Xl i s f o u n d b y l o c a t i n g t h e e n d p o i n t s

X l , x 2 o f th e n e x t s u b i n te r v a l w h e r e t h e c o n v e x e n v e l o p e b e c o m e s a l in e s e g m e n t .

T h i s c o r r e s p o n d s t o d r a w i n g a c o m m o n t a n g e n t t o f i n s i d e t h e i n t er v a l s [ d / l , d u l l

a n d [ d l i, d u i ] , i = 2 , . . . . a n d i s t h e s o l u t io n o f t h e fo l l o w i n g s y s t e m o f t w o

e q u a t i o n s :

f ( x 2 ) - f ( z l ) = f ' ( x l ) = : ' ( x 2 ) ,

X 2 - - X l

w h e r e X l E [ d l l , d u l l a n d x 2 E [ d l i , d u l l , i = 2 , . . . . A g a i n , t h e c o r r e c t s u b i n t e r v a l

[ d l i , d u i ] , i = 2 , . . . i s t h e n th e o n e f o r w h i c h t h e l in e c o n n e c t i n g t h e p o i n ts

( X l , f ( x l ) ) a n d ( x2 , f ( x 2 ) ) d o e s n o t c u t - o f f a n y p o r t io n o f t h e c u r v e f ( x ) . T h e

n e x t l i n e s e g m e n t is t h e n f o u n d b y i te r a ti v e ly s o l v in g t h e s y s t e m o f t w o e q u a t io n s

f o r lo c a t i n g t h e n e w p o i n t s x l , x 2 . T h i s t im e h o w e v e r , X l E [ d l i , d u i ] a n d x 2 E

[ d l j, d u j ] , j = i + 1 , . . . . T h i s is c o n t i n u e d u n t il t h e e n d p o i n t x = b i s m e t . B a s e d o n

t h is a n a l y s i s a n i t e r a t iv e p r o c e d u r e is d e f i n e d f o r c o n s t r u c t i n g t h e c o n v e x e n v e l o p e




of arbi trary univaria te no nc on ve x funct ions. In the nex t sect ion, an a l ternat ive

co n v ex lo w er b o u n d in g me th o d i s d i s cu s sed fo r p ro b lems in v o lv in g o n ly s ig n o mia l

terms.

4 . Co n v e x L o we r B o u n d in g o f S ig n o mia l P r o b le ms

A la rge num ber o f sys tem s of non l inea r equa l i t ies sub jec t to no n l inear inequa l i t ies

have o r can assum e a gen era l ized geom etr ic p rob lem formula t ion [24]:

m in st , s

w h e r e

su bjec t to G J ( t ) - G ~ ( t ) - s < 0 , j e A r S

-- G J ( t ) + G ~ ( t ) - s < 0 , j E A r E

G ) ( t ) - G ~ ( t ) < 0 , j e A r I

t i > 0 , i = l , . . . , N ,

N

v T ( t ) = c j k 1 - i i , j 6 . A l E , m = 1 , 2 , 3 , 4 .

kEK'~ i = 1

H e r e t = ( t l , . . . , tN ) is the posi t ive var iable vector ; G~n, m = 1,2 , 3, 4, j E ArE (3

A r /a r e p os i t ive posy nom ia l func t ions in t ; a i jk a re arbi trary rea l cons tan t expo-

nen ts ; wh ereas cjk are given positive coeff ic ients . Final ly , se ts K ~ , m = 1 ,2 , 3 , 4

co u n t h o w m an y p o s i ti v e ly /n eg a tiv e ly s ig n ed mo n o m ia l s fo rm th e p o sy n o m ia l s

G ~ , m = 1 ,2 , 3 , 4 respect ively . Clearly , the abo ve formulat ion correspon ds to a

h igh ly non l inear op t imiza t ion p rob lem wi th a non con vex cons tra in t se t and poss i -

b ly d is jo in t feasib le reg ion . H owe ver , a f te r app ly ing the t ransformat ion ,

t i = e x p x i , i = 1 , . . . , N

to the o r ig ina l fo rmula t ion we ob ta in the fo l lowing op t im iza t ion p rob lem w hich

invo lves cons tra in ts tha t a re the d i f fe rence o f two conv ex func t ions .

m i n sX~ S

subject to G l( x ) - G 2(x) - s _~ 0 , j e AlE

- G } ( x ) + G ~ ( x ) - s _ ~ 0 , j e A r E

G ~ ( x ) - G ~ ( x ) < 0 , j e A r x

x L <_ x i < _ x U , i = l , . . . , N



S O L U T IO N S O F N O N L I N E A R E Q U A T IO N S 159

w h e r e

G ~ ( x ) = ~ c j k e x p a i j k z i , j E A : E , m = 1 , 2 , 3 , 4 .kEKjm k i = l

A c o n v e x l o w e r b o u n d i n g f o r m u l a t i o n c a n b e o b t a in e d b y u n d e r e s ti m a t i n g e v e r y

s e p a r a b l e c o n c a v e f u n c t i o n w i t h a l i n e a r f u n c t i o n . A n a n a l y s i s o n t h e c o n v e x

l o w e r b o u n d i n g p r o c e d u r e as w e l l as o n a n u m b e r o f t e c h n i q u e s t h a t im p r o v e t h e

c o m p u t a t i o n a l e f f i c ie n c y o f t h e a p p r o a c h a r e d e s c r i b e d i n d e t a i l i n [ 24 ] .

5 . C o n v e x L o w e r B o u n d i n g U s i n g a B B

F o r ar b it r ar y n o n c o n v e x f u n c t i o n s f E C 2 " [ X L , X U ] ~ ~]'~, a co n v ex lo w er

b o u n d i n g f u n c t i o n / 2 o f f c a n b e d e f in e d b y a u g m e n t i n g f w i t h th e a d d i t io n o f a

s e p a ra b l e c o n v e x q u a d r a ti c f u n c t i o n o f x a s p r o p o s e d i n [ 23 ] a n d g e n e r a l i z e d t o

inc lu de equa l i ty and ineq ua l i ty cons t ra in t s in [4] .

w h e r e

_ - : < x / § - x ) " ( x . - x )

( 1 }i n A k ( x ) 9_> m a x 0 , - ~ k

x L < x < x U

N o t e t h a t a i s a n o n n e g a t i v e p a r a m e t e r w h i c h m u s t b e g r e a te r o r equa l to the

n e g a t iv e o n e h a l f o f t h e m i n i m u m e i g e n v a l u e o f f o v e r x L < x < x v . T h e

p a r a m e t e r a c a n b e e s t i m a t e d ei t h e r t h r o u g h th e s o l u t i o n o f a n o p t i m i z a t i o n p r o b l e m

o r b y u s i n g t h e c o n c e p t o f t h e m e a s u r e o f a m a t r i x . T h e e f f e c t o f a d d i n g t h e

e x t r a se p a r a b l e q u a d r a t i c te r m t o S i s t o m a k e L c o n v e x b y o v e r p o w e r i n g t h e

n o n c o n v e x i t y c h a r a c te r i st i cs o f f w i t h t h e a d d i t i o n o f t h e t e r m 2 a t o th e d i a g o n a l

e l e m e n t s o f i ts H e s s i a n m a t r ix . T h i s f u n c t i o n s d e f i n e d o v e r t h e r e c t a n g u l a r d o m a i n

x L , x V [ , a o f p r o p e r t i e s u s u s e a s a t i g h t c o n v e xn v o l v e s n u m b e r w h i c h enab le to

l o w e r b o u n d i n g f u n c t i o n o f f . T h e s e p r o p e r ti e s, w h o s e p r o o f i s g i v e n i n [ 23 ], a r e

a s fo l lows :

P R O P E R T Y 1. s i s a v a l i d u n d e r e s t i m a t o r o f f .

[ x L , x " ] , c ( x ) _ < : ( x ) .x

P R O P E R T Y 2 . / 3 m a t c h e s f a t a ll c o m e r p o i n t s .

P R O P E R T Y 3 . s i s c o n v e x i n [ x L , x U ] .




