E k o n o m i d a n K e u a n g a n I n d o n e s i a , v o l . 3 7 , n o . 2 , 1 9 8 9

Program Non Linier

S r i M u l y o n o

A b s t r a c t

The objective of this paper is to introduce the optimization of multivariate function with the emphasis on linear programing. 7 hp presentation of cases is aimed showing the importance of this wide a id difficult topic. Next the role of Kuhn-Tucker technique to improvp Lagrange Multiplier approach to solve optimization with inequalit es constraint will be discussed. Some way to studies Kuhn-Tucker methods easily, will be treated briefly. Further some illustration of the weakness of the Simplex method to solve Non-linear Programing will be high\ lighted. This paper will end with the treatment of some inefficiency pf Kuhn-Tucker method to solve the Non-linear Programing.

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2 1 ! )

Mulyono

Pengantar

D a l a m m a s a l a h p r o g r a m l i n i e r b a i k t u j u a n m a u p u n k e n d a l a - k e n d a l a n y a s e m u a b e r b e n t u k f u n g s i l i n i e r . A s u m s i h u b u n g a n b n i e r i n i m e r u p a k a n p e n d e k a t a n y a n g c o c o k a t a u s e k u r a n g - k u r a n g n y a c u k u p b a g u s u n t u k s u a t u i n t e r v a l n i l a i v a r i a b e l d a r i m a s a l a b t e r t e n t u . N a m u n , p a d a b e b e ­r a p a k a s u s k e a b s a b a n a p l i k a s i h u b u n g a n l i n i e r d a p a t d i p e r t a n y a k a n . S e h i n g g a , p e r l u d i b u a t h u b u n g a n n o n l i n i e r u n t u k m e n c e r m i n k a n d e n g a n t e p a t s t r u k t u r p e r s o a l a n n y a . K e r a n g k a s e p e r t i i n i a k a n d i t e m u i d a l a m p r o g r a m non linier. M e t o d e - m e t o d e y a n g d i g u n a k a n u n t u k m e n y e l e s a i ­k a n m a s a l a b i t u d i n a m a k a n a l g o r i t m a p r o g r a m n o n l i n i e r .

M a s a l a b p r o g r a m n o n b n i e r d i t a n d a i d e n g a n a d a n y a f u n g s i a t a u f u n g s i - f u n g s i n o n b n i e r d i a n t a r a t u j u a n d a n a t a u k e n d a l a - k e n d a l a n y a . B e n t u k n o n l i n i e r i t u m i s a l n y a :

, I n X , Ijx, ef, s i n ( x ) , t g ( x ) , d a n l a i n - l a i n .

K e t i d a k U n i e r a n j u g a d a p a t t i m b u l s e b a g a i a k i b a t d a r i i n t e r a k s i a n t a r a d u a a t a u l e b i b v a r i a b e l , s e p e r t i :

X j X j X j , X j I n f X j ) , xy, d a n l a i n - l a i n .

S e k a r a n g t a m p a k b a b w a m a s a l a b i n i l e b i b s u l i t d i b a n d i n g m a s a l a b p r o ­g r a m b n i e r . P e n g e n d o r a n a s u m s i k e l i n i e r a n t e l a b m e n y e b a b k a n k e s u l i t -a n d a l a m p e r b i t u n g a n , d a n r u a n g l i n g k u p t o p i k i n i m e n j a d i s a n g a t l u a s d a n l e b i b b e r v a r i a s i . S e j a k m u n c u l n y a a r t i k e l t e o r i d a s a r p r o g r a m n o n -l i n i e r o l e b K u b n - T u c k e r ( 1 9 5 1 ) , t e l a b b a n y a k p r o s e d u r a l g o r i t m i k y a n g d i k e m b a n g k a n u n t u k m e n y e l e s a i k a n m a s a l a b i n i , n a m u n , p r o s e d u r s o ­l u s i n u m e r i k y a n g e f i s i e n s a m p a i s e k a r a n g b e l u m d i t e m u k a n .

D i s i n i s a y a a k a n m e m b e r i k a n s u a t u p e n g a n t a r b a g i t o p i k u m u m p r o g r a m n o n b n i e r . T o p i k i n i t e r l a m p a u l u a s d a n s u U t d i b a n d i n g k a n d e n g a n p e r k e n a l a n s a m b i l l a l u s e p e r t i p a d a t u U s a n i n i . S a y a a k a n m e n g -a w a l i d e n g a n k e t i d a k b n i e r a n d a l a m b i d a n g e k o n o m i , d i t e r u s k a n d e n g a n p r a s y a r a t d a l a m m e m p e l a j a r i p r o g r a m n o n l i n i e r , o p t i m i s a s i t a n p a k e n ­d a l a d a n d e n g a n k e n d a l a p e r s a m a a n , d a n b e d a a n t a r a p r o g r a m l i n i e r d e n g a n n o n l i n i e r . A k b i m y a , s a m p a i p a d a t e o r i K u b n - T u c k e r d e n g a n t u j u a n m e m b e r i k a n p e n g e t a b u a n t e n t a n g o p t i m i s a s i n o n l i n i e r d e n g a n k e n d a l a p e r t i d a k s a m a a n .

I . KetidakUnieran dalam Ekonomi

K e t i d a k b n i e r a n d a l a m b i d a n g e k o n o m i d a p a t t e r j a d i l e w a t b e r b a g a i b e n t u k . C o n t o h p e r t a m a d i a n g k a t d a r i m a s a l a b s t r u k t u r b i a y a . B i a y a t o t a l p r o d u k s i a k a n m e n i n g k a t j i k a o u t p u t b e r t a m b a h . I n i a l a m i a b k a ­r e n a k e b u t u h a n a k a n t e n a g a k e r j a d a n a t a u i n p u t - i n p u t l a i n j u g a a k a n

220

Program Non Liniei

b e r t a m b a h . P e r t a n y a a n n y a a d a l a b a p a k a h p e n i n g k a t a n b i a y a t o t a l p r o p o r s i o n a l d e n g a n p e n i n g k a t a n p r o d u k s i ? D e n g a n k a t a l a i n a p a k a h b i a y j m e r u p a k a n f u n g s i l i n i e r d a r i k u a n t i t a s p r o d u k s i ? j a w a b i i y a a d a l a b Jidak k a r e n a b e b e r a p a a l a s a n s e p e r t i : p e m b e b a n i n p u t - i n p u t d a l a m j u m l a b l e ­b i b b a n y a k d a p a t m e n i n g k a t k a n b a r g a i n p u t , b e r t a m b a b n y a u k u r a n u n i t u s a b a d a p a t m e m b e r i k a n economies of scale k a l a u b u k a n diseconomies of scale, b m b a b i n d u s t r i y a n g b e r l e b i h a n d a p a t m e n i m b u l k a n external diseconomies. S e m u a f a k t o r i n i d a p a t m e n y e b a b k a n p e r l a m b a t a n d a n p e r c e p a t a n p e r u b a h a n b i a y a t o t a l , s e h i n g g a b i a y a t o t a l b u k a n l a g i f u n g s : l i n i e r d a r i j u m l a b o u t p u t .

S e b a g a i s u a t u i l u s t r a s i k a t a k a n s a m p a i t i n g k a t o u t p u t t e r t e n t u , m o d e l m a t e m a t i k b i a y a t o t a l ( Q m e r u p a k a n f u n g s i d a r i j u m l a b p r o d u k ­s i (Q), m i s a l n y a s a j a a d a l a b :

C = 1 0 + 2Q, y a n g m a s i b b e r b e n t u k l i n i e r . '

J i k a p r o d u k s i t e r u s b e r t a m b a h f u n g s i b i a y a d a p a t s a j a b e r u b a b , k a t a k a n m e n j a d i :

C = 1 0 + 1 1 1 ( 2 - 7 ( 2 ^ + 1 / 3 , y a n g n o n l i n i e r .

C o n t o h k e d u a d i a m b i l d a r i m a s a l a b p r o d u k s i . P e n e r i m a a n m e r u p a ­k a n p e r k a l i a n a n t a r a j u m l a b o u t p u t d e n g a n b a r g a o u t p u t p e r u n i t . J i k a u n i t u s a b a y a n g m e n g h a s i l k a n o u t p u t i t u b e r s i f a t monopoli, k e n a i k a n p r o d u k s i p a s a r d a p a t m e n y e b a b k a n t u r u n n y a b a r g a . I n i b e r a r t i p e n e r i ­m a a n b u k a n f u n g s i l i n i e r d a r i b a r g a . M i s a l k a n h u b u n g a n a n t a r a b a r g a ( P ) d e n g a n j u m l a b o u t p u t {Q), y a n g b i a s a d i n a m a k a n f u n g s i p e r m i n t a a n a d a l a b :

P = 1 0 0 - Q , m a k a f u n g s i p e n e r i m a a n (R) m o n o p o U s a d a l a b : R = Q.P

= Q ( 1 0 0 - G ) = l O O Q - G ^ , y a n g n o n l i n i e r . j

C o n t o h k e t i g a t e n t a n g p r e f e r e n s i k o n s u m e n . K a r e n a marginal rate of substitution s u a t u b a r a n g a d a l a b m e n u r u n m a k a indifference curve s e o r a n g k o n s u m e n t i d a k a k a n b e r b e n t u k l i n i e r m e l a i n k a n c e m b u n g k e -a r a b t i t i k a s a l . K e p u a s a n k o n s u m e n ( ( / ) y a n g m e r u p a k a n f u n g s i d a r i b a r a n g - b a r a n g y a n g d i k o n s u m s i {X, Y) y a n g d i t u n j u k k a n o l e b indiffer­ence curve, d e n g a n d e m i k i a n d a p a t b e r b e n t u k s e p e r t i b e r i k u t :

U = ( Z + 2 ) ( y + l ) = X y + Z + 2 y + 2 , y a n g n o n l l n i e r .

