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DISTRBJT£0 BY:

•

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I DOCI.MINT CONTROL OAT A . I & 0

tudi s at Austin

••· aero•• tc c u • · • • C\.•••• .. •c• ,,o.,.

Chara t rizataon f Ophmalit in onve Programming

..o . ... ' '"tt' -, _ ... --·· .... _ , . . tten· !l:r I . . l~n-T I

fl 8 r c- I t>por

. ••••• ••-• .. o••• ..,. __ ., -·- ..... ,.,.._ ......... , ppr v d for publi rt> I a . nd tot IP ; 1 t. di t ributeon

' ' ..................... •c•••••• Offece of ·aval e <"aar h (Code 434 ) Wa hincton, D. C.

r · ie~t C'Ondit ions f r pt im hty r giv n, for m , without onstraint qualificat ion. in t rm of

r gr m, whi h an hos n to b bili nc r.

I CAL SERVICf

DD .':-: •• 14 73 (P AG( I l nclusifi.ct

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F ( . i"'T "\i llTY IN COt V •. • PR ·r' MMl G

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YBERNETIC STUDIE~

u 1 l .hnc; . v · Tex~t

787 12 ( 5 ~' )4 -182 (5 1 ) 4 11 -489-9

5 12

I I

ABSTRACT

,, .. q . nd u fi i t: ond i tl n .. r o :: l- ; ( , .. l r .. r( l \f n.

" ., . : ~ ' '\<; c h •• r h le.m::. . \o:l t h o u ..: t r n ' l. l ; l ~ - "" .. ~ < · 1n m• - t l. al proqram. 1 .l C.. ~ a n ~ h o .. <. n ·-

' : ' .1 1. 'l

I

• ' l f'l i

.. . ~ !. oJ ~ • t t • t. r ~

'llh r• 0 t .t r ") 1 ") -r ' ' ' H r m . . o t. l ~ t 1r1nq on tr . 1n - lU"' &. "" n .

' v' - l n n t r -.. . a L 1n a r pr r"m xpr ' ~ :. u . .

' l ( :- <-~n..J , ... be. ... . .

H l- ,. I . I 'I <I 1nq ~ r t l t "'· wh h .j pcnj ~

. n c:. n 1 ->n~ t o be o~ Fo n a ~e

.. . . h n r ln f u n c- ti n !. ar trt c t ' I c on C'X 1n th it "<'~~ c tual"

r i\ .. ' lZ 1o n 0 pt una 1 L'j 1. "' 9 1V n ln . 'J . In .

. ,., r r . c.: t r1z p t. i""" lty h '- • tng l

:>r O< r m ( I •o ..., .l, Ex mp t bel

F r r. " £ : . t. " !" . "" l V .. Amp re u l 1n 3 . har-

. ... I - c:. r. I. s wher t. : .~ '-..- n.-t. ra 1nt f :.t n tlo n_

'i (. n : ' . : .;~.

n R • R, w d ion

. • . ) . :. . '7 .... n) not..- tn · 1nde>. -et

r <- ' ·~ X n w H .h , k ua 1 y d 1 end r

• k., a { Tht ,. Xl .... l ) ' • ( . l l

1 ~ J . e uch h l th t n tl n

(( ()- 1 ' c) ... , .... tr,) i _, not a nstant}.

r ' · I ~ ( Rn th ub.t l. l' X k, I. S obta1n ·d by jelet1ng the ~0--

Th . . " '1t r atr1ct1on . i. the unltion

: ptar(,-K ♦R oftaincJ by restricting ♦K to x

Ton. idrr the piogrammlnu problem

(P) «in r0(x)

ft .t. rk(x) * >< k c P » (l. 2 / ).

lc r i ea: ibl« -.oiat. on x*. i .«■..

( . i fk(K*) « 0. V c P.

vwr denote tne *et o: binding constraints by

U) P* ■ ik: k c P. tk(x*) ■ C) .

^ i Vi»i KM.« . i.!.'t ion -J ptimality it- 91/cr in «• .c f' '.lowing

Tru; < - . . itt

(I) th«. (:ol:.«m (r; ha/» convex functions (f : k c (0). P }

aasumrd Jiifprcntiable,

(II) x* br a rcasibl«- -c luti^n o* (P) at which the restrictions o;

rhc binding c o«"'r«int.- -. k c P», >rc ^trlctly or./«x ,

Cm! ^ : ,'ar,3 ä-*K b« «ny pubttive definite runction, i.e.

k , ^ card^k" . (2) > .. 0 ^ * C R

9 (0) • ( k c P>*.