P R O P E R T Y 4 . T h e m ax i m u m sep a ra t io n b e t w een s an d f i s b o u n d e d an d

propor t iona l to a and to the square o f the d iagona l o f the cur ren t box co n-

straints.

m ax ( f ( x ) - Z : ( x ) ) = 8 8 x U - xLl] 2.x L < x < x t r

P R O P E R T Y 5 . T h e u n d e r e s t im a t o r s co n s t ru c t ed o v e r su p e rse ts o f th e cu r r en t

se t a r e a lway s l e s s t i g h t t han the unde res t imator cons t ruc ted over the cu r ren t box

cons t r a in t s fo r ev ery po in t w i th in the cur ren t box cons t ra in ts .

P R O P E R T Y 6 . E co r re sp o n d s to a r e lax ed d u a l b o u n d o f t h e o ri g in a l f u n c t io nf .

This type o f conve x lower boun ding i s u t il i zed fo r a rb it r a ry no nco nve x func t ions

which l ack any spec i f i c s t ruc tu re tha t migh t enab le the cons t ruc t ion o f a more

cu s t o m i zed co n v ex l o w er b o u n d i n g f u n c ti o n .

6 . P r o c e d u r e f o r L o c a t i n g A l l S o l u t i o n s

6.1. DESCRIPTION

A de te rmin i s t ic g loba l op t imiza t ion approach i s p roposed fo r loca t ing a l l e - so lu t ions

of no n l inear sys tems o f equ a l i ti es sub jec t to no n l inear inequa l i ty cons tr a in ts (S ).

By in t roduc ing a s l ack var i ab le , the in i ti a l p rob lem (S) i s tr ansfo rme d in to a g lob a l

o p t im i za t io n p r o b l em ( P ) w h o se m u l ti p le g l o b a l m i n i m a ( i f an y ) co rr e sp o n d t o t h e

mu l t ip le so lu t ions o f (S ). A zero ob jec t ive func t ion va lue d eno tes the ex i s t ence o f

a so lu t ion w hereas a s t ri c tly pos i t ive ob jec t ive func t ion va lue impl ies tha t (S ) has

no so lu tions . Th i s def ines a one- to -one cor respo nden ce be tw een so lu t ions o f the

co n s t r a in ed sy s t em o f eq u a t io n s ( S ) an d m u l t ip l e g l o b a l m i n i m a w i t h an o b j ec ti v eva lue o f ze ro fo r p rob lem (P) . How ever , i t has been show n [14] tha t no a lgor i thm

can ex ac t ly loca te a l l m ul t ip le g loba l min im a of (P ) wi th a f in it e nu m ber o f func-

t ion eva lua tions . A cor ro la ry o f thi s r esu l t [14] i s tha t no a lgor i thm can a lwa ys

loca l i ze , wi th a f in i te num ber o f func t ion ev a lua t ions , a l l g loba l ly op t imal po in t s

b y co m p ac t su b r ec t an g le s i n o n e - t o - o n e co r r e sp o n d en ce w i t h t h em . T h e r e f o re , a

m ore t r ac tab le t a rge t, t han f ind ing a l l exac t g loba l m in im a of (P ) , i s to f ind a rb i-

t r ar i ly smal l d i s jo in t subrec tang les con ta in ing a l l g loba l ly op t im al po in t s o f (P ) ,

poss ib ly no t in a one- to -one cor responden ce .

These mul t ip le e -g loba l min ima of (P ) , ( i f any) can then be loca l i zed based

o n a b r an ch an d b o u n d p r o ced u r e i n v o l v in g t h e su cces s i v e r e f in em en t o f co n v ex

re laxa t ions (R) o f the in i ti a l p rob lem (P). Fo rmu la t ion (R) i s ob ta ined by r ep lac ing

th e n o n c o n v e x fu n c ti o n s '~3fin'~ --hn'~176 vk"n~ w it h t i g h t , c o n v e x l o w e r b o u n d i n g

func t ions ~ n o n c ~znone p,none by fo l lowing some of the t echn iques d i scussed in+ , J ~ - - , 3 ~ k '

t he p rev ious sec t ion . Because (R) i s convex , i t s g loba l min imum wi th in some






6.2. ALGORITHMIC STEPS

S T E P 0 - I n i t i a l i z a t i o nA s i z e t o l e r a n c e e , a n d a f e a s i b i l i ty to l e r a n c e e a r e s e l e c t e d a n d t h e i te r a t i o n

c o u n t e r I t e r i s s e t t o o n e . A p p r o p r i a t e g l o b a l b o u n J s X L B D ~ X U B D o n x a r e c h o s e n

a n d l o c a l b o u n d s X L ' I t e r , X U ' I t e r f o r t h e f i rs t i t e r a ti o n a r e s e t t o b e e q u a l t o t h e g l o b a l

o n e s . F i n a l l y , s e l e c t a n i n i t i a l p o i n t X c ' I t e r t h a t s a t i s f i e s t h e l i n e a r e q u a l i t i e s a n d

c o n v e x i n e q u a l it i e s o f ( P ) .

S T E P 1 - F e a s i b ~ i ty a n d C o n v e r g e n c e C h e c k

I f t h e m a x i m u m v i ~ a t i o n o f a ll n o n c o n v e x c o n s tr a in t s o f ( P ) c a l c u la t e d a t t h e

c u r r e n t p o i n t x c' I te~ f o r ( s = 0 ) i s l e s s t h a n e ,

m a x [ m a x h~ ~ max g ~ ~[ j~H.o .r ' keX.o.oi --

t h e n t h e p o i n t x r i s a e y - s o l u ti o n o f (S ) . F a t h o m c u r r e n t r e c t a n g l e i f it s d i a g o n a l

i s le s s t h a n e ~,

I Ix U , I t e r _ x L , I t e r l l < ,~ .

a n d G O T O S t e p 4 . O t h e rw i s e , c o n t i n u e w i t h S T E P 2 .

S T E P 2 - P a r t i ti o n i n g o f C u r r e n t R e c t a n g l e

Ix L , I t er , x U , It e~] i s p a r t i t i o n e d i n t o t h e f o l l o w i n g t w oT h e c u r r e n t r e c t a n g l e

r e c t a n g l e s ( r = 1 , 2 ) :

" L , I t e rX 1

L , I t e rX l I t e r

xU1 , I t e r

L , i t e r _ _ x U , I t e r , ~X l l t e r T l i f e r

2

L , I t e r U , I t e rX N X N

L , I t e r x U l , I t e rX 1

L ' I t e v - - x U ' I t e r ~X l i t e r " t- l i f e r ) U , t e r

2 X I I t e r

:

X ~ I t e r X ~ I t e r

w h e r e l I t e r c o r r e s p o n d s t o t h e v a r i a b l e w i t h t h e l o n g e s t s i d e i n t h e i n i t i a l

r e c t a n g l e ,

l i t er = a r g m a x ( x ~ , z te r _ z L , I t e r )

S T E P 3 - S o l u ti o n o f C o n v e x P r o b l e m s I n s i d e S u b r e c t a n g l e s

S o l v e t h e f o l lo w i n g c o n v e x o p t i m i z a t io n p r o b l e m ( R ) i n b o t h s u b r e c ta n g l e s

( r = 1 , 2 ) b y u s i n g a n y c o n v e x n o n l i n e a r s o l v e r ( e .g . M I N O S 5 . 4 [2 8 ]) . I f t h e

r , l ter i s n e g a t i v e t h e n , i t i s s t o re d a l o n g w i t h t h e v a l u e o f th e v a r i a b l e s xs o l u t i o n S so




r ,r ter . r r , I ter i s s t ri c tl y p o s i t i v e t h e n t h e e l e m e n t ( r , I t e r )t t h e s o l u t i o n p o i n t Xso t . xi Sso

i s f a th o m e d .

S T E P 4 - U p d a t e I te r at io n C o u n t e r I t e r a n d L o w e r B o u n d 8 L B D

T h e i te r a t io n c o u n t e r i s in c r e a s e d b y o n e ,

I t e r ~ I t e r + 1

a n d t h e l o w e r b o u n d 8 L B D i s u p d a t e d t o b e th e m i n i m u m s o l u t i o n o v e r t h e st o r e d

o n e s f r o m p r e v i o u s i t e ra t io n s . F u r t h e rm o r e , t h e s e l e c t e d s o l u t i o n i s e r a se d f r o m t h e

s t o r e d s e t .