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

2 2 1

Mulyono . c ; -

I I . Matriks dan Hubungannya dengan Optimisasi

B e b e r a p a k o n s e p d a l a m m a t r i k s d a n a l j a b a r s e p e r t i m a t r i k s H e s s i a n , k e c e k u n g a n , d a n s e t c e k u n g m e r u p a k a n p e n g e t a h u a n m i n i m u m y a n g d i h a r a p k a n d i k e t a b u i o l e b p a r a p e m b a c a y a n g i n g i n m e m p e l a j a r i p r o ­g r a m n o n l i n i e r . T a n p a p e n g e r t i a n a k a n k o n s e p - k o n s e p t e r s e b u t t i d a k m u n g k i n d a p a t m e m a b a m i m a s a l a b p r o g r a m n o n l i n i e r d e n g a n b a i k .

M a t r i k s H e s s i a n

M i s a l k a n t e r d a p a t s e b u a b f u n g s i n v a r i a b e l / ( X j , X 2 , . . . , x „ )

K e m u d i a n d i b u a t s u a t u m a t r i k s y a n g m e r u p a k a n t u r u n a n p a r s i a l k e d u a d a r i f u n g s i t e r s e b u t d e n g a n s u s u n a n s e p e r t i b e r i k u t :

hi hi

h n

h n

h i f r n l f . n n

m a k a m a t r i k s H d i n a m a k a n m a t r i k s H e s s i a n .

J i k a t e r d a p a t s u a t u m a t r i k s b e r u k u r a n ( n x n ) , m a k a principal minor ke k (k < n) a d a l a b s u a t u s u b m a t r i k s d e n g a n u k u r a n (kxk) y a n g d i p e r o l e h d e n g a n m e n g h a p u s (n—k) b a r i s d a n k o l o m y a n g b e r s e s u a i a n d a r i m a t r i k s t e r s e b u t . C o n t o h :

"1 2 3 "

( Q ) = 4 7

5 8

P r i n c i p a l m i n o r k e 1 a d a l a b e l e m e n - e l e m e n d i a g o n a l y a i t u 1 , 5 , 9 . P r i n c i p a l m i n o r k e d u a a d a l a b m a t r i k s - m a t r i k s ( 2 x 2 ) b e r i k u t :

" 1 2 " "1 3 " 5 6 " _4 5 _ _7 9_ _ 8 9 _

P r i n c i p a l m i n o r k e 3 a d a l a b m a t r i k s Q i t u s e n d i r i .

D e t e r m i n a n d a r i s u a t u p r i n c i p a l m i n o r d i n a m a k a n principal deter­minant. U n t u k s u a t u m a t r i k s b u j u r s a n g k a r («xn), t e r d a p a t 2 " " ^ prin­cipal determinant.

Leading principal minor k e k d a r i s u a t u m a t r i k s {nxn) d i p e r o l e h d e n g a n m e n g h a p u s {n-k) b a r i s t e r a k b i r d a n k o l o m y a n g b e r s e s u a i a n .

2 2 2

Program Non Linie r

D a r i m a t r i k s Q d i a t a s l e a d i n g p r i n c i p a l m i n o r k e 1 a d a l a h 1 ( h a p u s d q a b a r i s t e r a k b i r d a n d u a k o l o m t e r a k b i r ) . L e a d i n g p r i n c i p a l m i n o r k e a d a l a b :

"1 4

s e m e n t a r a y a n g k e 3 a d a l a b m a t r i k s Q i t u s e n d i r i . B a n y a k n y a l e a d i r f g p r i n c i p a l d e t e r m i n a t d a r i s u a t u m a t r i k s (nxn) a d a l a b n.

A d a c a r a p e n g u j i a n y a n g g a m p a n g u n t u k m e n e n t u k a n a p a k j j b s u a t u m a t r i k s a d a l a b positive definite, positive semidefinite, negative definite, negative semidefinite, a t a u indefinite. S e m u a p e n g u j i a n i t | u b e r l a k u b a n y a j i k a m a t r i k s n y a s i m e t r i s .

K e t e n t u a n u j i b a g i m a t r i k s positive definite a d a l a b : a . s e m u a e l e m e n d i a g o n a a l b a r u s p o s i t i f . b . s e m u a l e a d i n g p r i n c i p a l d e t e r m i n a n t b a r u s p o s i t i f . K e t e n t u a n u j i u n t u k m a t r i k s positive semidefinite a d a l a b : a . s e m u a e l e m e n d i a g o n a l n o n n e g a t i f . b . s e m u a l e a d i n g p r i n c i p a l d e t e r m i n a n t n o n n e g a t i f .

U n t u k m e m b u k t i k a n b a b w a s u a t u m a t r i k s a d a l a b n e g a t i v e d e f i n i t e ( s e m i d e f i n i t e ) , u j i n e g a t i f d a r i m a t r i k s i t u u n t u k p o s i t i f d e f i n i t e ( s e n i -d e f i n i t e ) . S u a t u u j i c u k u p b a g i s u a t u m a t r i k s m e n j a d i i n d e f i n i t e a d a b b b a b w a s e k u r a n g - k u r a n g n y a d u a d a r i e l e m e n d i a g o n a l n y a m e m i l i k i t a m j i a b e r l a w a n a n .

F u n g s i C e m b u n g d a n C e k u n g

S u a t u f u n g s i n v a r i a b e l / ( X ) , d i m a n a X = ( X j , X j , . . . , x ^ ) d i k a t a k s e b a g a i s u a t u f u n g s i c e m b u n g j i k a d a n b a n y a j i k a d u a t i t i k d a n A d e n g a n 0 < 0 < 1 , b e r l a k u :

/ [ 0 X ^ + ( 1 - 0 ) X J < 0 / ( X J + ( 1 - 0 ) / ( X ^ ) . S u a t u f u n g s i / ( X ) a d a l a b s u a t u f u n g s i c e k u n g j i k a d a n b a n y a j i ^ a

- / ( X ) a d a l a b s u a t u f u n g s i c e m b u n g . D a l a m p r a k t e k , u n t u k m e n g e t a b u i a p a k a h s u a t u f u n g s i a d a l a b

c e m b u n g a t a u c e k u n g , d i g u n a k a n p e n g u j i a n s e b a g a i b e r i k u t :

S u a t u f u n g s i / a d a l a b f u n g s i c e m b u n g j i k a m a t r i k s H e s s i a n d a r i f u n g s / a d a l a b d e f i n i t p o s i t i f a t a u s e m i d e f i n i t p o s i t i f .

2 3

Mulyono

S u a t u f u n g s i / a d a l a h c e k u n g j i k a m a t r i k s H e s s i a n d a r i f u n g s i / a d a l a h d e f i n i t n e g a t i f a t a u s e m i d e f i n i t n e g a t i f .

C o n t o h :

/ ( X j . X j . X g ) = 3 x j ^ + 2 x 2 - 2XjX2 — I X j X j + 2X2X3 - 6X3 — 4X2 - X3

M e m i l i k i 6 - 2 - 2 "

( H ) = - 2 4 2 - 2 2 2

d e n g a n l e a d i n g p r i n c i p a l d e t e r m i n a n t a d a l a h : = 6, H2 = 2 0 , d a n = 1 6 . S e h i n g g a ( H ) a d a l a h s u a t u m a t r i k s d e f i n i t p o s i t i f y a n g b e r a r t i

/ a d a l a b f u n g s i c e m b u n g . M e s k i p u n p e m b u k t i a n s e c a r a m a t e m a t i k t e r s e b u t b e r g u n a , l a n t a s

a p a a r t i n y a ? S e c a r a g e o m e t r i s b e r a r t i b a b w a j i k a s u a t u f u n g s i a d a l a b c e m b u n g ( c e k u n g ) d a n j i k a s u a t u g a r i s d i t a r i k a n t a r a s e t i a p d u a t i t i k p a d a p e r m u k a a n f u n g s i , g a r i s y a n g m e n g h u b u n g k a n t i t i k i n i s e l u r u h n y a t e r l e t a k d i a t a s ( d i b a w a b ) f u n g s i i t u . G a m b a r 1 m e n u n j u k k a n b e b e r a p a c o n t o h k u r v a f u n g s i c e m b u n g d a n c e k u n g .

G a m b a r 1 . Fungsi cekung dan cembung

m \ fix)

f u n g s i c e k u n g f u n g s i c e m b u n g f u n g s i y a n g t a k c e k u n g d a n t a k c e m b u n g

fix) fix) fix)

f u n g s i y a n g c e m ­b u n g d a n c e k u n g

f u n g s i c e k u n g m o n o t o n i s

X f u n g s i c e m b u n g d i s k o n t i n y u

2 2 4

Program Non LinNir

D a r i b e b e r a p a k u r v a d i a t a s d a p a t d i b u a t b e b e r a p a k e s i m p u l a i i p e n t i n g y a i t u : a . d e f i n i s i s u a t u f u n g s i c e m b u n g ( c e k u n g ) t i d a k t e r g a n t u n g a p a k a b s u a t | u

f u n g s i k o n t i n y u a t a u d i s k o n t i n y u . b . s u a t u f u n g s i d a p a t c e k u n g p a d a s u a t u d a e r a b d a n c e m b u n g p a d a w i l ^ -

y a b l a i n ,

c. s u a t u f u n g s i b n i e r a d a l a b c e m b u n g m a u p u n c e k u n g .