; S« n ..♦ 1^ opiimal ie «nd nnly if *» ■ D i« the apt ia»a 1 value o'

tnc pi >grani

This aosunption it «Mak« r than strict convexity o: t>e function.. {',t: K. C F^), units« [k3 ■ (1. •••. n) :or al] k € ^.

Xl -'>

( ., )

S )

m;lX • L " .. .... . .

j V f 0 tx•) + !') ,

l V : k ( x•) + I ~ ., , ) . , ..

. . \.of; ... r 1 t:i n~ .. uch t. "'

• • l

.. !. l • . " t t n non of -.' .

t . r pr ")p« rtic fo and {tr'k £ p•}

hat: th p i:~ lity 0 x• is equ1va l ent 2 t l h~ non-

·at: y1n

t.v o (x• ) < 0

Jt. ' (x • $ 0

. l . al.1ty rlr 1. ( , .. .. . Y. -

• n . • 1. • , let h

) . l

~ ... { - t 1 n (x • • .MX } }.

.., -.. T le a1. . are "' ~ 1n t ' proo of 1. Theore• 1 • •

( )

-·.

a n

r

' I

a

.. . • f

.

' ... . • l

. ~

Th

4

( x• )

k £ P•.

cam (~} ha < ?Os itive o~ t imal va l ue.

::' < :. ~ • { • 1, 1 a 1 o :: it. i

..t r. c a l ar ~r ~ a l. • t.nCJ (7 a (R • • <. •

d -~ <

~k( -(x * ) !I - d ~ c ,, . . p +, ., - c

-t v rk ( x •) 0 k( ark , > d .. • • 0

- k .. d[k .

.,. ' .a nee • }

pos i i ve ·"' in l LC .

- ;: i .. ... , .;. n (6) s howing t a~ x • 1 9 op l'na t .

o n ·; x . t • - s mptions 1n Thco m n 1 n r t:. " c

',• _..r· • c an f' ,. •.. I 1n the mann _!' 0

l . ' 1.11 r , • i r r n t1atn lity i :· I" t •! sent ia l t . ·r i n<.e

1.. L •• <.:a . bt.• l ..n term:· o di rt:c ti o n a Jet l. va l J£ ., .

J. i nc e d .. • .... .. 0 l. S a feasible s olut i on 0 (A ) , h e

l i'l "") ( ) !.. 3 iH ly non1"e9ativ . 1 no nz r l. . O~'l m.

l )'. n I . I \ . be bounded ( l. • d ... rf: ) }

r • ...: : lZ in<: i ,

( } - 1 ~ ~ l. l.

1 c

\ c tha .:>U :- u l t : t.

.. ! op mil hL ond1.t1 :2 (w l cl

q .l ~ "l) !a.l ; . T l S l 1. l . trated in

Th • :-> C ? 1 m 1. :'

H r -nci

. ..

. .....

- :c .. -

E

.

~

(

' · ~o

n

• l .

~ .

~ - • }· 4 •

{1. ') .. . ~ a

l l - ···•J

l. '

l

. t r <.:. )r" ·o n •J x.

.

1.

X -x ·. ) = • ~ + e 2 ...

( <) = r X l

-x . :::- e

) ( 2 = X . - 1) + X

2 :> { ;·) = X ... X .£.

( p. are . .. • { }. ~ 2 :

T h re trictions f _k ,

e ::: u i ... n s are

[ J X) > 0

U1 '- x• • l

v ') I: vr1 (x* ) ( ". ) • ) ""

. . . . n .

. l": .:l . s •he r

J requ1r -?IlK; c n _ tralnL

· ne f o l OWlng

, l

- i

~ I

-x .. ,

• { 2} • ~ 3 ~ • { 1 • .l}

k £ '· are .-. tr1. t ly

0 . t i' . ~. -\,- ..

G. \ ._ ~

not

hoo .. in.

l - n r~

(

, .. ;;Jt # ~· · '

L l\·

( 1

( . •.

po~i ti d

: ' r ""f

d

-2d

.. .. " : 'f ''~ [ '!

. ....

~"). s l ~

I ( i\) r \I I .. L

max

. . t.

i n it "'un ... ions k

c,p 0 Theore m l . the

= ' Z . I , P*. i 1

• 3

+ ,.,

+ . I I l

+ 'Y d I 2

+ ~c d 1 + ld I> 1 2

- d + e( ,dl I + I 2 I + d I)

3 3

f und, by · n ··pP c- t ion ,

t he positive _ 1 '

e n o a s t. .... ' ,,, ,.

