Confidence Regions for Linear Programs with Random Coefficients

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NOT FOR QUOTATION WITHOUT THE PERMISSION OF THE AUTHOR

CONFIDENCE REGIONS FOR

LTNEAR

PROGRAMS WITH RANDOM COEFFICIENTS

May 1 9 8 6 WP-06-20

Working Papers are interim r e p o r t s on work of t h e I n t e r n a t i o n a l I n s t i t u t e f o r Applied Systems Analysls a n d h a v e r e c e i v e d only limited review. Views or opinions e x p r e s s e d h e r e i n d o not n e c e s s a r i l y r e p r e s e n t t h o s e of t h e I n s t i t u t e o r of i t s National Member Organizations.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS 2361 Laxenburg. Austria

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FOREWORD

Statistical a p p r c a c h e s to stochastic optimization seem to b e suitable f r o m a p r a c t i c a l point of view since observed data a r e frequently t h e only information w e have on a stochastic optimization problem. In this p a p e r , t h e posslbllity of con- structing prediction regions in time s e r i e s is utilized f o r probabilistic conclusions on the behavior of t h e solution of t h e problem.

The r e s e a r c h w a s c a r r i e d out within the Adaptation and Optimization P r o j e c t of t h e System and Decision Sciences Program during t h e stay of t h e a u t h o r as a guest s c h o l a r in IIASA.

Alexander B. Kurzhanski Chairman System and Decision Sciences Program

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I would like t o thank Roger Wets who discussed with m e some problems from this work and gave me hints f o r further investigations.

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Assistant Professor TomdS Cipra from t h e Faculty of Mathematics and Physics.

Charles University, Prague works in the field of stochastic processes, stochastic programming and econometrics. H e wrote this paper during his stay a t IIASA in summer 1985.

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ABSTRACT

If random values in a l i n e a r p r o g r a m with random coefficients c a n b e p r e d i c t - ed using p r e v i o u s o b s e r v a t i o n s on them one c a n utilize t h e a p p r o p r i a t e p r e d i c t i o n r e g i o n a n d c o n s t r u c t a c o n f i d e n c e i n t e r v a l in which t h e optimal v a l u e of t h e o b j e c - tive function l i e s with a given p r o b a b i l i t y ( o r e v e n c o n s t r u c t a confidence r e g i o n for t h e optimal decision). I t i s a new s t a t i s t i c a l a p p r o a c h b a s e d o n p r o j e c t i o n of t h e o b s e r v e d d a t a i n t o t h e time p e r i o d of i n t e r e s t . The r e s u l t s are demonstrated by a numerical example.

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CONFIDENCE REGIONS FOR

LINEAR

PROGRAMS WITH RANDOM COEFFICIENTS

1. INTRODUCTION

Let u s h a v e a l i n e a r p r o g r a m

w h e r e A , b , c , z h a v e dimensions ( m , n ) , ( m , I ) , ( n , 1). ( n , I ) , r e s p e c t i v e l y . Let u s c o n s i d e r t h e usual situation when A a n d c h a v e known d e t e r m i n i s t i c values a n d b i s a random v e c t o r s u c h t h a t o b s e r v a t i o n s y

.

, y T of b in p r e v i o u s d i s c r e t e time p e r i o d s t

=

1 ,

.

, T are at o u r disposal. If o n e s u c c e e d s in c o n s t r u c t i n g a s t o c h a s t i c model g e n e r a t i n g t h e p r o c e s s [ y , o n e c a n a l s o usually c o n s t r u c t t h e p r e d i c t i o n

GT

⁺ _ofy T + f o r a time p e r i o d T

+

h in which t h e p r o g r a m (1.1) i s t o b e solved ( t h e most usual case i s h

=

1). Besides t h e point p r e d i c t i o n

GT

o n e c a n a l s o c o n s t r u c t a p r e d i c t i o n r e g i o n in which y T + l l e s with a p r e s c r i b e d p r o - bability ( t h e s e p r e d i c t i o n r e g i o n s which are quite analogical t o t h e confidence regions o r i n t e r v a l s in t h e o r y of s t a t i s t i c a l estimation are even p r e f e r r e d f o r p r a c - t i c a l p u r p o s e s in comparison with t h e point p r e d i c t i o n s ) .