8 L B D r l , I t e r t

= 8 s o l

r ' , I t e r I 9 r , Iw h e r e Sso I = m m S s o z, r = l , 2 , I : l , . . . , I t e r - 1 .

S T E P 5 - U p d a t e C u r r e n t P o i n t X c ' I t e r a n d C u r r en t B o u n d s X L ' I t e r , x U , I t e r

T h e c u r r e n t p o i n t i s s e l e c t e d t o b e t h e s o l u t io n p o i n t o f t h e p r e v i o u s l y f o u n d

m i n i m u m s o lu t io n in S T E P 4 ,

x C , I t e r r t , I t e r I

: X s o l

a n d t h e c u r r en t r e c t a n g l e b e c o m e s t h e su b r e c t a n g l e c o n t a i n i n g t h e p r e v i o u s l y f o u n d

s o l u t i o n ,

" x l L , I t e r t x U l , I te r t

x L , I t e r 'l i t e r !

L , I t e r IX N

L , I t e r IX 1

x L , I t e r t . l .x U , I t e r t

l i t e r ! ~ l i t e r t ]

2

U , I t e r IX N

I x , I t e r r _ U , I~ e r Il i t e r ! - I -X l l t er !

2

L , I t e r IX N

x U , I te r l

x U , I t e r 'I I t e r t

U , I t e r IX N

, i f r ' = 1 ,

, i f r l = 2 .

S T E P 6 - C h e c k f o r C o n v e r g e n c e

I F 8 L B D < O, t h e n r et u rn t o S T E P 1.




Otherw ise , t e rmina te .

M a t h em a t i ca l p r o o f t h a t t h e p r o p o sed p r o ced u r e is g u a r an t eed to co n v e r g e t o ase t o f d i s jo in t r ec tang les con ta in ing a l l g loba l m in im um so lu tions o f (P ) i s g iven

based on the ana lys i s o f a s t andard de te rmin i s t i c g loba l op t imiza t ion a lgor i thm

p r esen t ed in [1 5]. B ecau se t h e em p l o y ed b r an ch an d b o u n d t ech n i q u e fa t h o m s

only r ec tang les g u a r a n t e e d n o t t o c o n ta i n a n y g l o b a l m i n i m a o f (P) no solut ions

of (P ) which a re a t l eas t e r apar t a r e missed . By fo l lowing the p roof in [23] , a

suf f ic i en t cond i t ion fo r the p roposed b ran ch and bou nd a lgor i thm to be co nve rgen t

to the g loba l min ima , r equ i r es tha t the bound ing opera t ion must be c ons i s t e n t an d

the s e lec t ion opera t ion b o u n d i m p r o v i n g .

A boun ding opera t ion i s ca l led c o n s i s t e n t i f ( i) a t every s t ep any un fa thom ed

par t i t ion can be fu r ther r e f ined , and ( i i ) fo r any in f in i t e ly decreas ing sequence

o f su cces s i v e ly r e fi n ed p a r ti ti o n e l em en t s t h e g ap b e t w e en t h e l o w er an d u p p e r

bou nds g oes to ze ro as the i terat ions go to inf in ity. D ue to prop er t ies (1) , (2) , (3) o f

S ec t i o n 2 t h e g ap b e t w een t h e l o w e r an d u p p e r b o u n d f o r an y p a rt it io n e l em en t

goes to ze ro as the s i ze o f the par t it ion e lem ent goes to ze ro as w el l . Fur therm ore ,

the em ploy ed b i sec t ion sub d iv i s ion p rocess (b i sec t ion a long the longe s t s ide) is

exhaus t ive beca use the s i ze o f an in f in i te ly par t i tioned e le m ent goes to ze ro . There-

fo re , the bou nding opera t ion i s cons i s ten t . Al so , the em ploy ed se lec t ion opera t ion

is b o u n d i m p r o vi n g b ecau se t h e p a r t it io n e l em en t w h e r e t h e ac t u a l l o w er b o u n d

i s a t t a ined i s se lec ted fo r fu r ther par t it ion in the im m edia te ly fo l lowing i te r a tion .

Therefore acco rd ing to T heo rem IV.3 . in [15] the em ploy ed g loba l op t imiza t ion

a lgor i thm i s c o n v e r g e n t t o the g loba l min im a of (P ). In the nex t sec t ion the p ropose d

g loba l op t im iza t ion a lgor i thm i s app l i ed to a nu m be r o f exam ple p rob lems.

7. Com putational Results

In th is sec t ion , a num ber o f t es t p rob lem s a re add ressed wh ich a re a ime d a t de te r-m in ing the ab i l ity o f the appro ach to f ind a ll so lu t ions o f cons t r a ined sys tems o f

eq u a t io n s w i t h r ea so n ab l e co m p u t a t i o n a l r eq u ir em en t s . T h e p r o p o sed b r an ch an d

b o u n d co n v ex l o w er b o u n d i n g a lg o r i th m h as b een i m p l em en t ed i n G A M S [5 ] an d

com puta t iona l t imes a re repor ted fo r a ll exam ples on a HP -730 wo rks ta t ion w i th

s ize and f eas ib i l ity to le rances o f 10 -4 .

EX A M PL E 1 . The fir st exam ple invo lves the loca t ion o f a l l t he s t a tionary po in t s

o f the Him m elb lau func t ion as descr ibed in [33] .

4 x 3 + 4 X l X 2 + 2 x ~ - 42 xl - 14 = 0

4x32 + 2x 2 + 4 x l x 2 - 26x2 - 2 2 = 0

- 5 . 0 < X l < 5 . 0

- 5 . 0 < x 2 < 5 .0 .



SOLUTIONS OF NONLINEAREQUATIONS 165

F i rs t, t h e c h a n g e o f v a r ia b l e s

9 9

Y l = l . 0 + ~ ( X l + 5 . 0 ) , Y E = I . 0 + ( x 2 + 5 . 0 )

i s p e r f o r m e d w h i c h e n s u r e s t h a t a ll v a r i a b le a r e p o s i ti v e . T h i s r e su l t s i n t h e f o l l o w -

i n g s y s t e m o f e q u a t i o n s :

2 3 30 2 20 2 1 2 17- - + V ly2 + 7 v2 + E y l - 1 = o

2 y ~ _ 3 0 2 2 1 2 7 1 2 2 0121 ] - ~ Y 2 + i - ~ Y l Y 2 + " i- ~Y 2 + - ~ Y l - - T ~ Y l - 1 = o

1.0 _< yl ~ 10.01.0 < Y2 _< 10.0

T h e n t h e e x p o n e n t i a l v a r ia b l e t r a n s f o r m a t i o n , a s d e s c r i b e d i n s e c t i o n 4 , i s a p p l i e d .

T h e r e s u l t i n g p r o b l e m i s s o l v e d in 1 9 7 it e ra t io n s a n d 1 0 .8 9 s e c o n d s o f C P U t i m e .

A l l n i n e s o l u t i o n s a r e f o u n d a n d s h o w n i n T a b le I .

TABLE I. Nine solutions of

Example 1

# S o l x 1 x 2

1 -0.2709 -0.9230

2 -0 .1279 - 1.9538

3 3.5844 -1.8481

4 3.3852 0.0739

5 3.0000 2.0000

6 0 .0867 2 .8843

7 -2.8051 3.1313

8 -3.0730 -0.0814

9 -3.7793 -3.2832

E X A M P L E 2 . T h i s e x a m p l e a d d r es s e s t h e e q u il i b ri u m o f t h e p r o d u c t s o f a h y d r o -

c a r b o n c o m b u s t i o n p r o c e s s [ 25 ]. T h e p r o b l e m i s r e f o r m u l a t ed i n t h e ' e l e m e n t

va r iab le s ' space .