S e t C e m b u n g F u n g s i c e m b u n g d a n s e t c e m b u n g a d a l a b d u a k o n s e p y a n g b e r b e d a k a r e n a i t u j a n g a n d i k a c a u k a n . U n t u k m e m u d a b k a n p e n j e l a s a n a k a j n d i a w a b d e n g a n d e f i n i s i s e c a r a g e o m e t r i s t e n t a n g set c e m b u n g .

M i s a l k a n S a d a l a b s u a t u s e t t i t i k - t i t i k d a r i s u a t u b i d a n g ( 2 d i m e n s ) a t a u r u a n g ( 3 d i m e n s i ) . Jika untuk setiap dua titik pada set S, garis yarig menghubungkan dua titik itu seluruhnya terletak pada set S, maka d i k a t a k a n s e t c e m b u n g . P e r b a t i k a n c o n t o b - c o n t o b s e t c e m b u n g p a d l a G a m b a r 2 . S e m u a s e t p a d a G a m b a r 2 . a m e r u p a k a n s e t c e m b u n g d a i s e m u a s e t p a d a g a m b a r 2 . b a d a l a b b u k a n s e t c e m b u n g . S e c a r a u m u r i d a p a t d i k a t a k a n b a b w a s y a r a t s e b a g a i s e t c e m b u n g a d a l a b b a b w a s q t t i d a k m e m i U k i l u b a n g d a n b a t a s - b a t a s n y a b a r u s m u l u s .

G a m b a r 2 . Set

( a ) Cembung ( b ) bukan Cembung

D a l a m r u a n g b e r d i m e n s i 4 a t a u l e b i b , i n t e r p r e t a s i g e o m e t r i s m e n ­j a d i s u U t k a r e n a i t u d i p e r l u k a n d e f i n i s i s e t c e m b u n g s e c a r a a l j a b a r . U i -t u k t u j u a n i n i d i p e r l u k a n p e n g e r t i a n a k a n k o n s e p convex combination of vectors, y a n g m e r u p a k a n s u a t u j e n i s k b u s u s d a r i linear combinatio i S u a t u linear combination d a r i d u a v e k t o r U d a n V d a p a t d i t u l i s s e p e r t :

k , U + k , V ,

2; 15

Mulyono

d i m a n a /Cj d a n m e r u p a k a n s k a l a r . J i k a k e d u a s k a l a r t e r l e t a k p a d a i n t e r v a l t e r t u t u p [ 0 , 1 ] d a n j u m l a h n y a I , l i n e a r c o m b i n a t i o n d i k a t a k a n m e n j a d i convex combination d a n d i r u m u s k a n s e b a g a i : 0 U + ( 1 - 0 ) V , d i m a n a ( 0 < 0 < 1 ) .

S e b a g a i c o n t o b , k o m b i n a s i

s u a t u convex combination.

1 T 4

a d a l a b

K e m u d i a n , d e f i n i s i a l j a b a m y a a d a l a b : suatu set S adalah cembung jika dan hanya jika, untuk dua titik 13 GS dan WGS, dan untuk setiap skalar 0 £ [ 0 , 1 ] , adalah benar bahwa

[ w = 0 U + ( l - 0 ) V ] G 5

D e f i n i s i i n i d a p a t d i t e r a p k a n t a n p a m e m p e r b a t i k a n d i m e n s i r u a n g d i m a n a t e r d a p a t v e k t o r U d a n V .

I I I . Masalah Optimisasi

1. Optimisasi Tanpa Kendala

S u a t u n i l a i e k s t r i m d a r i f u n g s i m e n u n j u k k a n s u a t u n i l a i m a k s i m u m a t a u m i n i m u m d a r i f u n g s i i t u . T e o r i o p t i m i s a s i k l a s i k m e n g g u n a k a n k a l k u l u s d e r i v a t i f u n t u k m e n e n t u k a n t i t i k e k s t r i m b a i k u n t u k f u n g s i y a n g t a n p a k e n d a l a m a u p u n d e n g a n k e n d a l a p e r s a m a a n . S e c a r a m a t e m a t i k , s u a t u t i t i k X Q = ( X j , ^ 2 , . . . , x ^ ) a d a l a b m a k s i m u m j i k a

/ ( X Q + D ) < / ( X Q )

u n t u k s e m u a D = ( < i j , ^ 2 , . . . , d^) d i m a n a \d.\ c u k u p k e c i l u n t u k s e m u a / . D e n g a n c a r a y a n g s a m a , X ^ a d a l a b m i n i m u m ( u n t u k D y a n g d e d e f i n i s i k a n s e p e r t i s e b e l u m n y a ) j i k a :

/ ( X Q + D ) > / ( X o ) .

G a m b a r 3 . m e m p e r a g a k a n t i t i k m a k s i m a d a n m i n i m a s u a t u f u n g s i v a r i a b e l t u n g g a l / ( x ) d a l a m i n t e r v a l [a, b]. I n t e r v a l a < x < b t i d a k b e r a r t i m e n u n j u k k a n p e m b a t a s p a d a / ( x ) . T i t i k X j , X2, X3, X 4 , d a n Xg s e m u a n y a a d a l a b n i l a i e k s t r i m d a r i / ( x ) . T i t i k X j , X3 d a n Xg s e b a g a i n i l a i m a k s i m a d a n X2, x^ s e b a g a i m i n i m a . K a r e n a / ( X g ) = m a k s [ / ( X j ) , / ( x 3 ) , / ( x g ) ] , m a k a / ( X g ) d i n a m a k a n m a k s i ­m u m g l o b a l a t a u a b s o l u t , d a n / ( X j ) , / ( X 3 ) a d a l a b m a k s i m u m l o k a l a t a u r e l a t i f . S e c a r a s e r u p a / ( x ^ ) a d a l a b m i n i m u m l o k a l d a n / ( X 2 ) a d a l a b m i n i m u m g l o b a l . P e r l u d i t e g a s k a n b a b w a s u a t u o p t i m u m g l o b a l d e n g a n s e n d i r i n y a j u g a m e r u p a k a n o p t i m u m l o k a l .

226

I Program NonLiniir

G a m b a r 3 . N i l a i ekstrim suatu fungsi v a r i a b e l tunggal

fix)

a x^ X j X3 X4 X j X g /, X

M e s k i p u n Xj a d a l a b t i t i k m a k s i m u m , i a b e r b e d a d a r i m a k s i m u m l o k a l y a n g l a i n d a l a m b a l b a b w a n i l a i / u n t u k s e k u r a n g - k u r a n g n y a s a t u t i t i k d i s e k i t a r Xj s a m a d e n g a n / ( X j ) . D a l a m k a i t a n i n i , Xj d i n a m a k a n m a k s i m u m l e m a b d i b a n d i n g X3 m i s a l n y a , d i m a n a / ( x 3 ) m e n u n j u k k a n m a k s i m u m k u a t . S u a t u m a k s i m u m l e m a b b e r a r t i j u m l a b t a k t e r b a t a ;

( a l t e r n a t i f ) m a k s i m a . D e n g a n p e n a l a r a n y a n g s a m a , X4 a d a l a b m i n i m u n l e m a b . P a d a u m u m n y a a d a l a b m a k s i m u m l e m a b j i k a / ( X Q + D ) <

/ ( X Q ) d a n m a k s i m u m k u a t , j i k a / ( X Q + D ) < / ( X ^ ) , d i m a n a D d i d e f i n i • s i k a n s e p e r t i s e b e l u m n y a . S u a t u p e n g a m a t a n t e n t a n g t i t i k e k s t r i m y a n j m e n a r i k a d a l a b b a b w a t u r u n a n p e r t a m a d a r i / s a m a d e n g a n n o l p a d : t i t i k - t i t i k i n i . N a m u n c i r i m i t i d a k k b u s u s . C o n t o b n y a , t u r u n a n p e r t a m i ( s l o p e ) d a r i / ( X j ) a d a l a b n o l m e s k i p u n Xg b u k a n t i t i k e k s t r i m . T i t i k i n d i n a m a k a n s e b a g a i t i t i k b e l o k iinflection I saddle point).

D e n g a n T e o r i T a y l o r d a p a t d i b u k t i k a n b a b w a :

1 . S u a t u syarat perlu b a g i X ^ m e n j a d i s u a t u t i t i k e k s t r i m d a r i / ( X ) a d a l a b b a b w a g r a d i e n t v e c t o m y a a t a u V / ( X ^ ) = 0 , t i t i k y a n g d i p e r ­o l e h d a r i p e r s a m a a n i t u d i n a m a k a n t i t i k s t a s i o n e r .