T h e r e are v a r i o u s methods of p r e d i c t i n g . If w e confine o u r s e l v e s t o quantita- t i v e p r e d i c t i o n methods only ( s o t h a t t h e t e r m s "projection" o r "extrapolation"

should b e more s u i t a b l e t h a n "prediction" o r " f o r e c a s t " ) t h e n t h e most i m p o r t a n t a n d usual r e p r e s e n t a t i v e s are t h e p r e d i c t i o n method b a s e d on e c o n o m e t r i c modeling by means of systems of simultaneous equations (including t h e c l a s s i c a l r e g r e s - sion a p p r o a c h ) a n d t h e p r e d i c t i o n method in t h e framework of Box-Jenkins a p - p r o a c h . Both methods are d e s c r i b e d b r i e f l y in s e c t i o n 2 of t h i s p a p e r .

If w e h a v e c o n s t r u c t e d t h e a p p r o p r i a t e p r e d i c t i o n region f o r b concerning t h e time p e r i o d in which (1.1) i s t o b e solved o n e c a n make u s e of i t t o obtain a con- f i d e n c e i n t e r v a l with a p r e s c r i b e d confidence p r o b a b i l i t y f o r t h e optimal value of (1.1) o r e v e n (if p e r f o r m i n g m o r e d e t a i l e d analysis) t o o b t a i n a confidence region f o r t h e optimal decision in (1.1). I t i s obvious t h a t l i n e a r p a r a m e t r i c programming

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c a n b e obtained in some way outside t h e model, see discussion in [3, p . 1961 (e.g. in a simple r e g r e s s i o n situation i t c a n b e zit

=

t s o t h a t z ^ ( , ~ + ~

=

T

+

h ) . Then u n d e r g e n e r a l assumptions on t h e s t o c h a s t i c b e h a v i o r of t h e model in time T

+

h ( t h e r e must not b e a c h a n g e in t h e specification of t h e model in t h i s time) t h e optimal point prediction f o r t h e endogenous v a r i a b l e s in time T

+

h c a n b e con- s t r u c t e d as

with t h e (1

-

a ) 1 0 0 p e r c e n t p r e d i c t i o n region of t h e form

H e r e

i s t h e estimated c o v a r i a n c e matrix of t h e error

- cT

^{+ h}of t h e p r e d i c t i o n and F m V T -k - m + l ( a ) i s t h e tabuiated c r i t i c a l value of F i s h e r ' s distribution with t h e a p p r o p r i a t e d e g r e e s of freedom and t h e level of sienificance a (e.g. (2.5) is t h e 95% prediction region f o r a

=

0.05). S o called Hotelling's s t a t i s t i c h a s been used t o d e r i v e (2.5).

REMARK 2 Various multivariate t r e n d s c a n b e modeled by means of (2.1). E.g.

t h e multivariate polynomial t r e n d is modeled as

where pi ( t ) is a polynomial of a n o r d e r pi (it means t h a t t h e p r e d e t e r m i n e d vari- a b l e s a r e chosen as powers of time t ) .

REMARK 3 Hymans [4] d e r l v e d ( 1

-

a ) 1 0 0 p e r c e n t joint p r e d i c t i o n i n t e r v a l s f o r p a r t i c u l a r components of y T + in t h e i o r m

where sit is t h e i t h diagonal element of t h e matrix (Y'Y

-

~ U ( ' Y ) / (T

-

k ) and

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positive-definite m a t r i x a n d k ( a ) i s a known c o n s t a n t . Let us d e n o t e t h i s region a s P ( a )

=

[ b t R m : ( b

-

b ^ ) ' ~ ( b

- 5 )

⁵k ( a ) {

.

(2.16)

3. PROBLEM

OF

SOLVABILITY

Let us d e n o t e t h e r e g i o n of solvability of t h e p r o g r a m (1.1) as

S

=

[ b ^ERm : (1.1) h a s a n optimal solution

I

(3.1)

(i.e. t h e p r o g r a m (1.1) i s f e a s i b l e a n d bounded f o r a l l b ^ES ) a n d assume t h a t S i s nonempty. Then in o u r c o n t e x t t h e problem of solvability c o n s i s t s in t h e investiga- tion of t h e inclusion P ( a ) c S.

Wets

[ I l l

d e a l s with a g e n e r a l problem of t h i s t y p e when h e i n v e s t i g a t e s feasi- bility of s t o c h a s t i c p r o g r a m s with f i x e d r e c o u r s e . Using t h e o r y of p o l a r m a t r i c e s a n d c o n e o r a e r i n g h e c a n t r e a t cases with v e r y g e n e r a l r e g i o n s P ( a ) . In o u r case w e make u s e of t h e s p e c i a l elipsoid s h a p e of t h e p r e d i c t i o n r e g i o n P ( a ) a n d p r o c e e d in t h e following way.