YlY2 + Y l - - 3y5 = 0

2ylY 2 + Yl + 3R10Y22 + Y2y23 + RTY2Y3 + R9Y2Y4 + RsY 2 - R y5 = 0

2 y 2 y 2 + R 7 Y2 Y 3 + 2 R s y 2 + R 6 y 3 - 8y5 = 0

R 9 Y 2 Y 4 + 2 y 2 - 4 R y 5 = 0

Y lY 2 + Y l + R lo y 2 + Y 2Y ~ + R7Y 2Y 3 + R9Y 2Y 4

+ R a y s + R s y ~ + R 6 Y 3 + y ~ - 1 = 0




0 . 0 0 0 1 < y i < 1 0 0 .0 , i = 1 , . . . , 5

T h e v a l u e s o f t h e p a r a m e t e r s R , R i , i = 5 , . . . , 1 0 a r e s h o w n i n T a b le I I.

TABLE II. Parameters of Example 2

R R5 R6 /~ 7 /% /% R10

10 1.930 10 -1 4.106 10 -4 5.451 10 -4 4.497 10 -7 3.407 10 -5 9.615 10 -7

U s i n g t h e e x p o n e n t i a l v a r i a b l e t r a n s f o r m a t i o n d e s c r i b e d i n S e c t i o n 4 , t h e s i n g le

s o l u t i o n o f t h e p r o b l e m i s f o u n d a f t e r 6 3 1 i t e r a t io n s a n d 3 1 . 7 s e c o n d s o f C P U t i m e(see Tab le I I I ).

TABLE III. Solution of Example 2

X1 X~ ~3 X4 ~5

0.00311410 34.59792453 0.06504178 0.85937805 0.03695186

E X A M P L E 3 . T h i s e x a m p l e [6 ] a d d r es s e s a b a d l y s c a l e d s y s t e m s o f e q u a ti o n s:

1 0 4 X l X 2 - 1 = 0

e x p ( - x l ) + e x p ( - x 2 ) - 1.001 = 0

5.49 01 0 -6 <_ Xl < 4 .553

2 . 1 9 6 1 0 - 3 < x 2 < 1 8 . 2 1 0

T h e b i l i n e a r t e r m s XlX2, -x lx2 are u n d e r e s t i m a t e d b a s e d o n t h e a n a l y s i s in

S e c t io n 3 , a n d th e te r m s e x p ( - X l ) , e x p (-x2) are c o n v e x , h o w e v e r , - e x p ( - X l ) ,

- e x p ( - x 2 ) a r e u n i v ar ia te c o n c a v e te r m s a n d a r e c o n v e x l o w e r b o u n d e d w i th al i n e s e g m e n t .

A f t e r 3 2 i te r a t io n s a n d 1 .5 s e c o n d s o f C P U t i m e , it i s s h o w n t h a t

( x ~ , x ~ ) = ( 0 . 0 0 0 0 1 4 5 0 6 7 , 6 . 8 9 3 3 5 2 8 7 )

i s a u n i q u e s o l u t i o n t o t h e p r o b l e m . N o t e t h a t, t h e s e c o n d s o l u t io n ( X l , x 2 ) =

( 0 . 0 0 0 0 1 0 9 8 , 9 . 1 0 6 ) r e p o r t e d i n [6 ] d o e s n o t s a t i s f y t h e n o n l i n e a r e q u a t i o n s .

E X A M P L E 4 . T h i s te s t e x a m p l e [ 11 ], i n v o l v e s a b l e n d o f t r ig o n o m e t r ic a n d

e x p o n e n t i a l t e r m s .

0.5 s in (XlX2) - 0.25 x2/T r - 0.5X l = 0

(1 - 0 . 2 5 / - ) ( e x p ( 2 X l ) - e ) + ez21 - 2 e 1 = 0

0 . 2 5 < x l < 1

1.5 < x2 < 6 .28.



SOLUTIONSOF NON LINEAREQUATIONS 167

T h e o ~-b ased u n d e r e s t i m a t i o n , d e s c r i b e d i n S e c t i o n 5 , w a s c h o s e n t o a d d r e s s t h e

c o n v e x l o w e r b o u n d i n g o f th e t er m s i n ( x l x 2 ) . T h e e i g e n v a l u e s o f th i s te r m a r e

e q u a l t o :

9 1 ,2 = _ 1 ( X l + x 2 ) s i n ( X lX 2 )

q - l ~ 4 - 8 X l X 2 s i n ( x l x 2 ) c o s ( x l x 2 , + [ ( x 2 + x 2 ) 2 - - 4 ] s i n ( X l X 2 ) 2 2 .

A l o w e r b o u n d o n t h is e x p r e s s i o n i s t h e n :

A m in > - m a x [ ( x L ) 2 , (x lU ) 2 ] -- m a x [ ( x L ) 2 , ( x U ) 2 ] .

T h e r e f o r e ,

m a x [ ( x L ) 2 , (x V ) 2 ] + m a x [ ( x L ) 2 , ( x U ) 2 ]

2

T w o s o l u t i o n s a r e fo u n d f o r t hi s p r o b l e m i n 4 5 i te r a ti o n s a n d 2 . 0 s e c o n d s o f C P U

t i m e ( s e e T a b l e I V ) . N o t e t h at , b o t h s o l u t i o n s w e r e m i s s e d i n [ 6 ].

TAB LE IV. Solutions of E xample 4#So l z~ z~

1 0 . 2 9 9 4 4 8 6 9 2 .8 3 69 2 77 7

2 0 . 5 0 0 0 0 0 0 0 3 .1 4 15 9 26 5

E X A M P L E 5 . T h i s t es t p r o b l e m is B r o w n ' s a l m o s t li n e a r s y s t e m [ 17 ].

2Xl+x2+x3+x4--]-z5-6 = 0 ,

x l + 2 x 2 + x 3 + x 4 + x s - 6 = 0 ,

X l + X 2 + 2 x 3 + x 4 + x 5 - 6 = 0 ,

X l + x 2 + x 3 + 2 x 4 + x 5 - 6 = 0 ,

Xl X2X3X4X5 - 1 = 0

- 2 < x i < 2 , i = 1 , . . . , 5 .

TAB LE V. Solutionsof Example 5

Sot# z l z2 z~ z4 zs

1 1.000 1.000 1.000 1.000 1.000

2 0 . 91 6 0 . 9 1 6 0 . 9 1 6 0 . 91 6 1.418



168 C.D. MARANASAND C. A. FLOUDAS

TABLE VI. Computational require-

ments for Example 5

a Iterations CPU (sec)

1000 352 22.16

100 112 7.42

10 37 2.26

5 12 0.69

1 7 0.35

This system exhibits two solutions: shown in Table V. The o~ parameter was used

to convex lower bound the last and only nonconvex constraint. The computational

requirements for different values of c~ are shown in Table VI.

Note that the total number of iterations remains relatively small even for very large

values of c~. Moreover, a value of o~ of as small as one appears to be sufficient.

EXAMPLE 6. This example addres sesar obot kinem atics probl em[17 ].

4.73110-3xlx3 - 0.3578x2x3 - 0.1238xl + x7

-1 .63710-3x2 - 0.9338x4 - 0.3571 = 0

0.2238XlX3 + 0.7623x2x3 + 0.2638xl - x7

-0. 077 45x2 - 0.6734x4 - 0.6022 = 0

x 6 x 8 + 0.3578Xl + 4.73110-3x2 = 0

-0 .7 62 3x - 1 + 0.2238x2 + 0.3461 = 0

Xl + x ~ - 1 = 0

+ l = 0

x 2 + x ~ - 1 = 0

x72 + x82 - 1 = 0

- l_ <x i < 1, i= 1 , . . . , 8

The only nonconvex terms in the formulation are the bilinear terms x x x 3 , x 2 x 3 , x 6 x 8

and are convex lower bounded based on the analysis of Section 3. All distinct 16

solutions of this problem are found in 2188 iterations and 109.58 seconds of CPU

time.

EXA MPLE 7. This example involves the solution of a circuit design problem with

extraordinary sensitivities to small perturbations [32] leading to the following set



SOLUTIONSOFNONLINEAREQUA~ONS 169

o f e q u a t i o n s .