2 . S u a t u syarat cukup u n t u k s u a t u t i t i k s t a s i o n e r X ^ m e n j a d i e k s t r i m a d a l a b d e n g a n m e n g e v a l u a s i m a t r i k s H e s s i a n , H , p a d a X ^ , s e p e r t i y a n g t e l a b d i t u n j u k k a n o l e b E n d e l b a u m b e r i k u t i n i : a . S u a t u t i t i k s t a s i o n e r m e n j a d i s u a t u n i l a i m i n i m u m a d a l a b c u k u p

j i k a l e a d i n g p r i n c i p a l d e t e r m i n a n t H.^, H^, . . . , s e m u a n y a p o s i t i f ( p o s i t i f d e f i n i t ) .

227

Mulyono

b . S u a t u t i t i k s t a s i o n e r m e n j a d i s u a t u n i l a i m a k s i m u m , a d a l a b c u k u p j i k a s e m u a l e a d i n g p r i n c i p a l d e t e r m i n a n t g e n a p a d a l a b p o s i t i f d a n s e m u a l e a d i n g p r i n c i p a l d e t e r m i n a n t g a n j i l a d a l a b n e g a t i f ( n e g a t i f d e f i n i t ) .

H ^ . < 0 u n t u k / = 1 , 3 , 5 , . . . H ^ > 0 u n t u k 7 = 2 , 4 , 6 , . . . J i k a s y a r a t - s y a r a t d i a t a s t i d a k s e c a r a t e p a t d i p e n u b i , m a k a t i t i k i t u m u n g k i n a t a u m u n g k i n t i d a k m e r u p a k a n s o l u s i o p t i m u m . D a l a m b a l i n i , u j i d e n g a n t i n g k a t l e b i b t i n g g i d i p e r l u k a n a t a u s e m u a t i t i k s t a s i o ­n e r d i t e b t i .

P e r b a t i k a n c o n t o b b e r i k u t :

1 . S y a r a t p e r l u V / ( X ^ ) = 0 m e m b e r i k a n / i = 1 - = 0 /2 = Z3 - 2 ^ 2 = 0

/3 = 2 + ^ 2 - 2 ^ 3 = 0 S o l u s i p e r s a m a a n s i m u l t a n i n i a d a l a b : X ^ = ( 1 / 2 , 2 / 3 , 4 / 3 )

2 . S y a r a t c u k u p d i t e t a p k a n d e n g a n :

r / 1 1 A 2 ~ - 2 0 Q " iH) = A l A 2 A s 0 - 2 1

_ A i A 2 A 3 _ 0 1 - 2 _

S e b i n g g a l e a d i n g p r i n c i p a l d e t e r m i n a n t a d a l a b : Hy= —2, H^^ 4, d a n 7 /3 = - 6 .

I n i b e r a r t i X ^ m e r u p a k a n t i t i k m a k s i m u m .

2 . O p t i m i s a s i d e n g a n K e n d a l a P e r s a m a a n

K e n d a l a i t u d a p a t b e r b e n t u k p e r s a m a a n a t a u p e r t i d a k s a m a a n . B a g i a n b e r i k u t i n i m e m b i c a r a k a n o p t i m i s a s i f u n g s i k o n t i n y u d a n d a p a t d i t u r u n ­k a n d e n g a n k e n d a l a p e r s a m a a n . U n t u k m e n c a p a i m a k s u d i t u t e r s e d i a b e b e r a p a m e t o d e o p t i m i s a s i , d i s i n i b a n y a a k a n d i b i c a r a k a n p r o s e d u r L a g r a n g e a n . T e k n i k m a t e m a t i k Lagrange multiplier t e l a b d i k e m b a n g ­k a n u n t u k m e n g a t a s i m a s a l a b o p t i m i s a s i d e n g a n k e n d a l a p e r s a m a a n d a l a m s u a t u b e n t u k d e m i k i a n b i n g g a s y a r a t p e r l u b a g i m a s a l a b o p t i m i ­s a s i t a n p a k e n d a l a m a s i b d a p a t d i t e r a p k a n . H a l i n i t e n t u b a n y a d a p a t d i c a p a i d e n g a n m e n c i p t a k a n m a s a l a b b a r u s e p e r t i y a n g a k a n d i t e r a n g -k a n b e r i k u t .

2 2 8

Program Non Lini it

B a y a n g k a n m a s a l a h m a k s i m i s a s i s u a t u f u n g s i k o n t i n y u d a n d a p i t d i t u r u n k a n y ^ = / ( x ^ , X j , . . . , x ^ ) d e n g a n k e n d a l a g ( X j , X j , . . . , x^ ) = b, di m a n a g (X) j u g a k o n t i n y u d a n d a p a t d i t u r u n k a n . K o n d i s i l i a t a s m e n y a r a n k a n b a h w a d a p a t d i p i l i b v a r i a b e l x ^ p a d a k e n d a l a d< n m e n y a t a k a n d a l a m v a r i a b e l y a n g l a i n s e b i n g g a , x ^ = / f ( X j , X j , . . . , x ^ _ j ) . K e m u d i a n d i s u b s t i t u s i k a n k e f u n g s i t u j u ­a n u n t u k m e n d a p a t k a n :

Po = / [ ^ 1 . ^ 2 ' • • • ' \ - V f i ( . h ' ^ 2 '

D a l a m b e n t u k i n i , m e t o d e k l a s i k d a p a t d i t e r a p k a n k a r e n a f u n g s i n y t a n p a k e n d a l a . S u a t u s y a r a t p e r l u u n t u k t i t i k e k s t r i m a d a l a b d e n g a | n m e n g b i l a n g k a n s e m u a t u r u n a n p e r t a m a .

a y O -dx

= 0 , d i m a n a / = 1 , 2 , . . . , « — 1 . D e n g a n dalil rantai 1

^ i Z + ^ _ ^

3 x , 3 x , 3 x • 3 x , .

D a r i g ( X j , x^, . . . , x ^ ) = b , dg M 1 1 +

9 x . 3 x / 3 x .

S e b i n g g a :

K a r e n a i t u :

a y , 3 x , 3 x ,

0 ,

a g

d i m a n a j = 1,2, . . . ,n—

d i p e r o l e h

d i m a n a / = 1 , 2 , . . . , n —

3 x 7 a x . j i k a

7

3 / 3 g 3 g _ ^ 3x„ ~ 3x„- 3 x .

a g - T — 5 ^ 0 , u n t u k

K / = 1 , 2 , n-\

j = 1 , 2 , . .. ,n-

J i k a v e k t o r s o l u s i y a n g d i p e r o l e h a d a l a b v e k t o r m a k s i m u m , m a k | a X 3 * , X 2 * , . . . , x ^ * a d a l a b n i l a i m a k s i m u m . D e n g a n m e n g g a n t i

a / a g 3x„ 9x„ n n

- 3 x . " '

m a k a

d i m a n a / = 1 , 2 , . . . , / .

d e n g a n s y a r a t g ( x ^ , X 2 , . . . , x ^ ) = b

27 9

Mulyono

I n g a t b a h w a s e k a r a n g t e r d a p a t n + 1 p e r s a m a a n d a n n + 1 v a r i a b e l t a k d i k e t a b u i . I n i a d a l a b s y a r a t p e r l u u n t u k s u a t u o p t i m u m . K o n d i s i i n i d a l a m p r a k t e k d a p a t d i p e r o l e h d e n g a n m u d a b , m e l a l u i h u b u n g a n b e r i k u t :

J i k a s u a t u f u n g s i t u j u a n f (x^, x ^ , . . . , x ^ ) d e n g a n k e n d a l a g ( X j , X 2 , . . . , x ^ ) = b , m a k a f u n g s i L a g r a n g e a n n y a a d a l a b :

L = / ( X j , X 2 , . . . , x ^ ) - X [ g ( X j , X 2

d a n k e m u d i a n s y a r a t p e r l u u n t u k s u a t u n i l a i s t a s i o n e r a d a l a b :

d L d f dg ^ ' ^. . , ^ ^ = - ^ . - ^ ^ r ^ ' d i m a n a / = l , 2 , . . . , n .

d L d X

= g ( X j , X 2 , . . . ,x^)-b = 0

y a n g p e r s i s s a m a d e n g a n k o n d i s i s e b e l u m n y a y a n g d i t u r u n k a n u n t u k o p t i m i s a s i .

C o n t o b :

C a r i l a b n i l a i e k s t r i m d a r i / ( X ) = 3 X j ^ +-^2^ 2XjX2 + 6Xj + 2X2

d e n g a n s y a r a t : 2Xj — X2 = 4 F u n g s i L a g r a n g e a n y a n g d i p e r o l e h :

L ( X j , X 2 , X ) = 3 X j 2 + X 2 ^ + 2 x j X 2 + 6 X j + 2 X 2 - X ( 2 X j - X 2 - 4 )

U n t u k s u a t u n i l a i s t a s i o n e r , s y a r a t p e r l u n y a a d a l a b :

Z , j = 6 x j + 2x2 + 6 - 2 X = 0

Z,2 = 2x2 + 2 X j + 2 + X = 0

L A = 2 x i - X 2 - 4 = 0

T e r d a p a t 3 p e r s a m a a n d e n g a n 3 v a r i a b e l y a n g d i p e c a b k a n s e c a r a s i m u l t a n m e n g h a s i l k a n : X* = 7 / 1 1 , x * = - 3 0 / 1 1 X = 2 4 / 1 1 f u n g s i t u j u a n n y a m e m b e r i k a n n i l a i / ( X ^ ) = 8 5 , 7 . P e r b a t i k a n b a b w a p a d a n i l a i e k s t r i m , n i l a i f u n g s i L a g r a n g e a n s a m a d e n g a n n i l a i f u n g s i t u j u a n , L ( x ^ , X 2 , X ) = / ( X j , X 2 ) .