The solvability r e g i o n S i s a convex p o l y h e d r a l c o n e with t h e v e r t e x in t h e origin ( s e e e . g . [9]. [lo]), i.e.

The e x p l i c i t numerical form of t h i s c o n e (i.e. t h e v e c t o r s h l ,

. . .

^,h N ) c a n b e found by means of v a r i o u s algorithmic p r o c e d u r e s (e.g. 121, [7, p. 2761).

In o r d e r t o simplify t h e solution of o u r problem l e t us t r a n s f o r m t h e coordi- n a t e system in Rm so t h a t t h e elipsoid (2.16) t r a n s f e r s t o a s p h e r e in R m . The positive-definite m a t r i x V from (2.16) can be decomposed as

v =

C'C , (3.3)

w h e r e C i s a n u p p e r t r i a n g u l a r matrix with positive elements on t h e main diagonal ( s o called Cholesky decomposition). If w e d e f i n e t h e transformation of t h e s p a c e Rm as

z -+z*

=

CZ. z E Rm _(3.4)

( t h e a s t e r i s k will always d e n o t e t h e t r a n s f o r m e d value) t h e n t h e elipsoid P ( a ) will b e obviously t r a n s f o r m e d t o t h e f o r m

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4. CONSTRUCTION OF CONFUIENCE REGIONS Let us d e n o t e

f o r b E S . The function q i s convex, continuous a n d piecewise l i n e a r o n S . More explicitly, t h e r e e x i s t v e c t o r s g l.

. . .

, g, c R m such t h a t

According t o t h e b a s i s decomposition t h e o r e m ( s e e [9]) t h e definition r e g i o n S of t h e function q c a n b e decomposed to a finite number of convex p o l y h e d r a l c o n e s St with t h e v e r t i c e s in t h e origin 0 s u c h t h a t t h e i n t e r i o r s of Si are mutually disjunct a n d q i s l i n e a r on e a c h of St ( t h e s e r e g i o n s c o r r e s p o n d t o p a r t i c u l a r b a s e s Bi in A s u c h t h a t

c;'

Bi-l A S c ' , w h e r e i s t h e s u b v e c t o r of c c o r r e s p o n d i n g t o B 1 ) The e x p l i c i t f o r m of t h e c o n e s Si a n d t h e function q c a n b e a l s o found by means of t h e mentioned algorithms [2] or [7].

Now l e t u s t r y t o d e t e r m i n e t h e maximal a n d minimal values of q o v e r P ( a ) . In o t h e r words, w e s h a l l c o n s t r u c t t h e ( 1

-

a ) 1 0 0 p e r c e n t confidence i n t e r v a l f o r t h e optimal value of t h e o b j e c t i v e function in (1.1).

THEOREM 2 L e t P ( a ) C S. T h e n i t h o l d s

G-T-

r n a x I q ( b ) : b € P ( a ) l = max [ 5 j ( g * + & g j ) l , j = 1 ,

. . . .

^r llgj

II

a n d

m i n f q ( b ) : b € P ( a ) j 2 max f ~ j ( c * - = ~ ~ ) i , j = 1 , . . . , T ll9j

I1

w h e r e

a n d f o r g j

= gj =

0 w e p u t

PROOF W e can w r i t e

m a x l q ( b ) : b € P ( a ) { = max

I

max j g j b l j 6 € P ( a ) j = I ,

. . .

, T

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REMARK 7 This work i s not t h e f i r s t one dealing with c o n f i d e n c e r e g i o n s in l i n e a r p r o g r a m s with random coefficients. E.g., r e s u l t s h a v e b e e n o b t a i n e d by means of p r o j e c t i o n of r e c t a n g u l a r s in which t h e values (A. R , c ) lie with a given probability ( s e e [ 6 , s e c t i o n

13.11

o r [a]). These r e c t a n g u l a r s are defined by means of t h e mean values a n d s t a n d a r d deviations of t h e random components of (A, b , c ) a n d d o not make use of t h e c o r r e l a t i o n s t r u c t u r e ( r e l a t i o n s among p a r t i c u l a r r a n - dom components) how i t i s t h e c a s e when p r o j e c t i n g elipsoids.