(1 - XlX2)X3 { e x p [x 5 ( g l k - g 3/ex 71 0 - 3 - g 5 k x 8 1 0 - 3 ) ] - 1 }

- - g5k+g4kx2=O, k = 1 , . . . , 4

( 1 - X l X 2 ) X 4 ( e x p I x 6 ( g l k - g 2 k - g 3 k X 7 1 0 - 3 --k g 4 k x 9 1 0 - 3 ) ] - - 1 }

-g5kx l + g 4 k = 0 , k = 1 , . . . , 4

X l X 3 - - X 2 X 4 ~ 0

O < x i < l O , i = 1 , . . . , 9

w h e r e

k = 1 k = 2 k = 3 k = 4

g l k 0 . 4 8 5 0 0 . 7 5 2 0 0 . 8 6 9 0 0 . 9 8 2 0

g 2k 0 . 3 6 9 0 1 . 2 5 4 0 0 . 7 0 3 0 1 . 4 5 5 0

g 3k 5 . 2 0 9 5 1 0 . 0 6 7 7 2 2 . 9 2 7 4 2 0 . 2 1 5 3

g 4k 2 3 . 3 0 3 7 1 0 1 . 7 7 9 0 1 1 1 . 4 6 1 0 1 9 1 . 2 6 7 0

g s k 2 8 . 5 1 3 2 1 1 1 . 8 4 6 7 1 3 4 . 3 8 8 4 2 1 1 . 4 8 2 3

T h e a p a r a m e t e r w a s u t i li z e d t o c o n v e x l o w e r b o u n d t h e v a r io u s n o n l i n e a r t e rm s .

T h e s i n g l e s o l u t i o n o f t h e p r o b l e m ,

(~ = 0 . 8 9 9 9 9 9 , x ~ = 7 . 9 9 9 6 9 3 /

x ~ = 0 . 4 4 9 9 8 7 , x ~ = 5 . 0 0 0 0 3 1

* = 1 . 0 0 0 0 0 6 , x ~ = 0 . 9 9 9 9 8 83x ~ = 2 . 0 0 0 0 6 9 , x ~ = 2 . 0 0 0 0 5 2

x ~ = 7 . 9 9 9 9 7 1 ,

w a s f ir st r e p o r t e d i n r e f e r e n c e [ 32 ]. C o m p u t a t i o n a l r e q u i r e m e n t s f o r v a r i o u s v a l u e s

o f a a r e s h o w n i n T a b l e V I I . N o t e t h a t t h e s e C P U r e q u i r e m e n t s a r e o n l y a s m a l l

f r a c t i o n o f th e o n e s r e p o r t e d i n [ 3 2 ].

TAB LE V II . Computa tional require-ments for Exam ple 7

a I tera t io ns CPU (see)

0.1 1645 987.910.01 212 143.41

F u r t h e r m o r e , a f t e r r e l a x i n g t h e v a r i a b l e b o u n d s t o

- 1 0 _< x i _< 1 0, i = 1 , . . . , 9




a secon d so lu t ion wa s found

/:~ = 0 .823226 , x~ = -2 .7 65 09 2 '~

x ~ = - 0 . 5 5 3 2 8 6 , x~ = 6 . 04 6 6 4 6 ]x ~ = 0 .6 7 1 8 78 , x ~ = 0 . 9 7 5 9 4 0 |

x ~ = - 0 . 9 9 9 6 7 7 , x ~ = - 1 . 7 0 8 4 8 9 ]

x~ = 8 .854525. /

w hich w as m issed in a l l p rev ious a t t empts a t so lv ing th i s p rob lem.

8 . S u m m a r y a n d C o n c l u s io n s

I n t h is p ap e r a d e t e r m i n is t ic b r an ch an d b o u n d t y p e a l g o r it h m w as p r o p o sed f o r

loca t ing a ll e -g loba l so lu t ions o f ce r t a in c l asses o f cons t r a ined sys tem s o f non l inear

eq u a ti o n s. T h e ap p r o ach i s b a sed o n t h e o n e - t o -o n e co r r e sp o n d en ce b e t w een t h e

m u l t i p le so l u ti o n s o f t h e n o n l i n ea r sy s t em s an d t h e m u l t i p le g l o b a l m i n i m a w i t h a

zero ob jec t ive va lue fo r the r esu l t ing no nco nve x op t im iza t ion p rob lem. Al l m ul -

t ip l e e - g l o b al m i n i m a o f t h e n o n co n v ex o p t i m i za ti o n p r o b l em a r e l o ca l iz ed b a sed

o n a co n s t r u c t i o n o f u p p e r b o u n d s w i t h f u n c t i o n ev a l u a t i o n s an d l o w er b o u n d

o n t h e g l o b a l m i n i m u m so l ut io n t h r o u g h t h e co n v ex r e l ax a ti o n o f t h e co n s t r a in t

s e t an d t h e so l u ti o n o f co n v ex m i n i m i za t io n p ro b l em s . B ased o n t h e f o r m o f t h e

par t ic ipa ting func t ions , a nu m ber o f a l t e rna tive t echn iques fo r co ns t ruc t ing th i scon vex r e laxa t ion a re p roposed . In par t i cu la r , by t ak ing advan tage o f the p roper t i es

o f p r o d u c t s o f u n i v a r ia t e f u n c ti o n s , cu s to m i zed co n v ex l o w er b o u n d i n g f u n c ti o n s

are in t roduced fo r a l a rge nu m ber o f express ions tha t a re o r can be t r ansfo rmed

in to p roduc t s o f un ivar i a te func t ions . The u t il i ty o f these con vex low er bound ing

func t ions tr anscends the spec i fi cs o f the roo t f ind ing p rob lem beca use they can be

i n co r p o ra t ed i n an y co n v ex l o w er b o u n d i n g a l g or it h m . A l t e r n a ti v e co n v ex r e l ax -

a t io n p r o ced u r e s i n v o l v e e i t h e r th e d i f fe r en ce o f t w o co n v ex f u n c t i o n s em p l o y ed

in o~BB [23] o r the expon en t i a l var iab le t r ansfo rmat ion ba sed underes t im ator s

em p l o y ed f o r g en e r a l iz ed g eo m e t r ic p r o g r am m i n g p r o b l em s [2 4]. T h e p r o p o sed

branch and bo und approac h i s guaran teed to loca l i ze a l l e -so lu tions o f (S ) wi th in

arb it ra r ily sm al l r ec tang les in a f in i te num ber o f i te r a tions . A n um be r o f exam -

p l e p r o b lem s f r o m m an y a r ea s o f re sea r ch h av e b een ad d r e s sed an d i n a l l ca se s ,

co n v e r g en ce to a ll m u l t i p le so lu t io n s w as ach i ev ed w i t h r ea so n ab l e co m p u t a t io n a l

e f fo r t. Fu r thermore , in ce r t a in cases ne w so lu tions we re iden t if i ed .

A p p e n d i x AC o n v e x L o w e r B o u n d i n g E x a m p l e s o f U n i v a r ia t e F u n c t i o n s

I n t hi s ap p en d i x a n u m b er o f co n v ex l o w er b o u n d i n g s it u at io n s a r e ex am i n ed .

( 1 ) B i l in e a r t e r m s

T h e co n v ex u n d e r e s ti m a t i o n o f b i l in ea r te r m s x y i ns ide the r ec tangu la r r eg ion

[ x L , x U ] x [ y L ,y U ] c a n b e h an d l ed b y i n v o k i n g T h e o r em ( 1 ) an d se tt in g f ( x ) = x




a n d g ( y ) = y :

x y > _ m ax { x L y + y L x - - x L y L ,

x V y + y V x - x V y U } .

Note tha t , t he lower bounding procedure can be appl i ed to a nega t ive ly-s igned

bi l inear term - x y by set t ing f ( x ) = - x and g ( y ) = y :

- x y > _ m ax { - x V y - y L x + x U y L ,

- -xL y - - y V x + x L y U } .