'I;

-;

4

• ~

e , • ' ·

•• l ' .

6

-::hoi e i ~ l h l - n -- rm ( i ) • • • wh1< ..

th fo • ow ins ta .L ne ar proq .. a r.

..,

,t V f 0 (>.*) • ~ « 0

t v k(x*) l: I l I + " ~ k ( p•.

ic" ~ 1

1

· - 1 o r r i x .,, are in <lt:l

c1 •• • n - r p roblem (A: o Th · - · ·- ·

i t . o.rm, 1. c ., .1 .<-~.near program, i» where

i •

( . ~

n • .

'· y .. ,. !...

l ~~, 0

{ a!- - {x•j ~X ~ 0}

J

o· the type

: q !

1 • '""! Ua • _ 1 ne ed be <.. h ·

y k

. ?"1

-..

n ( ) , an

· • : , L • r ror equivalent to the non

dt 'lf0 (x•) < G

k v· (x*) < 0, k £ , ••

p ••

• ' 1

'I \.1 ,. t: l ror .m tne <llternativc i ~ qu~val~nl t~ Ln~

• l'tt 1

n it1.on.

• 0

~ {) , l E { } ' ' P l.

1.n • tr • l. o .1..t1on ( 3) i ~ a -..pct:ia.. a .. l. ~·- Lrl <-

) •i t I L L:e ... the optl l . ·• ol I ' , .>n f l. !. 'I < • • ll. w~ • ~no .. ., • n :.. tr .1..11 IIJ il I Lt:<ll n _,

8

*-fuci guarin.tt the nccc-»ity of the Kuhn-Tucker condxticn.

The 'oUawlmi theorc« qiveu an alternative characteri.ati

:>: op'-imaiity. I u , roo will be umxucd. • mcc i: nwi'... -iftLic

th« .)r^o' o-' Theorem 1.

Th« o > .a 2. UrJtr \,t\e a function: o" "•hcorcn: 1, the ecsibl« .„latiür.

* i >4itii.>a1 *. , ani ^r.i> i:, tor e/er> po. itiv« •», t>.t i»p> *nia;

R IVK • 'riiowin: ilia is zero.

(P.o trn i1 7^(x*)

J .t. (8) dt 7- (>:•) ♦ *♦ f^(dr ^ * 0. > t f*.

O) -1 * di - i. - n.

acnark.

'. >v poisib'« advAntaqe o problra (B.-») ovar the pr«viou& ly

• i/e KioLi«.in (*). - tlwit the direction ounJ here i* o btecp«.r

ii. cent.

Per any - > C, .et d(-») denote an Dpti»al »o^Uwioi :

(8.^). Clearly

^1 < -»2 * dC^j) 7r0(x*) - dl^)1 ? ''(x*).

'ihui- t.i« optimality oC x* is cqux/alent to

(7) lia *n; J(^) ?f0(ic«) - C

.. "'• .a y ....

: · · · 1.. i n Thc o rc· ~ wh re the f un t i on· ( · k : k ~ P *)

. , • tr

• l .

• •• • • • lh l ( • .... z - ,... .. d i .. c · • · l. • t • ; f • 0 • •

<a: o ...

J

3 .

( .

(' ,

.f, . 7 •

1 • Th1 .a.

?r •

lt ... t ~r thltt p r obl c• (P' o .... ) 1 t. • 1

in

.. . r.

0 • 0

dt ' ~, •• ,

. l 9 ek ( ~•) ~ , + ~ 1 •

1( ... ~

. "' d . ~ 1, .. n o .. .. . ...

1

(8. f") beca.e .

,.)t ' ~, •• ,

- 1 c d • 1 , 1

\ ' ,. i • • . .•• no

h.· • n !lt • , i 1 v• ue i . " r

} : - uc:kcr c:andi i

~

~ .l . )..

v ,.. ( ~:•) • ' ( .• • . kcP

' ,.

1.n ar

k • ,. _

-.

1-.fen

t r

uri ~ ti pr edure Cor checking th opti~l1ty o a

• 1 : t 1. l • is:

) .. o ve th in a r pr~;ra (8.0).

c) 1 it. .. opt1• : .,alu£ h zero then. by the prev1ou

c ) I! ( 8)

for o.e ... .... ,.

j

. n n ., 'la • ;, J J (,. ) 1 a Jir c t i n "~ d S '- n • 0

(

-onab

(B.-.1

) fOI' -.