If w e c a r r y o u t a m o r e d e t a i l e d analysis of (1.1) as t h e problem of p a r a m e t r i c programming with t h e ( v e c t o r ) p a r a m e t e r b (i.e. if o n e finds explicitly t h e decomposition of t h e solvability region S to t h e cones S f ) t h e n i t i s even possible t o con- s t r u c t a confidence r e g i o n f o r t h e optimal decision in (1.1). W e s h a l l show such c o n s t r u c t i o n including t h e application of Theorem 1 a n d CorolIary of Theorem 2 in t h e following example which i s simple enough t o d e m o n s t r a t e d e a r l y t h e p r e v i o u s t h e o r y . The a p p l i c a t i o n f o r r e a l examples assumes t h e exploitation of s o f t w a r e from s t a t i s t i c s a n d l i n e a r p a r a m e t r i c programming.

EXAMPLE The a u t h o r s of [7] investigated t h e following problem

In t h i s case i t i s

a n d

~ ( b )

=

max 10,

- -

1 b1,

-

b,.

- -

³₄b l - - b 3 ³4 ,

f o r b ^tS. Table 1 contains t h e d e s c r i p t i o n of a l l c o n e s Sf from t h e decomposition of S including t h e c o r r e s p o n d i n g f o r m s of rp a n d t h e optimal b a s e s Bf

.

Let t h e ( 1

-

^a)1 0 0 p e r c e n t p r e d i c t i o n r e g i o n (2.16) h a v e t h e following f o r m

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According t o (4.12) t h e vectors h i from (3.2) are

and according t o (4.13) t h e vectors g, from (4.2) are

The left-hand-sides o f t h e inequalities (3.8) ( o r equivalently o f (3.9) or (3.10)) are

235.03, 512.86, 125.76, 1010.72, 599.61

.

^(4.18)

Since each o f these values is non-negative t h e elipsoid (4.14) is t h e subset o f t h e solvability region (4.12). The corresponding confidence region f o r t h e optimal value o f t h e objective function is according t o (4.9) or (4.10)

If t h e vector

c

is replaced by t h e vector

then t h e problem stays solvable and t h e confidence interval is

so that t h e objective function has t h e optimal value 0 with t h e probability at least 1

-

^a.

The cones Si can be written formally as

where Hi are (3.3) matrices, e.g. it is

Then it is not d i f f i c u l t t o v e r i f y that t h e (1

-

a)100 per cent confidence region f o r t h e optimal decision in (4.11) can be taken as

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5. APPLICATION OF BUNCHING METHOD

The method of bunching [I21 and especially i t s t r i c k l i n g down modification in combination with Schur-complement b a s e s u p d a t e s

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i s t h e e f f i c i e n t tool f o r solving l i n e a r p r o g r a m s with v a r i a b l e right-hand-sides ( a bunch i s s u c h s u b s e t of a given set of rieht-hand-side v e c t o r s which c o r r e s p o n d s t o t h e same optimal b a s i s of t h e p r o g r a m ) .

In o u r c a s e t h e bunching method will enable t o solve in a n e f f i c i e n t way a l o t of problems of t h e t y p e

( t h e points z k are c h o s e n f r o m t h e elipsoid P ( a ) ) without performing explicitly t h e decomposition of t h e solvability r e g i o n S to t h e c o n e s St.

A s t h e c h o i c e of t h e points z k i s c o n c e r n e d o n e c a n u s e v a r i o u s s t r a t e g i e s . E.g., i t is possible t o c h o o s e t h e points z k randomly from t h e s u r f a c e of t h e elipsoid P ( a ) . If w e t r a n s f o r m t h e c o o r d i n a t e s a c c o r d i n g t o (3.4) t h e n w e c a n g e n e r a t e t h e s e points uniformly from t h e s u r f a c e of t h e s p h e r e (3.5) taking

z;

= 6; +

c o s c o s 192 c o s 193

. . .

^COSI9, -1

z;

= g; +

+sin c o s 1 9 ~ c o s 1 9 ~ .

.

. c o s 19, _ I

z ;

= gi +

+sin 1 9 ~ c o s f14 c o s *5

. . .

^COS

*, -,

- -.

2;-1 -bm -1

+

m s i n 19, - 2 c o s 1 9 ~

z,

=&A +

,

w h e r e 0 S 19~ 5. 2 n ,

-

n / 2 5. 192 5. n / 2 ,

. . .

,

-

n / 2 S 19, S n / 2 are independent random v a r i a b l e s with uniform d i s t r i b u t i o n s o n t h e i r r a n g e s .