B ecause in t h is ca se f ( x ) , g ( y ) a r e l i near and therefore conc ave func t ions , we havef rom T heo rem (2) tha t t he ob ta ined convex low er bou nding func t ions a re iden t ica l

to the co nvex en ve lop es as we re f ir st der ived by [3] .

( 2 ) F r a c t i o n a l t e r m s

Co nve x lower bou nding o f the l inear frac tiona l te rm x / g i ns ide the rec tangular

reg ion [ x L , x U J x [yL , yU ] can a l so be accom pl i shed based on Theo rem (1 )b y

select ing f ( x ) = x a n d g ( y ) - 1 .

x > m a x { r 1 6 2

Y

r x

N ote that, r =xy-r, r x = -tr an d

rf i ( Y ~ + v ~ - g i f z s < o '

yLyU

r = x~b(Y L+Y ~- v) i f x s < O .yLyU

Therefore ,

7 - 7 -> m a x -V + i f x L > 0-- x ~ x LY y-~- - -k -~7~ i fx L < 0

[ . : ] }+ 7 - 7 i f z v >

x ~ _ ~ x U

7 - + 7 i fx U <



1 7 2 C.D. MARANAS AND C. A. FLOUDAS

T h e s a m e a p p r o a c h c a n b e u s e d f o r n e g a t i v e l y -s i g n e d l i n ea r f r a ct i o n al t e r m s . I n

t h is c a s e , h o w e v e r , w e h a v e f ( x ) = x , g ( y ) = - 1 / y . A f t e r f o l l o w i n g t h e s a m e

a n a l y s i s w e o b t a i n :

9 O o ]x > m a x Y Y ~ + ~ i f x L <-

y -- _ y_ _~ + y ~ _ ~ i f x L >

y ~ + ~ i f z g <_ 0

- - ~ r + ~ - @ - ~-~ if x g > 0 "

(3) Trilinear t e r m s

F r o m T h e o r e m ( 3) w e k n o w t ha t a p o s s i b l e c o n v e x l o w e r b o u n d i n g f u n c t io n o f x y z

i n s i d e t h e re c t a n g u l a r r e g i o n [ x L , xU ] x [ yL yU ] x [ z L , z U ] w i t h x L , y L , ZL > 0

is :

x y z > s 0 = m a x { x L s l + x s L - - x L s L , x U s l + X S f - - x U s T } ,

w h e r e S l = m a x { y L z + y z L -- y L z L , y U z + y z U - - yU z U }

a nd s L = y L z L , s ~ = y U z U .

H o w e v e r , T h e o r e m ( 4 ) s ta t es t h at t h e r e e x i st th r e e d i f fe r e n t c o n v e x l o w e r b o u n d i n g

s c h e m e s f o r tr i li n e a r t e rm s . T h e s e o t h e r t w o a l t e r n a t iv e s a re :

x y z >__ so = m a x { y L s l + y s L - -y L s L I ,y U s l + y s U - - y U s U } ,

w h e r e S l = m a x { x L z + x z L - - x L z L , x U z + x z U - - x U z U }

and sL 1 = x L z L , 8 ~ = x v z ~ ,

x y ~ _> ~0 : m a x { z ~ , + ~ 4 - z ~ 4 , ~1 + ~ 7 - Z ~ l ~ } ,

w h e r e S l = m a x { x L y + x y L - - x L y L , x U y + x y U -- x U y U }

and s ~ = x Z y z , s~= xV y v .

A f t e r e l i m i n a t i n g s o , s l a n d s u b s t i tu t i n g f o r s L , s ~ w e o b t a i n f o r t h e t h r e e d i ff e r e n t

c o n v e x l o w e r b o u n d i n g s c h em e s :

x y z > m a x { x y L z L -t- x L y z L + x L y L z -- 2 x L y L z L ,

x y U z U + x U y z L + x U y L z _ x U y L z L _ x U y U z U



SOLUTIONSOFNONLINEAREQUA~ON$

x y L z L + x L y z U + x L y U z -- x L y U z U _ x L y L z L ,

x y V z v + x V y z U + x V y U z _ 2 x V y U z U } ,

173

x y z > m a x { x y L z L + x L y z L + x L y L z -- 2 x L y L z L ,

x y U z L + x U y z U + x L y U z -- x L y U z L _ x U y U z U '

x y L z U + x L y z L + x U y L z -- xU y L z U _ x L y L z L

x y U z v + x V y z v + x V y V z -- 2 x V y V z V } ,

x y z > m a x { x y L z L + x L y z L + x L y L z -- 2 x L y L z L ,

x y L z U + x L y z U + x U y U z -- x L y L z U _ x U y U zU ~

x y U z L + x U y z L + x L y L z -- xU y U z U _ x L y L z L '

x y U z v + x V y z v + x U y U z -- 2 x V y V z V } .

The combination of ~1 three convex low ~ bounding alternatives ymlds the fol-

lowing eight linear functions in x, y, z whose maximum is a t ig~ convex low~

bounding ~nction ~ r x y z :

x y z > m a x { x y L z L + x L y z L + x L y L z - - 2 x L y L z L ,

x y U z U + x U y z L + x U y L z _ x U y L z L _ x U y U z U

x y L z L + x L y z U + x L y U z _ x L y U z U _ x L y L z L

x y U z L + x U y z U + x L y U z -- x L y U z L _ x U y U z U '

x y L z U + x L y z L + x U y L z -- x U y L z U -- x L y L z L ,

x y L z U + x L y z U + x U y U z _ x L y L z U _ x U y U z U

x y U z L + x U y z L + x L y L z _ x U y U zU _ x L y L z L '

x y U z U + x U y z U + x U y U z -- 2 x U y U z U } .

( 4) F r a c t i o n a l t r i l in e a r t e r m s

From Theorems (3), (4) we have that the three convex lower bounding alterna-

tives for ~ inside the rectangular region [ x L , x U ] x [y L, y U ] x [ zL , z g ] with

X L , y L ZL ~ 0 are:

xy > so

z= max { x L 81 --~ x s f - x L s f , x U 81 + x 8 U - x U 8 7 } ,

{ y L y y L y U y y U }

+ ZL ZLhere sl max + z U z v ' z




a n dy L S U y U

8 L = Z-'-ff~ = Z"--~

x y > so = m a x { y L s l + y s L - - y L s L , y U 81 + y 8 U __ y U 8 U } ,z

wh e r e 8 1 = m a x{ xL x x L X U x X U }

+ Z U Z U~ Z + Z L Z L

x L x U

4 = = 7

_ _ f z L z z L z U z z Ux y > s o = m a x + +z - ~ - ~ s U s U ' s l s L s L !

w h e r e S l = m a x { x L y + x y L - - x L y L , x U y + x y U - - x U y U }

a n d s lL = x L y L , s v = x U y U .

A g a i n , a f t e r e l i m i n a t i n g s o , 81 a n d r e p l a c i n g s L , s ~ w e o b t a in :

x y L x L y x L y L x L y L

X y z m a x ~ + 7 - + - - z 2 - ~ ,

x y L x L y x L y U x L y U x L y L

z u + - - J - + z - z- - T - - z- - ~ ' '

x y U x U y x U y L x U y L x U y U

Z L + - ~ + - - zU Z L

xy U xUy xVy v xVy v

ZL + - z - + - T - 2 - ~ ,

x y L x L y x L y Lxy > max + + - 2 xLyL

z - 7 - - 7 ~ ~ u ,

x y U x U y x L y U x L y U x U y Uz-'--U- "}- - ~ + - z z v z L '

x y L x L y x U y L x U y L x L y L

Z U "}- ~ + -- Z Z L Z U

xy U xUy xUy U xUy U

7 + - T + - - - i - 2 7 ~ ,




z y

Zf xy L xLy

m a x [ - ~ - - + -)-6- + - -

xy L xLy

z L + - - - ~ - + - -

xy U

z U +

xy U

z L +

xLy L 2 xLy L

Z Z U

x U y U x L y L x U y U

Z Z L Z L

x U y x L y L x U y U x L y L

+ - - z z U z U

xVy xVy v xgy v ]- ~ + - - ; - 2 - 7 - ] 9

A f t e r c o m b i n i n g a l l t h r e e c o n v e x l o w e r b o u n d i n g a l te m a t i v e s , w e o b t a i n t h e f o l-

l o w i n g e i g h t c o n v e x f u n c t i o n s i n x , y , z w h o s e m a x i m u m is a ti g h t c o n v e x l o w e rb o u n d i n g f u n c t i o n fo r xyz:

~x y L xLy xLy L xLy Lx y > m a x + + - - 2 ~ ,

z - [ - i v - - - ; v -

xy L xLy xLy U xLy U xLy L

z U q - " ~ ' 4 - - - Z Z n Z U '

xy U xVy xUy L xVy L xVy v

z L + - - F + - - z z v z L 'xy U xUy xLy U xLy U xUy U

z U -'[- - ' ~ " l- - - Z Z U Z L '

xy L xLy xUy L xUy L xLy L

Z-- -~ "{" ~ "{- - - Z Z L Z v '

xy U xUy xLy U xLy g xUy U

z v + - - Z - + - - Z Z U Z L

xy L xLy xUy L xUy L xLy L

Zu -[" - - ~ "4- --Z ZL ZU 'xy u xVy xUy U xUy U ]

~ L + - -; z- + - - - - ; - - 2 - - 7 ~ ~ 9

A p p e n d i x B

C o n v e x i t y /C o n c a v i t y I d e n t i f ic a t io n o f G e n e r a l i z e d P o l y n o m i a l T e r m s

I n t h i s a ppend ix , neces s a ry a nd s u f f i c i en t co nd i t io ns a re pro v ided f o r co n v ex i -

t y /c o n c a v i ty o f g e n e r a l iz e d p o l y n o m i a l t e rm s o f t h e fo rm :

z d li , d iEN, i= l , . . . ,N .

G e n e r a l i z e d p o l y n o m i a l t e rm s a r e a s p e c ia l c a s e o f p r o d u c t s o f u n i v a r ia t e f u n c t io n s

b y s e l e c ti n g fi(xi) = x~~.Fi r s t , t he tw o va r i ab l e c a se f = x'~yb i s c o n s i d e r e d .



1 7 6 C . D . M A R A N A S A N D C . A . F L O U D A S

T H E O R E M 5 . I f o n e o f t he f o l l o w i n g c o n d i ti o n s h o l ds ,

(1 ) x , y > 0 .

(2 ) a , b ar e eve n i n t eg er s .

(3 ) a , b ar e o d d i n t e g e r s, a n d x y > O .

(4 ) a , b a r e i n t eg er s , a i s o d d , b is even , a n d x > O .

(5 ) a , b a r e i n t eg er s , a i s ev en , b i s o d d , a n d y > 0

th en (a) f = x a y b i s c o n v e x i n ( x , y ) i f o n e o f th e f o l l o w i n g i s tr u e :

( i ) a < 0 , b < 0 .

( i i ) a g 0 , 1 - a - b < 0 .

( i i i ) b _ < 0 , 1 - a - b < 0 .a n d (b) f = x a y b i s c o n c a v e in ( x , y ) i f

( i ) a >_ O ,b >_ O ,a + b <_ 1.

P r o o f . T h e f u n c t i o n f = x a y b i s c o n v e x i n ( x , y ) o n l y i f a l l t h e e i g e n v a l u e s o f

t h e H e s s i a n m a t r i x H o f f a r e p o s i ti v e . T h e H e s s i a n m a t r i x H i n c l u d e s t h e s e c o n d

o r d e r d e r i v a t i v e s o f f w i t h r e s p e c t to x a n d y .

: ~ x ~

= Ox 2 = a (a - 1) x a - 2 y b

- ~ u w h e r e f x y = O x O y O y O x -

f y u = 0 2 f = b ( b - 1) x a y b -2 .O y 2

T h e e i g e n v a l u e s A o f H a r e t h e r o o t s o f t h e c h a r a c t e r i s ti c e q u a t i o n :

: - + + = 0,

T h i s e q u a t i o n a c c e p t s o n l y p o s i t i v e o o t s i f a n d o n l y if:

f x x + f y y >_ O,

f ~ f y y - f ~ > o.

A f t e r s u b s t i t u t in g t h e e x p r e s s i o n s f o r f z x , f y v , f~ y w e h a v e ,

x ~ b - 2 [ a e a - 1 ) y 2 + b eb - 1 ) : ] > 0 ,

y Z b - 2 x 2 a - 2 [ab (1 - a - b)] >_ O.

N o t e t h a t x 2 a - 2 y 2 b-2 is a lw a y s p o s i t iv e a n d f u r t h e r m o r e x a -2 y b -2 iS a l s o p o s i t i v e

i f o n e o f t h e c o n d i t i o n s ( 1 ) - ( 5 ) i s tr u e . I n th i s c a s e , th e c o n d i t i o n s f o r p o s i t i v i t y o f

e i g e n v a l u e s c a n b e r e w r i tt e n a s :

a ( a - 1 ) y 2 + b ( b - 1 ) x 2 > 0 ,

a b ( 1 - a - b ) > O .



SOLUTIONS OF NONLINEAREQUATIONS 1 7 7

T h e s e c o n d i t i o n s a r e s a t is f i ed f o r e v e r y x , y E ~ o n l y i f a l l t h e f o l l o w i n g i n e q u a l -

i t i e s a re sa t i s f ied :

(i) a ( a - 1) _> O,

(ii) b ( b - 1) > O,

( i i i ) ab (1 - a - b) > 0

T h e s e i n e q u a l it i e s d e c o m p o s e i n t o t h e f o l l o w i n g t h r e e d is j o i n t s u ff ic i en t c o n d i t i o n s

f o r c o n v e x i t y o f f = x a y b a s s u m i n g t h a t o n e o f t h e r e q u i r e m e n t s ( 1 ) - - ( 5 ) i s t r u e .

( i ) a < O , b < O .

( ii) a g O , l - a - b _ < O .( i i i ) b g O , l - a - b _ < O .

N o t e th a t th e r e q u i r e m e n t s b > 1 i n (i i) a n d a > 1 i n (i ii ) a r e i m p l i e d b y t h e o t h e r

t w o i n e q u a l it i es , a n d t h e r e fo r e a r e n o t i n c l u d e d .

T h e s a m e a n a l y s is a p p l i es f o r c h e c k i n g c o n c a v i t y o f f = xa y b. T h e c h a r a c t e r -

i st ic e q u a t i o n a c c e p t s o n l y n e g a t i v e r o o ts i f a n d o n l y i f

f x x q - f y y <_ O,

f x x f y y - >_ o .

A f t e r s o m e a l g e b r a w e o b t a i n th e f o l l o w i n g s e t o f c o n d i ti o n s f o r c o n c a v i t y o f f .

( i ) a ( a - 1 ) ~ O ,

( ii) b ( b - 1) ~ O,

( i i i ) ab (1 - a - b)

w h i c h a l t er n a t iv e l y

( i ) a _> O, b _>

>0

c a n b e w r i t t e n a s

0 , a + b < 1 []

N o t e t h a t i f t h e term x a - 2 y b -2 i s a l w a y s n e g a t i v e , t h e c o n d i t i o n s f o r c o n v e x i -

t y / c o n c a v i t y a r e r e v e r s e d .

T H E O R E M 6 . I f o n e o f t h e f o l l o w i n g c o n d i t io n s h o l ds ,

(1 ) a , bare odd i n t eger s , andxy < O.

( 2 ) a , bare i n t eger s , a i s odd , h is even , a nd x < O.

( 3 ) a , bare i n t eger s , a i s even , h i s odd , andy < 0

then (a) f = x a y b is c o n c a v e i n ( x , y ) i f o n e o f t h e f o l l o w i n g i s tr u e :

( i ) a < O , b _ < O .

( i i ) a _ < O , l - a - b < O .

( i i i ) b _ < O , l - a - b < O .




a n d (b) f = x a y b i s c o n v e x i n (x , y ) i f

(i) a >_ 0, b > O, a + b < 1

T h e p ro o f o f T h eo rem 6 i s co mp le t e ly eq u iv a len t w i th th at o f T h e o rem 5 an d

therefore i t is om it ted.