:.. toppinq rule.

. .:. I tc.:.

~·

7. ot that 1.t 1 po6Sl.bl or (17) t' hold. ever. t .ou~h

To 11 uatrat th1s c•n u e any exa..,J where

;. nn Tu ker '-on 1.tion. clo not hold at an optiMl po1.nt x• •

• t. :1 c c

1

- d • 0 2

- .t.d. .. 0

- cl • )

-1 ~ d c 1 l

1 • 1. 2. 3.

t . o :ut1. n h r • d(o) • (0. 1. 0) \ itb opt~\ ., .,}u • -1.

. .,

· . .._ · r

·'' . r

..

r

( ~ . )

, . .,.

r

. 'l ! • 1. . Al n ~h ·.~ rob lea

IU. r, -o(x)

. . -. X) c o. k E p •

th -un t :.. n (fk: }.. E { ) . ' , ) ace conv~x. but w~thout

:.. 1on.· th~ r striction~ r k

• tl i ut ion x•. t.~ 0 (.

• ..

.. .

. ' 0~ •

. ~

-'1

'· 9 ..

1 '1 " · s:. K

. . , -only J E D • [ , .. \JA 1ty 1 . . ...

tP"c ·y o1 ~ at. )1,•.

v , E ro.

• Y •

vee. t.hi con i- ~uit .. n ~-~

k • ) • r (A.. x + 1 .. ) • a ... .

--k . "'

~ • .. ~ .a • l &.n C ft " '>C t 1 I

• C R •

l l

.._ ).

inlroduc~td a d

k For t~ ~un~tion f g1ven

the t.c n 1. . l mply

• ~ ( ~ 3 ok • Nf •• th nu l l ~ p~C'- 0

. ' h ( m ... l) x n IM~t' ix

a., uNependef'lt ~· of x•.

~ •

Th .. u &~ I t to Theore• l ~

( . l . ~ v • a c n"'-'X ot JrCtl.V ' ur t L.&.o ~ c afllc con ve );

• "1 r "l l ., _ ( 2 ) 0 <I l

t.l b

) n~ pos itive d :i~i~e funtt ona. ~ E p •

Th• n •

- • (l l. .. t: opt1 .. . va u• o f the probl ..

m •• ,..

' • t. .

+ ~~-( ~ d) • 0 ~

• r ~c.. •

l l uw

a T r • tv. l. 1:1t: reau t. .• of ~2 ca" .. illi l ar l y be adaJ•t• - o th

o n v _ · ,. .

Rr.F'f:. ~NCE

'. . St.:· -Til , 1\ . 8e.,-1sra a r u .• Z I obe~. • hara~t~ri z~t . on

o ~ opt~ma l lt y 1n ·~n ~ x ~ r~ ra~inq ~it

Ol.il tl.f&. a "•'" '1' hn&. c•n' reprint . r~ t.

Ju t ; l 7 •• TPchn& "· lMra ... l in .· titul.

ul ccn~tr int "o. Allft'- .. 5 ,

· Tc hno ) _ y,

. r ·k r. '"Nor. : 1 . · a.. 1 ' OCJ .r ; mnn n·_ • ' P r <. •

.n, Be r k ' y . y mp. Oft ~tbc~ t ~~ ~ l ~l « ll l , ., · r ~l •• t-1 11 '- / • ~-n, c d 1 tor • n v j or n • rr :•.:> , Berkeley, 195 1 •

.J . J. Ponaat~&.n. •.eve n k&.nd .i ·.> CO"\ Jt:~1t }' , " !A.J'H R '!tW, Vo i . C)

N •• .Januar) 1967. 115- 119 •

. : . T. R c k ...... ,.'ar. "OI"II.n:t:-• Con·J ~ :t 1 · ..._ r · m ,._ 1t.hout.., ~ua ~ 1 t)' ~ap,'" .J. Opti iz. Theorx 1\pp1., Vol. 7, Ro. 3,

CJ7 • 143- .... .-.

T. Rock~ !'c 1 l ~ r • ·~ o llll! c nv x 1- CfJ' ramh .., ho:.c i n ,. y t. , •• ~~ d . • in Nonli n a r 1 r

.!:)> .l . 1 • •• .t"-. t: •uver 1ty o r. • ~· -n :, &. . • J . ~. c .-n. o. L . Ma :- · · ~t · · · rian, a •• lh tt t l.' .

~ • ... • It 1 ' • · . , ~ w Yo r . • i ., 7 •

/.