The t r i c k l i n g down p r o c e d u r e c a n b e s t a r t e d i n t h e point

6*

( t h e c e n t e r of t h e s p h e r e P ( a ) ) . Let B;~, b e t h e optimal b a s i s f o r t h i s point b; (it holds obviously BT1)

=

CB(l). w h e r e B(l) i s t h e optimal b a s i s f o r t h e point

b^

b e f o r e t h e t r a n s f o r - mation (3.4) s i n c e t h e problem (1.1) c a n b e written equivalently as [min c'z : CAz = Cb, z 2 01). Let z l *

=

(z:*,

. .

^., zh*)' b e t h e f i r s t point gen- e r a t e d a c c o r d i n g t o (5.2). By using trickling down p r o c e d u r e (i.e. t h e p r o p e r se-

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REFERENCES

1 Box, G.E.P., Jenkins, G.M. Time S e r i e s Analysis, F o r e c a s t i n g and Control. Hol- d e n Day, S a n F r a n c i s c o 1970.

2 Gal. T., Nedoma, J. Multiparametric l i n e a r programming. Management S c i e n c e 1 8 , 1972, 406-422.

3 G r a n g e r , C.W.J., Newbold, P . F o r e c a s t i n g Economic Time S e r i e s . Academic P r e s s , N e w York 1977.

4 Hymans, S.H. Simultaneous confidence i n t e r v a l s in econometric f o r e c a s t i n g . Econometrica 36. 1968. 18-30.

5 Johnston, J. Econometric Methods. McGraw-Hill. N e w York 1972.

6 Judin. D.B. Mathematical Methods of Control u n d e r Incomplete Information (Problems a n d Methods of S t o c h a s t i c Programming). Sov. r a d i o . Moskva 1974 (in Russian).

7 ~ozi:ka, F., Guddat. J . , Hollatz, H.. Bank. B. Theorie d e r l i n e a r e n p a r a m e t r i s c h e n Optimierung. Akademie-Verlag. Berlin 1974.

8 Theodorescu. R. Random p r o g r a m s . Math. O p e r a t i o n s f o r s c h . S t a t i s t . 3. 1972.

19-47.

9 Walkup, D., Wets, R. Lifting p r o j e c t i o n s of convex p o l y e d r a . P a c i f i c J. Mathem.

2 8 , 1969, 465-475.

1 0 Wets, R. Programming u n d e r uncertainty: t h e equivalent convex p r o g r a m . J.

SIAM Appl. Math. 1 4 , 1966, 89-105.

11 Wets, R. S t o c h a s t i c p r o g r a m s with f i x e d r e c o u r s e : t h e equivalent deterministic p r o g r a m . SIAM Review 1 6 , 1974, 309-339.

1 2 Wets, R. S t o c h a s t i c programming: solution techniques and approximation schemes. In: Mathematical Programming: The S t a t e of t h e A r t (A. Bachem, M . G r o t s c h e d , B. K o r t e eds.). S p r i n g e r , Berlin 1983, 566-603.

13 Wets, R. L a r g e s c a l e l i n e a r programming techniques in s t o c h a s t i c programming.

IIASA Working P a p e r . WP-84-90. Laxenburg, Austria 1984.

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may b e convenient tool f o r t h i s p u r p o s e . In t h e initial s t a g e of s u c h a n a l y s i s b a s e d on p r e d i c t i o n r e g i o n s o n e should investigate w h e t h e r t h e p r e d i c t i o n r e g i o n f o r b with a p r e s c r i b e d confidence probability (e.g. 95%) i s t h e s u b s e t of s u c h r e g i o n f o r 6 in which t h e problem (1.1) i s solvable (so called solvability region). This problem i s discussed in s e c t i o n 3 while t h e c o n s t r u c t i o n of t h e confidence r e g i o n s i s d e s c r i b e d in s e c t i o n 4 including a simple example which d e m o n s t r a t e s it. A possible application o f s o c a l l e d bunching method [12], [13] f o r t h e c o n s i d e r e d situation i s suggested in s e c t i o n 5.

REMARK 1 Although t h e simplest case with random 6 only i s c o n s i d e r e d in t h i s work t h e method could b e e x t e n d e d in p r i n c i p a l t o more g e n e r a l situations.

The case with random c only i s equivalent to t h e case discussed h e r e due t o duall- t y .