A s imi la r se t o f condi t ions fo r convex i ty /concav i ty can be ob ta ined fo r the

genera l n -d im ens io na l case . For the sake o f s impl ic i ty , w e assum e tha t x i > 0, i =

1 , . . . , N , w h ich can a lway s b e ach iev ed wi th s imp le r e sca l in g o f va r iab le s .

d iT H E O R E M 7 . T h e f u n c t io n f " ~ N __, ~ + , f ( x ) = I ~ N = I x i i s ( a ) i s c o n v e x i n

x E ~ N i f o n e o f t h e f o l l o w i n g c o n d i t i o n s i s t r u e:

(i) d i _< 0 , Vi = 1 , . . . , N ,

(ii) 3 j s u c h t h a t d j > 1 - ~ . i g # j d i , a n d d i < 0 , V i # j , i = 1 , . . . , N

a n d (b) i s c o n c a v e in x E ~ N i f

(i) d i > O, Vi = 1 , . . . , N , a nd ) ~ N = l d i < 1

diP r o o f . T h e second order der iva t ives o f f = I IN=I x i are equ al to:

{

i # J

f i j = O x i O x j ~ f , i = j "x i

The ex pans ion o f the Hess ian mat r ix o f f y ie lds the fo l low ing charac te r is t ic equa-

tion:

/~ N + C N _ I ( d , x ) / ~ N - 1 + . . . + C l(d, x)A + Co(d, x) = 0 ,

w h e r e

N d i ( 1 - d i ) fC N - l ( d , x ) = ~ x/2

i = 1

N - 1 N

C y - 2 ( d , x ) = E Ei = 1 j - -- - -i + 1

: _ _ :

k1-[ dij

C N - k ( d , x ) = ~ J= l

d i d j (1 - d i - d j ) f 22 2

X i X j

1 - ~ d i j

k j = l f k

r Ij = l




1 I d i 1 - ~ d i

C o ( d , x ) = i=1 i= l f N .N

11,4i=1

N o t e t h a t t h e s e ts ~P k, k = 1 , . . . , N c o n t a i n al l p o s s i b le w a y s t h a t k e l e m e n t s o f

t h e s e t N c a n b e s e l e c t e d .

{ i : 1 < i < N } ,

{ ( i l , . . . , h r : i j E A f , j = 1 , . . . , k

i l < i2 < . . . < i k } , k = l , . . . , N .

a n d

A s u f f i c ie n t c o n d i t i o n f o r t h e c h a r a c t e r i s t ic e q u a t i o n n o t t o a c c e p t a n y n e g a t i v e r o o t s

i s tha t a l l t e rm s C N - k ( d , x ) A N - k , k = 1 , . . . , N m a i n t a in c o n s ta n t s ig n f o r e v e r y

A < 0 a n d f o r e v e r y x E ~ N . M o r e s p e c if ic a ll y , a ll t e r m s C N _ k ( d , x ) A N - k , k =

1 , . . . , N m u s t b e p o s i ti v e w h e n N i s e v e n a n d n e g a t i v e if N i s o d d . T h i s is s a ti sf ie d

if ,

_< 0 , i f k i s od d V xV k = 1 . . . , N , CN-k(d,k) _> O, if k is ev e n E

~ N .

T h e s e r e la t io n s m u s t b e s a ti sf ie d f o r e v e r y p o s i t iv e x , t h e r e f o r e t h e y c a n b e w r i t t e n

e q u i v a l e n t l y a s:

V k = 1 . . . , N , d i j 1 - ~ d ij > O, i f k i s ev enj = l

f o r a l l ( i l , . . . , i k ) E P k . N o t e t h a t i f d i <_ O , i = 1 , . . . , N t h e n

V k = 1 . . . , N , 1 - Z d i~ >_ 0 a n d d i j > 0 , i f k i s ev enj = l = 1 - -

f o r a l l ( i l , . . . , i a ) E 7 : ' k , w h i c h i m p l i e s t h a t t h e c h a r a c t e r is t ic e q u a t i o n a c c e p t s

on ly pos i t i ve roo t s fo r a l l x E !}rN w h e n d l _< 0 , i = 1 , . . . , N .

I f w e n o w a l lo w o n l y o n e e x p o n e n t to h e p o s i t iv e d i I _> 0 , a n d d i s <_ O , V j =

2 , . . . , N th e n w e h a v e

V k = 1 . . . , N , d is < 0 , i f k i s e v e n " '

j = l

w h i c h m e a n s t h a t t h e c h a r a c t e r i s t i c e q u a t i o n a c c e p t s o n l y p o s i t i v e r o o t s f o r a l l

x E ~+N on ly i f

V k = 1 . . . , N , 1 - ~ di~ < 0 , V ( i l , . . . , i k ) E 7:'k.

j = l



18 0 c .D . MARANASAND C . A. FLOUDAS

O r e q u i v a l e n t ly ,

k

di l >_ l - ~ d i j .j= 2

F i n a l ly , it w i l l b e s h o w n t h a t t h e s e r e q u i r e m e n t s c a n n o t b e s a ti s fi e d i f m o r e t h a n

o n e d i , i = 1 , . . . , N i s p o s i t i v e . L e t d i ~ , d i 2 > 0 , t h e n b e c a u s e ( i l , i : ) E P l w e

h a v e : -

d i l ( 1 - d , ~ ) < 0 a n d d , , ( 1 - d ~ ) < 0 .

H o w e v e r , d il , d i2 > O , t h e r e f o r e

d i a _ > l a n d d i2 _ > 1 .

F u r t h e r m o r e , ( i l , i 2) E P 2 s o

(d i~ d i : ) (1 - d i l - d i 2 ) ~ 0

o r

( 1 - d i l - d i 2 ) > _ 0

w h i c h is in c o n t r a d i c t i o n w i t h d i l , d ! ~ _> 1 . T h e r e f o r e , a s s u m i n g c o n d i t i o n s ( i ) o r( ii ) t h e n f i s c o n v e x i n f o r a l l x E ~ + .

B y f o l l o w i n g t h e s a m e l i n e o f t h o u g h t , f i s c o n c a v e i f a l l t e r m s C N - k ( d , x ) / ~ N - k ,

k = 1 , . . . , N m a i n t a i n c o n s t a n t s ig n f o r e v e r y A > 0 a n d f o r e v e r y x E ~ N . T h i s

i s tru e i f ,

V k = I . . . , N , C N - k ( d , k ) > 0 , V x E ~ N .

T h i s c a n b e w r i t t e n e q u i v a l e n t l y a s ,

V k = 1 . . . , N , d i~ 1 - E d i~ >_ 0 ,

j = l j = l

F o r ( k = 1 ) w e d e d u c e th a t ,

d i ( 1 - d i ) > _ O o r 0 < d i < 1 ,

T h i s i m p l i e s t h a t ,

V k = 2 . . . , N ,

w h i c h s i m p l if i e s in t o

N

~ d i < 1 .

i= l

k

1 - ~ di j >_ 0,j = l

' x / ( i l , . . . , i k ) E P k .

V i = 1 , . . . , N .

V ( i l , . . . , i D E P k ,




T h e r e f o r e , i f c o n d i t i o n ( i i i) i s t r u e t h e n f i s c o n c a v e i n f o r a l l x E N +N . [ ]

A c k n o w l e d g e m e n t s

F i n a n c i a l s u p p o r t f r o m t h e N a t i o n a l S c i e n c e F o u n d a t i o n u n d e r G r an ts C B T - 8 8 5 7 0 1 3 ,

C T S - 9 2 2 1 4 1 1 , A i r F o r c e O f f ic e o f S c ie n t if ic R e s e a rc h , a s w e l l a s E x x o n C o . , A m o -

c o C h e m i c a ls C o . , M o b i l C o . , a n d T e n n e s s e e E a s tm a n C o . i s g ra t ef u ll y a c k n o w l -

e d g e d .

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Maranas and Floudas, JOGO, 1995