M e s k i p u n t e l a b d i p e r o l e h s u a t u t i t i k s t a s i o n e r , s e s u n g g u b n y a t a k a d a j a m i n a n b a b w a v e k t o r s o l u s i i n i a d a l a b s a t u - s a t u n y a y a n g d i c a r i . K e n y a t a a n n y a , s e t i a p v e k t o r s o l u s i y a n g d i p e r o l e h d e n g a n t e k n i k o p t i -m a s i i n i d a p a t b e r u p a s u a t u n i l a i m a k s i m u m , m i n i m u m , a t a u t i t i k b e l o k . T u g a s s e l a n j u t n y a a d a l a b m e n g u j i s i f a t t i t i k s t a s i o n e r , d e n g a n k a t a l a i n

2 3 0

Program Non Linier

d i p e r l u k a n s u a t u s y a r a t c u k u p u n t u k d a p a t m e n g a t a k a n a p a k a h s u a t p n i l a i s t a s i o n e r a d a l a h s u a t u m a k s i m u m a t a u m i n i m u m .

S e p e r t i p a d a k a s u s o p t i m i s a s i t a n p a k e n d a l a , s y a r a t c u k u p p a d a k a s u s i n i j u g a d i e k s p r e s i k a n d a l a m b e n t u k d e t e r m i n a n . P o s i s i d e t e r m i i a n m a t r i k s H e s s i a n p a d a m a s a l a b o p t i m i s a s i d e n g a n k e n d a l a p e r s a m a a i d i g a n t i k a n d e n g a n a p a y a n g d i s e b u t Bordered Hessian. S y a r a t c u k u p u i b a n y a d a p a t d i t e r a p k a n s e t e l a b s y a r a t p e r l u d i p e n u b i .

U n t u k m e n g o p t i m i s a s i s u a t u f u n g s i / ( x ^ , X j , . . . , x „ ) d e n g a b k e n d a l a g ( X j , x^, • • .,xf) = 6 , m a k a d a p a t d i b e n t u k f u n g s i L a g r a n g e a n

L = / ( X J ) - X [ g ( X j ) - 6 ] , d e n g a n s y a r a t p e r l u :

L j = L 2 = . . . = L ^ = L x - 0 .

s y a r a t c u k u p d a p a t d i e k s p r e s i k a n d a l a m s u a t u B o r d e r e d H e s s i a n , (H d a l a m d u a c a r a , y a i t u :

(H) =

"11

"21

"12

"22

^nl ^ n l h Si

"In

"2n

^1

Si

S„

a t a u

0

S,

S„

Si L 11

^nl

Si L 12

^n2

"In

S y a r a t i t u b a r u s d i e k s p r e s i k a n d a l a m bordered principal minor. D a j i Bordered Hessian d i a t a s , m a k a b o r d e r e d p r i n c i p a l m i n o r n y a a d a l a b :

" 0 s.

Si L 11

Si L

Si ^21 12

"22

0 Si Si S3 ^1 h i h i h3 Si h i h i h3 S3 h i h i h3

d a n s e t e r u s n y a . K e m u d i a n ,

J i k a H 2 , Hy . . . , H^ = H < 0, b o r d e r e d H e s s i a n a d a l a b p o s i t i f d e f i n i t y a n g m e r u p a k a n s y a r a t c u k u p u n t u k s u a t u m i n i m u m .

J i k a H2 > 0, Hj < 0 , H^ > 0 , d a n s e t e r u s n y a , b o r d e r e d H e s s i a n a d a l a b n e g a t i f d e f i n i t y a n g m e r u p a k a n s y a r a t c u k u p u n t u k m a k s i m u m . I n g a | t b a b w a p e m e r i k s a a n d i m u l a i d e n g a n / ( j , d a n b u k a n H^. M e l a n j u t k a n c o n t o b d i a t a s d i p e r o l e h :

Mulyono

0 2

- 1

2 6 2

- r 2 2

= - 2 2

yang berarti fungsi L adalab pada nilai minimum.

3. Pembedaan Program Linier dengan Non Linier

Suatu metode umum yang dinamakan metode simplex dapat digunakan untuk menyelesaikan masalab program bnier baik yang mebbatkan dua atau lebib variabel. Metode itu berlaku karena dalam setiap masalab program linier berlaku bal-bal seperti berikut yang merupakan syarat bagi bekerjanya metode simplex: (1) solusi layak merupakan set cem­bung dengan jumlab titik pojok (ekstrim) yang terbatas, (2) jika fungsi tujuan terbatas, nilai optimalnya akan terjadi pada salab satu titik pojok dari ruang solusi, mungkin pada lebib dari satu titik pojok, (3) suatu optimum lokal adalab juga optimal global dari fungsi tujuan.

Dalam program non bnier, beberapa atau semua bal di atas mung­kin tidak berlaku. Sebagai suatu ilustrasi, perbatikan masalab-masalab berikut yang untungnya dapat diselesaikan secara grafis karena banya mebbatkan dua variabel. Contob : Minimumkan

dengan syarat z =

^1 =

^ 2 = h = « 4 =

X

3x -2x

2x 2x

X

Ini adalab suatu masalab program Unie

+ + +

^2 2x,

- X , < <

+ 3x, >

12 3 4 6 0

dengan nilai minimum z sebesar 13/4 seperti yang ditunjukkan pada gambar 4.

Gambar 4.

232

Program Non Linie r

r u a n g s o -

M i s a l k a n s e k a r a n g k e n d a l a t e t a p d i p e r t a h a n k a n , t e t a p i m e n g u b 4 h , f u n g s i t u j u a n m e n j a d i n o n b n i e r , k a t a k a n : z = ( X j - 4 ) + ( X j - 3 ) ' ' N i l a i m i n i m a l f u n g s i z t i d a k l a g i t e r j a d i p a d a t i t i k p o j o k d a r i l u s i , t e t a p i p a d a s u a t u t i t i k p a d a b a t a s , d e n g a n X j = 3 4 / 1 3 , X 2 = 2 7 / 1 3 d a n z = 4 6 8 / 1 6 9 , s e p e r t i d i t u n j u k k a n p a d a g a m b a r 5 . J a d i j e l a s b a b w m e t o d e s i m p l e x ' y a n g b e r g e r a k d a r i s a t u t i t i k p o j o k k e t i t i k p o j o k y a n b e r d e k a t a n , d i s i n i t a k d a p a t d i g u n a k a n .

G a m b a r 5 .

Z = 4 9 3 / 1 0 0

J i k a f u n g s i t u j u a n i t u b e r u b a b m e n j a d i : z = ( X j - 9 / 5 ) ' + ( X 2 - if d a n k e n d a l a - k e n d a l a n y a t e t a p s a m a , n i l a i m i n i m a l z t e r j a d i p a d a t i t i k

X j = 9 / 5 , X 2 = 2 y a n g m e r u p a k a n s u a t u t i t i k i n t e r i o r ( b a g i a n d a l a m | ) d a r i r u a n g s o l u s i . J e l a s b a b w a z = 0 a d a l a b n i l a i m i n i m u m , s e p e r t i d t u n j u k k a n p a d a g a m b a r 6 .

G a m b a r 6

9 / 5 ^

G a m b a r 6 j u g a m e n u n j u k k a n b a b w a s u a t u o p t i m u m l o k a l t i d a k s e l a l m e r u p a k a n o p t i m u m g l o b a l . F u n g s i t u j u a n y a n g d i s e b u t k a n t e r a k b i | r m e n c a p a i s u a t u m a k s i m u m l o k a l p a d a t i t i k B , t e t a p i i n i b u k a n m a k s m u m g l o b a l , y a n g t e r j a d i p a d a t i t i k D .

233

Mulyono

P a d a c o n t o h d i a t a s s e m u a k e n d a l a b e r u p a f u n g s i l i n i e r , s e b i n g g a r u a n g s o l u s i n y a m e r u p a k a n s e t c e m b u n g . J i k a b e b e r a p a a t a u s e m u a k e n ­d a l a t i d a k b n i e r , r u a n g s o l u s i m u n g k i n b u k a n m e r u p a k a n s e t c e m b u n g s e p e r t i d i t u n j u k k a n o l e b w i l a y a b y a n g t e r p i s a b p a d a g a m b a r 7 , d i m a n a k e n d a l a - k e n d a l a n y a a d a l a b :

X j • < 1 . X j 2 + xf- > 4 X j . X j > 0

G a m b a r 7 . Ruang solusi bukan Set Cembung

^2

K e c e m b u n g a n , b e s e r t a b e b e r a p a a s u m s i l a i n , m e n g h a s i l k a n s u a t u t e o i i y a n g m e n y a t a k a n b a b w a s u a t u m i n i m u m l o k a l a d a l a b j u g a s u a t u m i n i ­m u m g l o b a l . S e b i n g g a , t a n p a k e c e m b u n g a n k e s u l i t a n - k e s u b t a n y a n g d i -b a d a p i d a l a m m e n y e l e s a i k a n m a s a l a b i t u p a d a u n l u m n y a j a u b l e b i b b e s a r .