2. CONSTRUCTION OF PREDICTION REGIONS IN PRACTICE

In t h i s s e c t i o n t h e both mentioned methods of q u a n t i t a t i v e p r e d i c t i o n are re- minded:

a ) P r e d i c t i o n b a s e d on e c o n o m e t r i c modeling ( s e e e.g. [5]) i s a g e n e r a l method which includes as a s p e c i a l case e.g. t h e p r e d i c t i o n b a s e d on t h e c l a s s i c a l r e g r e s - sion analysis. If using t h l s method w e must e.g. h a v e at o u r disposal t h e estimated model of simultaneous equations in t h e r e d u c e d form

H e r e y t

=

( y l t . .

.

. , y m t ) ' i s a v e c t o r of endogenous v a r i a b l e s in time t which is explained by a v e c t o r of p r e d e t e r m i n e d v a r i a b l e s zt

=

(zit

. . . .

z k t ) ' in time t ( t h e o b j e c t of t h e p r e d i c t i o n are t h e endogenous v a r i a b l e s ) ,

n

^{i s a}^{( m ,}^{k )}matrix of p a r a m e t e r s a n d v t

=

( v l t ,

. .

. , v m t ) ' i s a v e c t o r of d i s t u r b a n c e s in time t . One assumes t h a t E v t

=

0 , E v t v i

= zvv

(a positive-definite matrix) a n d Ev, v i

=

0 f o r s

+

^{1 .}The model c a n b e summarized f o r a l l t as

w h e r e Y

=

( y l ,

. . . .

y T ) ' , X

=

( z l , . . . , z T ) ' and V

=

( v l ,

.

. .

vT)'. The c l a s s i c a l OLS (Ordinary Least S q u a r e s ) e s t i m a t o r of

Il

h a s t h e f o r m

L e t

2,

⁺^, b e a v e c t o r of p r e d i c t e d p r e d e t e r m i n e d v a r i a b l e s f o r time T

+

^hwhich

(15)

b ) P r e d i c t i o n in t h e framework of Box-Jenkins a p p r o a c h ( s e e [I], 131) i s ex- ploited by many s t a t i s t i c i a n s and econometricians as a v e r y f l e x i b l e a n d f r u i t f u l p r e d i c t i o n mothod. Similarly as in t h e econometric modeling one must c o n s t r u c t a n a p p r o p r i a t e model a t f i r s t . Box-Jenkins methodology utilizes s o called ARMA (p, q ) models ( o r t h e i r v a r i o u s modifications) of t h e f o r m

w h e r e y t i s t h e modeled m-dimensional p r o c e s s . A l ,

.

^{. .},

ilp

a n d B 1 . .

. .

^,^Bqare ( m , m ) m a t r i c e s of p a r a m e t e r s and i s t h e m-dimensional white noise, i.e.

E z t

=

0 , E z t E ;

= C

( a positive-definite matrix), E E ~ E ; = O f o r s # t . Then u n d e r g e n e r a l conditions t h e optimal point p r e d i c t i o n f o r time T

+

h c a n b e writ- t e n a s

w h e r e t h e ( m , m ) m a t r i c e s Ci fulfill t h e following power s e r i e s equation

The c o r r e s p o n d i n g ( 1

-

a ) 1 0 0 p e r c e n t prediction r e g i o n h a s t h e f o r m

where

i s t h e c o v a r i a n c e matrix of t h e e r r o r y T +

- ^jT

⁺ of t h e p r e d i c t i o n a n d X:(a) i s t h e t a b u l a t e d c r i t i c a l value of chi-squared d i s t r i b u t i o n with m d e g r e e s of freedom and t h e level of significance a.

The following conclusion c a n b e drawn f r o m t h e p r e v i o u s t e x t . In both prediction methods (and a l s o in o t h e r l e s s important o n e s ) t h e a p p r o p r i a t e ( 1

-

a ) 1 0 0 p e r c e n t p r e d i c t i o n r e g i o n h a s t h e geometric form of a n elipsoid. This elipsoid c a n be written generally f o r t h e p r o g r a m (1.1) as

(b

-

6 ) ' ~ ( b

-

6 ) h k ( a ) , (2.15)

w h e r e

6

i s a known v e c t o r ( t h e c e n t e r of t h e elipsoid), V i s a known

(16)

which is a m-dimensional s p h e r e with t h e c e n t e r

bi

and t h e r a d i u s

m.

^The

transformed solvability r e g i o n S h a s t h e f o r m

where

If t h e i n t e r i o r of t h e elipsoid P ( a ) (denoted as int P ( a ) ) contains t h e z e r o v e c t o r 0 ( o r equivalently if t h e z e r o v e c t o r 0 l i e s in i n t P . ( a ) ) t h e n t h e problem discussed in t h i s s e c t i o n h a s t h e following simple solution.