T a k a d a s e b u a b a l g o r i t m a p u n , s e p e r t i a l g o r i t m a s i m p l e x d a l a m m a s a l a b - m a s a l a b p r o g r a m l i n i e r , y a n g d a p a t d i t e r a p k a n t e r b a d a p s e l u r u b m a s a l a b p r o g r a m n o n l i n i e r y a n g b e g i t u l u a s . B e r m a c a m - m a c a m t e k n i k d a n m e t o d e p e r b i t u n g a n t e l a b d i g u n a k a n u n t u k m e n y e l e s a i k a n m a s a l a b -m a s a l a b y a n g b e r l a i n a n , d a n s u a t u r i n g k a s a n y a n g b a i k d a r i m e t o d e -m e t o d e i t u d a p a t d i t e m u k a n p a d a C o n v e r s e ( 1 9 7 0 ) . T u U s a n m i t i d a k a k a n m e m b a b a s m e t o d e - m e t o d e i t u , n a m u n b a n y a a k a n m e m b i c a r a k a n b e b e r a p a i d e a n a l i t i s y a n g m e n a n d a i s u a t u s o l u s i , k h u s u s n y a t e o r i K u b n -T u c k e r . T e o r i i n i m e n u n j u k k a n b a g a i m a n a m e n g e t a b u i a p a k a b s u a t u

234

Program Non Unit r

s o l u s i o p t i m a l a t a u t i d a k , t e t a p i t i d a k m e m b e r i k a n t e k n i k p e r b i t u n g a i u n t u k m e m p e r o l e b s o l u s i i t u , k e c u a i i m u n g k i n , d a l a m k a s u s - k ^ s u s y a n i s a n g a t s e d e r b a n a .

4. M a s a l a h O p t i m i s a s i K e n d a l a P e r t i d a k s a m a a n ( P r o g r a m N o n L i n i e r )

P a d a b a g i a n t e r d a b u l u , t e l a b d i j e l a s k a n b a b w a Lagrangean multiplie d a p a t d i g u n a k a n d a l a m m e n y e l e s a i k a n m a s a l a b - m a s a l a b o p t i m i s a s i d e n g a n k e n d a l a p e r s a m a a n . K u b n d a n T u c k e r ( 1 9 5 1 ) t e l a b m e m p e r l u a s t e o r i i n i u n t u k m e n y e l e s a i k a n m a s a l a b p r o g r a m n o n l i n i e r u m u m b a i l ; d e n g a n k e n d a l a p e r s a m a a n m a u p u n p e r t i d a k s a m a a n .

S y a r a t P e r l u K u h n - T u c k e r

S y a r a t p e r l u K u b n - T u c k e r y a n g d i b i c a r a k a n d i s i n i b e r t u j u a n m e n g i d e n -t i f i k a s i t i t i k s t a s i o n e r d a r i s u a t u m a s a l a b n o n l i n e a r d e n g a n k e n d a l a p e r ­t i d a k s a m a a n . D a l a m b a t a s - b a t a s t e r t e n t u s y a r a t - s y a r a t i n i j u g a m e r u p a ­k a n s y a r a t c u k u p . P i k i r k a n m a s a l a b b e r i k u t :

M a k s i m u m k a n z = / ( X ) d e n g a n s y a r a t g (X) < 0

K e n d a l a - k e n d a l a p e r t i d a k s a m a a n d a p a t d i u b a b m e n j a d i p e r s a m a a i d e n g a n m e n a m b a b k a n v a r i a b e l slack n o n n e g a t i f . U n t u k m e m e n u b s y a r a t n o n - n e g a t i f , m i s a l k a n 5*? ( > 0 ) m e r u p a k a n k u a n t i t a slack y a n g d i t a m b a b k a n p a d a k e n d a l a k e i, g. ( X ) < 0 . D e f i n i s i k a n S = ( s j , . . . , ^ ) d a n 5 ^ = {sf, sf, - - - . 5^^) d i m a n a m a d a l a b b a n y a k n y a k e n d a l a p e r t i d a k s a m a a n . S e b i n g g a f u n g s i L a g r a n g e - n y a a d a l a b

L ( X , S , X ) = / ( X ) - X [ g ( X ) + sA

d e n g a n s y a r a t g ( X ) < 0 .

S u a t u s y a r a t p e r l u u n t u k o p t i m a l i t a s a d a l a b b a b w a X n o n n e g a t i ( n o n p o s i t i f ) u n t u k s e m u a m a s a l a b m a k s i m i s a s i ( m i n i m i s a s i ) . P e m i k i r a n n y a a d a l a b s e p e r t i b e r i k u t . P i k i r k a n k a s u s m a k s i m i s a s i d i a t a s . D i s e b u t ­k a n b a b w a X m e n g u k u r t i n g k a t p e r u b a h a n / d a l a m m e n a n g g a p i g, y a i t i X = 3 / / 3 g ( d a r i b a g i a n 3 . 2 ) . J i k a s i s i k a n a n k e n d a l a g ( X ) < 0 m e n i n g k a d i a t a s n o l , r u a n g s o l u s i m e n j a d i k u r a n g d i b a t a s i , s e b i n g g a / t a k d a p a t u r u n . I n i b e r a r t i b a b w a X > 0 . D e n g a n c a r a y a n g s a m a , u n t u k m i n i m i s a s i j i k a s u m b e r d a y a b e r t a m b a h , / t a k d a p a t n a i k y a n g b e r a r t i X < 0 J i k a k e n d a l a b e r u p a p e r s a m a a n y a i t u g ( X ) = 0 , m a k a X m e n j a d i t a k t e r b a t a s d a l a m t a n d a .

P e m b a t a s a n X y a n g d i b e r i k a n d i a t a s b a r u s d i p e n u b i s e b a g a i b a g i a i d a r i s y a r a t p e r l u K u b n - T u c k e r . P e m b a t a s a n X i n i d i k e n a l s e b a g a i con straint qualification y a i t u k o n d i s i t a k t e r a t u r d a l a m r u a n g s o l u s i t e r u t a

2 3 : :

Mulyono

m a p a d a t i t i k s t a s i o n e r . K o n d i s i i n i d i a s u m s i k a n t a k a k a n t e r j a d i . P e m -b i c a r a a n constraint qualification y a n g l e n g k a p d i l u a r j a n g k a u a n t u l i s a n i n i . S y a r a t - s y a r a t p e r l u y a n g l a i n a k a n d i j e l a s k a n b e r i k u t i n i .

D e n g a n m e n g a m b i l t u r u n a n p a r s i a l ( t e r b a d a p X, S d a n X ) d i p e r ­o l e h :

= V / ( X ) - X V g ( X ) = 0

L . = - 2 X . S. = 0, i = l,2,...,m

Lx= ~ [g{X) + Sf] = 0

P e r s a m a a n k e d u a m e n y a t a k a n : 1 . J i k a X. l e b i b b e s a r d a r i n o l , S? = 0 . I n i b e r a r t i b a b w a s u m b e r d a y a y a n g b e r s a n g k u t a n a d a l a b l a n g k a , d a n k o n s e k u e n s i n y a i a d i b a b i s k a n s e l u r u h n y a ( k e n d a l a p e r s a m a a n ) . 2 . J i k a sf > 0 , X . = 0 . I n i b e r a r t i s u m b e r d a y a y a n g k e i t i d a k l a n g k a d a n k o n s e k u e n s i n y a i a t i d a k m e m p e n g a r u h i n i l a i f u n g s i t u j u a n ( X . . = df/dg. = 0 ) .

P e r s a m a a n k e d u a d a n k e t i g a s e l a n j u t n y a m e n y a t a k a n b a b w a

X . • g. ( X ) = 0 , k a r e n a j i k a X . > 0 m a k a 5 ^ = 0 y a n g b e r a r t i g , ( X ) = 0 . I I * * — *

D e n g a n p e n a l a r a n s e r u p a , j i k a g - ( X ) < 0 , b e r a r t i Sf > 0 , m a k a X ^ = 0 .

S y a r a t p e r l u K u b n T u c k e r u n t u k X d a n X u n t u k m e n j a d i t i t i k s t a s i o n e r d a r i m a s a l a b m a k s i m i s a s i d a p a t d i r i n g k a s s e p e r t i b e r i k u t :

X > 0 V / ( X ) - X V g ( X ) = 0

X j g . ( X ) = 0 , / = 1 , 2 , . . . , m g ( X ) < 0

A n d a d a p a t m e m b u k t i k a n b a h w a s y a r a t - s y a r a t i n i b e r l a k u u n t u k k a s u s m i n i m i s a s i , k e c u a i i b a b w a X b a r u s n o n p o s i t i v e . D a l a m m a k s i m i ­s a s i m a u p u n m i n i m i s a s i , L a g r a n g e m u l t i p l i e r y a n g b e r b u b u n g a n d e n g a n k e n d a l a - k e n d a l a p e r s a m a a n b a r u s t a k t e r b a t a s .

J a n g a n l u p a , j i k a a n d a m e m b e n t u k f u n g s i l a g r a n g e s e p e r t i b e r i k u t :

Z , ( X , X , S ) = / ( X ) + X [ g ( X ) + 5 f ]

U n t u k m a s a l a b m a k s i m i s a s i , s e b a g a i s y a r a t p e r l u X b a r u s n o n - p o s i t i v e ( b u k a n l a g i n o n n e g a t i O k a r e n a X d i s i n i d i d e f i n i s i k a n s a m a d e n g a n - dfldg ( b u k a n dfldg).