LEMMA 1 Let 0 E i n t P ( a ) . T h e n t h e i n c l u s i o n P ( a ) c S i s t r u e

tP

a n d o n l y

i p s ^{= R m .}

PROOF Lemma is obvious s i n c e S i s a cone with t h e v e r t e x in 0 and P ( a ) is a n elipsoid.

General solution of t h e considered problem i s given in t h e following theorem.

THEOREM 1 The i n c l u s i o n P ( a ) c S i s t r u e u a n d o n l y q i t h o l d s

w h e r e

11 (1

i s t h e u ~ u a l E u c l e i d i a n norm in

R m

PROOF P ( a ) C S is equivalent t o P. ( a ) c S and t h i s l a s t inclusion holds iff t h e s p h e r e P ( a ) with t h e c e n t e r b; and t h e r a d i u s l i e s in all half-spaces f b * 6

R m

: &b*

a

_01,_i

=

1.

. . . ,

N . This l a s t condition i s obviously equivalent t o (3.8).

REMARK 4 The inequalities (3.8) c a n b e written in t h e equivalent form

hi b^ - ^dk

^{( a )}

hi v-lh, ^a

^{0 ,}i

=

1.

. . .

, N

.

(17)

-

max

l

max 1 9 ; ' b * i { = max

I

max I G j b * I I 6 - E P ' ( ~ ) j = I , .

. .

^{, r} ^{j = i ,}. . . , + b a € P . ( a )

Now t h e r e l a t i o n (4.3) i s p r o v e d s i n c e i t obviously holds

G -

max j 3 b * 1 = i j ( b ; + - - U g j )

b' EP' ( a ) _Ilij

II

(we maximize t h e l i n e a r function $ b * o v e r t h e s p h e r e with t h e c e n t e r b; a n d t h e r a d i u s

w,

t h e maximal value i s a c h i e v e d in t h e p o i n t w h e r e t h e v e c t o r d i r e c t - e d f r o m b; as t h e g r a d i e n t

ej

of t h e function

Fj

b* c r o s s e s t h e s u r f a c e of t h e s p h e r e , i.e. in t h e point b;

+

( m / l l G j l l ) i j ) .

A s t h e r e l a t i o n (4.4) i s c o n c e r n e d w e h a v e

m i n l r p ( b ) : b ~ P ( a ) l = min

I

max j b * max

I

min c j b * { l , b ' € P ( a ) j = l ,

. . . .

r j = l , .

. .

, r b ' € P m ( a )

The l a s t inequality holds s i n c e i t i s

max

j

b

>

max f min

f i j

b*

( i

j = l , . . . , r j = I , .

. .

, r 6 . C P s ( m )

f o r e a c h 6- r P ( a ) s o t h a t w e c a n r e p l a c e t h e left-hand-side of (4.7) by i t s minimal v a l u e o v e r b* G F"@ ( a ) . The proof i s finished b e c a u s e o n e c a n d e r i v e in t h e same way as (4.6) t h a t

G -

min

f6j

b*

I = 5;

(6*

-

- - U g j ) .

6' E P m ( a )

lli, II

COROLLARY Let P ( a ) ^CS . T h e n t h e i n t e r v a l of t h e f o r m

[ max

1;;

(c*

- 4Z-T- 4Z-T-

j = I , .

. .

^{, 7} g

IIC, 11

j j =I,. max . . a r

i ~ j

(c*

+

A g j ) 1 ]

IIQ; II

^(4.9)

i s t h e c o n f i d e n c e i n t e r v a l f o r t h e o p t i m a l v a l u e of t h e o b j e c t i v e f u n c t i o n i n (1.1) w i t h t h e c o m d e n c e p r o b a b i l i t y a t l e a s t 1

-

a.

REMARK 5 The i n t e r v a l (4.9) c a n b e w r i t t e n again in t h e equivalent f o r m

REMARK 6 S i n c e t h e f u n c t i o n rp(b) a t t a i n s t h e v a l u e

+ -

o u t s i d e t h e set S ( s e e e . g . [ I l l ) o n e c a n omit t h e assumption P ( a ) c S in t h e p r e v i o u s C o r o l l a r y and formulate i t in s u c h a way t h a t t h e optimal o b j e c t i v e v a l u e l i e s in t h e i n t e r v a l (4.9) o r i s e q u a l t o

+

-with t h e p r o b a b i l i t y at l e a s t 1

-

a.

(18)

Table 1 The analysis of the linear parametric problem (4.11)

~ ( b )

I (numbers of columns of A ) 0

(19)

where

Zi

d e n o t e s such s u b v e c t o r of t h e v e c t o r z E R' which c o r r e s p o n d s to t h e basis B i . E.g. if w e u s e t h e prediction region (4.14) t h e n t h e s e t with t h e index i

=

1 in t h e union (4.23) h a s t h e form

and t h e s e t with t h e index i

=

2 h a s t h e form

I t is obvious t h a t t h e d e s c r i b e d method of t h e construction of t h e confidence regions exploits substantially t h e p r o c e d u r e s of l i n e a r p a r a m e t r i c programming s o t h a t i t s p r a c t i c a l applicability i s Limited if t h e dimensions of t h e problem a r e l a r g e (one must a l s o k e e p in mind t h e f a c t t h a t f o r increasing m t h e construction of t h e prediction elipsoid becomes more and more difficult). The method seems t o b e suit- a b l e in such c a s e s when o n e solves a lot of problems (1.1) with t h e same values A and c f o r v a r i o u s t r a j e c t o r i e s ly,

1

so t h a t t h e tedious and expensive p r o c e d u r e s of p a r a m e t r i c programming will bring a n e f f e c t (such situations may b e usual in r o u t i n e p r a c t i c a l problems). On t h e o t h e r hand, t h e method cannot b e recommend- e d f o r a single use in l a r g e s c a l e problems f o r which more e f f e c t i v e p r o c e d u r e s should b e suggested. One of s u c h potential p r o c e d u r e s based on t h e bunching method i s s k e t c h e d in t h e following s e c t i o n .

(20)

q u e n c e of dual simplex s t e p s exploiting Schur-complement u p d a t e s ) o n e will find t h e c o r r e s p o n d i n g s e q u e n c e B ( l ) *

.

^.

^.^{, B} of t h e b a s e s which i s ended by t h e basis B(,) optimal f o r t h e point z l * . Let us c a l c u l a t e t h e values *

where

(C"(,) i s t h e s u b v e c t o r of c c o r r e s p o n d i n g t o t h e b a s i s B ( , ) ) . * The same p r o c e d u r e

will b e p e r f o r m e d with t h e second g e n e r a t e d point z2* producing t h e values 1' a n d u 2 , etc. lf p r o c e e d i n g in t h i s way w e o b t a i n a t r e e r o o t e d at t h e basis B ( l ) ( s e e * [13]) t h e p a t h s of which are ended by t h e couples (1'. u l ) , (12, u 2 ) , . . . , ( l K , u K ) . The ( 1

-

a ) 100 p e r c e n t p r e d i c t i o n i n t e r v a l f o r t h e optimal o b j e c t i v e value c a n b e t h e n approximated by t h e i n t e r v a l

[

^max L ( ~ ) , max u ( ~ ) ]

k = l , . . . . K k = l ,

. . . .

K

The stopping r u l e by means of which t h e number K i s found can b e p r e s c r i b e d in such a way t h a t t h e l a s t L couples ( l K -L

".

^UK-L^{t i}⁾^,^{. .}

^.

, ( l K , u K ) will s a t i s f y

max I'

-

max l k ^{< E}, k = l , . . . .K k = 1 ,

. .

. . L -K

where a n i n t e g e r L a n d a sufficiently small c

>

0 are chosen a p r i o r i .

More complicated s t r a t e g i e s can b e suggested but t h e p r e v i o u s one seems t o b e s u i t a b l e in s p i t e of i t s simplicity. The i n a c c u r a c i e s which can o r i g i n a t e when using t h e g e n e r a t i n g formulas (5.2) d o not r e d u c e t h e e f f i c i e n c y of t h e method s i n c e t h e points z l ' . z 2 '

....

are used only t o determine t h e c o r r e s p o n d i n g optimal b a s e s a n d t h e s e b a s e s d o n o t usually v a r y in t h e neighborhoods of p a r t i c u l a r r i g h t - hand-side v e c t o r s . Moreover, when t h e components of t h e v e c t o r a r e l a r g e ( a s i t i s f r e q u e n t in p r a c t i c e ) t h e n usually only small number of t h e c o n e s St from t h e decomposition of S h a v e nonempty i n t e r s e c t i o n s with t h e elipsoid P ( a ) s o t h a t t h e mentioned t r e e from t h e t r i c k l i n g down p r o c e d u r e h a s small number of p a t h s which r e d u c e s t h e computing e f f o r t .