2 3 6

Program Non Linier

Syarat Cukup Kuhn-Tucker Syarat perlu Kuhn Tucker juga merupakan syarat cukup jika fungri tujuan dan ruang solusi memenuhi syarat-syarat tertentu yang berkaita i dengan kecekungan dan kecembungan. Syarat-syarat ini diringkas padp tabel 1.

Tabel 1. Syarat C u k u p K u h n - T u c k e r

Jenis syarat yang diperlukan optimisasi fungsi tujuan ruang solusi

Maksimisasi cekung set cembung Minimisasi cembung set cembung

Lebib mudab membuktikan babwa suatu fungsi adalab cekun atau cembung daripada membuktikan babwa suatu ruang solusi adalah merupakan set cembung atau bukan. Karena alasan ini, akan diberika: i suatu daftar tentang syarat-syarat yang lebib mudab diterapkan dalar i praktek dalam arti babwa kecembungan ruang solusi dapat ditetapkai i dengan memeriksa langsung kecekungan atau kecembungan fungsi-fungsi kendala. Untuk memberikan syarat-syarat ini, misalkan didefinisikap masalab program non-linear sebagai berikut:

maximumkan atau minimumkan

dengan syarat :

z = AX)

g, . (X)<0, i = 1,2, ...,r g. ( X ) > 0 , i = r+l p g.(X)= 0, i = p+l,...,m

L (X. s, X) = / ( X ) - i X (g.(X) + sf)- X igfX) - sf)

di mana X̂ . adalab lagrange multiplier yang berbubungan dengan kendala ke /. Syarat-syarat untuk menetapkan syarat cukup Kubn Tucker dapa diringkas pada tabel 2 .

Kondisi-kondisi dalam tabel 2 banya merupakan suatu sub set dar i syarat-syarat pada tabel 1 . Alasannya adalab babwa suatu ruang solus dapat cembung tanpa memenuhi syarat-syarat seperti yang ditentukai pada Tabel 2 tentang fungsi kendala gfX).

2 3 7

Mulyono

T a b e l 2 . Syarat Cukup Kuhn-Tucker

J e n i s s y a r a t y a n g d i p e r l u k a n o p t i m i s a s i

/ (X) g f X ) \ M a k s i m i s a s i c e k u n g c e m b u n g > 0 ( l < i < / - )

c e k u n g < 0 ( r + 1 < z < p ) l i n e a r u n r e s t . z < m)

M i n i m i s a s i c e m b u n g c e m b u n g < 0 ( 1 < / < r ) c e k u n g > 0 < / < p ) l i n e a r u n r e s t . ( p + 1 < i < m)

B e r i k u t i n i a k a n d i t u n j u k k a n b e b e r a p a c o n t o b c a r a m e n y e l e s a i k a n m a s a l a b p r o g r a m n o n l i n i e r d e n g a n s y a r a t p e r l u d a n s y a r a t c u k u p K u b n -T u c k e r .

C o n t o b 1 :

T e n t u k a n x ^ , d a n x ^ s e b i n g g a

M a k s i m u m k a n Z^-xf -xf -xf + 4 x ^ + 6 x 2

d e n g a n s y a r a t : X j + X 2 < 2

2 x j + 3 X 2 < 1 2

X j , X 2 > 0

K a r e n a f u n g s i t u j u a n c e k u n g d a n s e l u r u b f u n g s i k e n d a l a a d a l a b c e m b u n g m a k a s y a r a t p e r l u j u g a m e r u p a k a n s y a r a t c u k u p u n t u k s u a t u n i l a i m a k s i m u m . S y a r a t - s y a r a t p e r l u i n i d i p e r o l e b d e n g a n t u r u n a n p a r ­t i a l f u n g s i L a g r a n g e .

L = -xf - xf - xf + 4 x j + 6 X 2 - \ ( X j + X 2 - 2 )

- X 2 ( 2 x j + 3 x 2 - 1 2 ) .

S y a r a t p e r l u :

\ , X 2 > 0 (1)

L j = - 2 x j -h 4 - X J - 2 X 2 = 0 ( 2 a )

L 2 = - 2 x 2 + 6 - X J - 3 X 2 = 0 ( 2 b )

L 3 = - 2 X 3 = 0 ( 2 c )

X J ( X j + X 2 - 2 ) = 0 ( 3 a )

X 2 ( 2 x j + 3 x 2 - 1 2 ) = 0 ( 3 b )

2 3 8

Program Non Link r

A ^ " ^ ^ 2 - 2 ) = 0 2 x j +3x2 - 1 2 < 0 ( 4 1 )

Pada umumnya, setiap formulasi Lagrange dari suatu masala h program non linier dengan n variable keputusan dan m kendala dapi it mengbasilkan 'f*" kemungkinan kombinasi solusi. Dalam kasus ini terdapat tiga variabel dan dua kendala, sebingga terdapat 2^*^ kemun |-kinan kombinasi solusi.

Agar suatu kasus solusi memenubi syarat sebagai calon solusi opti­mal, ia barus memenuhi syarat perlu Kubn-Tucker. Karena itu setij p kasus solusi harus diperiksa dengan cara mensubstitusikan ke dala n syarat perlu. Beberapa kemungkinan kombinasi akan diperiksa seperti ditunjukkan berikut ini.

Kasus 1 X j = 0 X 2 = 0 . Dari persamaan 2a, 2b, 2c dihasilkan X j = 2, X 2 = 3 Solusi ini menyimpang dari kondisi 4a dan 4b.

X3 = 0 .

Kasus 2 : X J = 0 dan vt 0 Syarat nomor 3 menghasilkan 2Xj + Sxj = 12, syarat 2a dan 2b meng hasilkan — 2xj + 4 = 2X3, — 2xj + 6 = 3X2. Solusi persamaan simult tan ini menghasilkan Xj = 24/13, Xj = 36/13 dan Xj = 2/13 > 0 syarat nomor 2c memberikan X3 = 0. Namun, solusi ini menyimpajig dari 4a. Karena itu solusi ini dibilangkan.

+ Xj = 2 dan 2Xj + Sx^ = l. . Kasus 3 : X J # 0 . h 4^ 0

Syarat 3a dan 3b menghasilkan Xj Kedua persamaan ini mengbasilkan Xj = 6 dan Xj = 8. Sebingga 2 2b, dan 2c menghasilkan X 3 = 0, Xj = 68 dan Xj = -26 Solusi ini juga dibilangkan karena Xj = —26 menyimpang dari syarat nomor 1 .

Kasus 4 : X J # 0 dan X J = 0 Syarat 3a menghasilkan Xj + X 2 = 2. Ini bersama dengan 2a, 2b meih berikan Xj = 1/2, Xj = 3/2 dan Xj = 3 > 0. Di samping itu dari 2 diperoleb X3 = 0. Setelab ditebti, solusi ini tidak meny syarat perlu manapun. Sebingga solusi optimum (maksimum) glotja untuk masalab program non bnier adalab : Xj = 1/2, Xj = 3/2, X3 = 0 dengan Xj = 3, Xj = 0 Nilai maksimum fungsi tujuan adalab Z = 17/2.

739

Mulyono

C o n t o h 2 :

S e l e s a i k a n m a s a l a h p r o g r a m n o n l i n i e r b e r i k u t :

M i n i m u m k a n Z = xf + 5xf + lOxf - Ax^x^ + 6 x ^ X 3 - I 2 X 2 X ,

- 2 x j + l O X j + 5 x 3

d e n g a n s y a r a t X j + 2 X 3 + X 3 > 4 X j , X j , X 3 > 0

P e r t a m a , u b a b p e r s a m a a n d i a t a s s e b i n g g a m e n j a d i f u n g s i d e n g a n t u j u a n m a k s i m i s a s i :

B e n t u k f u n g s i L a g r a n g e a n n y a a d a l a b :

L = -xf - Sxf - lOxf + 4 x j X 2 - 6 x j X 3 + 1 2 X 3 X 3 + 2 x

- l O x j - 5 X 3 - X ( — X j - 2 X 3 - X 3 + 4 )

S y a r a t p e r l u n y a :

D i s i n i t e r d a p a t 2 v a r i a b e l d a n 1 k e n d a l a , s e b i n g g a d i h a s i l k a n 2^"^^ = 1 6 k e m u n g k i n a n k o m b i n a s i s o l u s i . S a t u - s a t u n y a c a l o n s o l u s i o p t i m a l y a n g m e m e n u b i s y a r a t p e r l u a d a l a b : X = 1 , 7 6 4 , X j = 2 , 9 4 1 , ^ 2 = 0 , 5 2 9 4 d a n = 0 , d e n g a n n i l a i Z = 3 , 2 3 5 .

K a r e n a f u n g s i t u j u a n a d a l a b f u n g s i c e m b u n g d a n f u n g s i k e n d a l a a d a l a b l i n i e r , m a k a m e m e n u b i s y a r a t c u k u p u n t u k m a s a l a b m i n i m i s a s i , s e b i n g g a n i l a i s o l u s i i t u m e r u p a k a n n i l a i m i n i m u m g l o b a l .

L j = - 2 X j - 4 x 3 - 6 x 3 + 2 + X = 0

L j = - l O X j + 4 X j + 1 2 x 3 - 1 0 + 2 X = 0

Z,3 = - 2 0 x 3 - 6 X j + 1 2 x 3 - 5 + X = 0

X ( - X j - 2 x 3 - X 3 + 4 ) = 0

- X j - 2 x 3 - X 3 + 4 < 0

X > 0 ( 1 ) ( 2 a ) ( 2 b ) ( 2 c )

( 3 ) (4)

240

Program Non Linier

C o n t o h 3 :

P i k i r k a n m a s a l a h m a k s i m i s a s i n o n l i n i e r b e r i k u t

M a k s i m u m k a n

d e n g a n s y a r a t

Y = 8xf + 2x3^

g = xf Axf X, , X ,

< 9 > 0

B e n t u k f u n g s i L a g r a n g e a d a l a b : L = 8xf + 2x3^ - \ ( X j 2 yxf' - 9 )

K e m u d i a n s y a r a t p e r l u d i s a j i k a n s e b a g a i b e r i k u t

X > 0 L j = 1 6 X j - 2 X X j = 0

= 4X3 - 2XX2 = 0

2 + X 2 2 - 9 ) = 0

< 0

( 2

U n t u k k a s u s i n i , t e r d a p a t 2 = 8 k o m b i n a s i s o l u s i u n t u k X j , d a n X y a n g b a r u s d i p e r t i m b a n g k a n . K o m b i n a s i n i l a i i t u d i r i n g k a s p a | d a T a b e l 3 .

T a b e l 3 .

a ) ( 2 b )

3:

K a s u s X j X2 X

1 = 0 = 0 = 0 2 = 0 = 0 > 0 3 = 0 > 0 = 0 4 = 0 > 0 > 0 5 > 0 = 0 H 0 6 > 0 = 0 > 0 7 > 0 > 0 = 0 8 > 0 > 0 > 0

D a r i 8 s o l u s i t e r n y a t a b a n y a t e r d a p a t 3 c a l o n y a n g m e m e m j b i s y a r a t p e r l u K u b n - T u c k e r y a i t u k a s u s 1 , 4 , d a n 6 .

141

Mulyono

T a b e l 4.

K a s u s ( X j , X j , X ) b a s i l q j i n i l a i s o l u s i s y a r a t p e r l u (Y)

1 ( 0 . 0 , 0 ) X J = 0 , X 2 = 0 X = 0 0 2 ( 0 , 0 > 0 ) m e n y i m p a n g 3 ( 0 , > 0 , 0 ) m e n y i m p a n g 4 ( o , > o ; > o ) X j = 3 , X = 2 , X j = 0

m e n y i m p a n g 1 8

5 0 0 , 0 , 0 ) X j = 3 , X = 2 , X j = 0 m e n y i m p a n g

6 O 0 , 0 , > 0 ) X j = 3 , X = 8 , X j = 0 7 2 * 7 O O O O , 0 ) m e n y i m p a n g 8 ( > 0 , > 0 , > 0 ) m e n y i m p a n g

K a r e n a f u n g s i t u j u a n m e r u p a k a n f u n g s i d e f i n i t p o s i t i f ( f u n g s i c e m b u n g ) m a k a s y a r a t p e r l u t i d a k m e m e n u b i s y a r a t c u k u p . S e b i n g g a p e n y e l e s a i a n p e r s a m a a n s i m u l t a n d a r i s y a r a t p e r l u a k a n m e n g b a s i l k a n o p t i m u m l o k a l , y a i t u :

P a d a t i t i k s t a s i o n e r , t a n g e n t f u n g s i k e n d a l a s a m a d e n g a n t a n g e n t f u n g s i t u j u a n ,

-gi I gi - h I h

2 X j / 2 X 3 = 1 6 X j / 4 X 3 , a t a u 4 X j = X j , b e r a r t i X j = 0 . D a r i s y a r a t p e r l u n o m o r 2 b d i p e r o l e b b a b w a X = 2 . K e m u d i a n d i s u b s t i t u s i k a n k e s y a r a t p e r l u n o m o r 3 d i p e r o l e b X 3 = 9 , j a d i X j = 3 . V e k t o r s o l u s i i n i a k a n m e n g b a s i l k a n n i l a i m a k s i r u m s e b e s a r 1 8 , y a n g t e r b u k t i b a n y a m e r u p a k a n m a k s i m u m l o k a l . D a l a m k a s u s s e m a c a m i n i n i l a i o p t i m u m g l o b a l d i c a r i d e n g a n m e m e r i k s a s a t u - p e r s a t u s e l u r u b k e m u n g k i n a n k o m b i n a s i s o l u s i s e p e r t i y a n g d i p e -r a g a k a n p a d a t a b e l 3 d a n 4 , d a n a k b i m y a d i p e r o l e b n i l a i m a k s i m u m g l o b a l 7 2 .

A k b i m y a , d a p a t d i t u n j u k k a n b a b w a d a r i s y a r a t p e r l u K u b n - T u c k e r s u l i t u n t u k m e n e m u k a n s e c a r a e k s p l i s i t s o l u s i n y a . I n i b e r a r t i p r o s e d u r i t u j u g a t i d a k e f i s i e n u n t u k p e r b i t u n g a n n u m e r i k . N a m u n , t e o r i i n i m a m p u m e n g i d e n t i f i k a s i k b a r a k t e r s o l u s i m a s a l a b u m u m p r o g r a m n o n b n i e r , a p a k a b i a o p t i m u m g l o b a l a t a u b a n y a l o k a l .

I V . K e s i m p u l a n

B i l a a n t a r v a r i a b e l d a l a m b i d a n g e k o n o m i b e r b u b u n g a n d e n g a n p o l a n o n l i n i e r , m a k a p r o g r a m n o n l i n i e r a k a n l e b i b t e p a t u n t u k m e n g e k s p r e -

2 4 2

Program Non Link r

s i k a n p e r s o a l a n i t u d i b a n d i n g p r o g r a m l i n i e r . H i l a n g n y a a s u m s i k e U n i e i -a n m e m b u a t p e r b i t u n g a n s o l u s i m a k i n s u l i t , d a n l i n g k u p t o p i k p r o g r a i j i n o n l i n i e r m e n j a d i s a n g a t l u a s d a n h e r v a r i a s i . T i d a k s e p e r t i a l g o r i t m s i m p l e x d a l a m p r o g r a m l i n i e r , d a l a m p r o g r a m n o n b n i e r b e l u m d i t e m u ­k a n s u a t u a l g o r i t m a y a n g e f i s i e n u n t u k s e l u r u b m a s a l a b i n i y a n g b e g i t q l u a s .

T e o r i d a s a r p r o g r a m n o n l i n i e r K u b n - T u c k e r m e m b e r i k a n i d e a n a ­l i t i s y a n g m e n a n d a i s u a t u s o l u s i . T e o r i i n i j u g a t i d a k e f i s i e n d a l a m m e n c a r l v e k t o r s o l u s i , n a m u n , m a m p u m e n y e b u t k a n k a r a k t e r s o l u s i a p a k a l i b e r s i f a t g l o b a l a t a u b a n y a l o k a l . J i k a d a r i p e r s a m a a n s i m u l t a n y a n g d i -b a s i l k a n s y a r a t p e r l u K u b n - T u c k e r s u b t d i s e l e s a i k a n , m a k a v e k t o r s o l u s d i l a c a k d e n g a n p r o s e d u r c o b a - c o b a d a r i k a s u s - p e r k a s u s . J i k a s y a r a c u k u p K u b n - T u c k e r t i d a k d i p e n u b i , m a k a s o l u s i y a n g d i p e r o l e b b e r s i f a o p t i m u m l o k a l , d a l a m k a s u s i n i o p t i m u m g l o b a l b i s a d i t e m u k a n d e n g a r c a r a m e m e r i k s a s e l u r u b k e m u n g k i n a n k o m b i n a s i s o l u s i .

2 4 3

Mulyono

Referensi

C h i a n g , A l p h a C , Fundamental Methods of Mathematical Economics, M c G r a w -H i U , S i n g a p o r e , 1 9 8 4 .

C o n v e r s e , A . D . , Optimization, Holt, R i n e h a r t a n d Winston, N e w Y o r k , 1 9 7 0 . K o o , Delia, Element of Optimization : with Aplications in Economics and Business,

S p r i n g e r V e r l a g , N e w Y o r k , 1 9 7 7 . K u h n , H . W . , d a n A . W . T u c k e r , Non Linear Programming, d a l a m J . N e y m a n , i Y o -

ceedings of the Second Berkeley Simposium on mathematical Statistics and Probability, U n i v e r s i t y o f C a l i f o r n i a Press , B e r k e l e y , C a l i f o r n i a , 1 9 5 1 , h a l . 4 8 1 - 4 9 2 .

L e e , S M . , L . J . M o o r e , d a n B . W . T a y l o r , Management Science, W m . C . B r o w , I o w a , 1 9 8 1 .

M u l y o n o , S r i , Operations Research, P u s a t A n t a r U n i v e r s i t a s - E k o n o m i - U n i v e r s i t a s I n d o n e s i a , J a k a r t a , 1 9 8 8 .

P h i l l i p s , D . T . , A . R a v i n d r a n , d a n J . S o l b e r g , Operations Research : Principles and Practice, J o h n W i l e y & S o n s , N e w Y o r k , 1 9 7 6 .

S w a m p , K . , P . K . G u p t a , d a n M a n M o h a n , Operations Research, S u l t a n C h a n d & S o n s , N e w D e l h i , 1 9 8 0 .

T a h a , H . A . , Operations Research : an introduction, M a c m i l l i a n P u b l i s h i n g , N e w Y o r k , 1 9 8 2 .

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