Computing Bounds for the Solution of the Stochastic Optimization Problem with Incomplete Information on Distribution of Random Parameters

NOT FOR QUOTATION WITHOUT THE PERMISSION OF THE AUTHOR

COMPUTING BOUNDS FOR THE SOLUTION OF THE STOCHASTIC OPTIMIZATION PROBLElW WITH

INCOMPLFI'E INFORMATION ON D I r n U T I O N OF RANDOM PARAWWERS

A. G a i v o r o n s k i

November 1986 WP-86-72

Working P a p e r s are interim r e p o r t s on work of t h e International Institute f o r Applied Systems Analysis and have received only limited review. Views or opinions e x p r e s s e d h e r e i n d o not necessarily r e p r e s e n t those of t h e Institute or of i t s National Member Organizations.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS A-2361 Laxenburg, Austria

FOREWORD

The p a p e r d e a l s with t h e solution of a s t o c h a s t i c optimization problem u n d e r incomplete information. I t i s assumed t h a t t h e d i s t r i b u t i o n of p r o b a b i l i s t i c param- eters i s unknown a n d t h e only available information comes with o b s e r v a t i o n s . In ad- dition t h e set to which t h e p r o b a b i l i s t i c p a r a m e t e r s belong i s a l s o known. Numeri- c a l techniques are p r o p o s e d which allow t o compute u p p e r a n d lower bounds f o r t h e solution of t h e s t o c h a s t i c optimization problem u n d e r t h e s e assumptions. These bounds are updated s u c c e s s i v e l y a f t e r t h e a r r i v a l of new o b s e r v a t i o n s . The r e s e a r c h r e p o r t e d in t h i s p a p e r was p e r f o r m e d in t h e Adaptation a n d Optimization P r o j e c t of t h e Systems a n d Decision S c i e n c e s P r o g r a m .

Alexander B. Kurzhanski Chairman System a n d Decision S c i e n c e s P r o g r a m

CONTENTS

1 Introduction

2 Problem Formulation

3 The Solution of t h e Inner Problem 4 Incorporating Additinal Information

5 Numerical Techniques f o r Finding t h e Upper Bound f o r t h e Solution of Optimization Problems Acknowledgment

R e f e r e n c e s

COMPUTING BOUNDS

FOR THE

SOLUTION OF

THE

STOCHASTIC 0 ~ 1 7 A T I O N PROBLEM WITH INCOMPLETE 1NM)RbIATION ON DISTRIBUTION

OF RANDOM P A R A h E E R S A. Gaivoronski

During r e c e n t y e a r s c o n s i d e r a b l e e f f o r t w a s made t o develop numerical t e c h - niques f o r a solution of s t o c h a s t i c programming problems. These problems c a n b e formulated q u i t e g e n e r a l l y as follows:

minimize

s u b j e c t t o c o n s t r a i n t s

w h e r e x E Rn is a v e c t o r of decision v a r i a b l e s , o E I) C Rm i s a v e c t o r of random v a r i a b l e s which belong t o some p r o b a b i l i t y s p a c e , H

-

some p r o b a b i l i t y m e a s u r e a n d E , d e n o t e s e x p e c t a t i o n with r e s p e c t t o o. Algorithms f o r solving t h i s problem as well as v a r i o u s a p p l i c a t i o n s t o o p e r a t i o n s r e s e a r c h a n d systems analysis c a n b e found in Ermoliev [7], [8], Kall [13], P r e k o p a [21], Wets [23] [24] w h e r e o n e c a n find f u r t h e r r e f e r e n c e s . I t w a s assumed usually t h a t p r o b a b i l i t y d i s t r i b u t i o n H i s known. This d i s t r i b u t i o n is o b t a i n e d from t h e set of o b s e r v a t i o n s tol,

. . .

, o,,

...I

of random v e c t o r o t h r o u g h application of a p p r o p r i a t e s t a t i s t i c a l techniques.

However, information contained in o b s e r v a t i o n s i s o f t e n insufficient t o identi- fy unique d i s t r i b u t i o n H. Then i t i s o f t e n possible t o define some set G t o which dis- t r i b u t i o n belong and a p p l y minimax a p p r o a c h (worst case analysis) to t h e problem (1). This a p p r o a c h war; s t u d i e d by DupaEov6 [4] [5], Ermoliev [6], Golodnikov [12], Ermoliev, Gaivoronski and Nedeva [9], Birge a n d Wets [I], Gaivoronski

[lo]

f o r t h e c a s e when t h e set G of admissible d i s t r i b u t i o n s is defined by c o n s t r a i n t s of moment t y p e a n d by Gaivoronski [I l] for- t h e c a s e when t h e m e a s u r e H i s contained between

some u p p e r and lower measures. In t h i s case however some unspecified s t a t i s t i c a l methods should b e used to obtain t h e s e bounding measures or moment constraints.

The resulting set G a s well as t h e solution of t h e problem (1) and i t s a c c u r a c y will heavily depend on t h e s e techniques.

This p a p e r p r e s e n t s t h e attempt t o solve t h e problem (1) s t a r t i n g d i r e c t l y from t h e finite set of observations [al,

. . .

, o,

1.

In this case i t is possible only to identify some u p p e r and lower bounds on t h e optimal value of t h e objective func- tion and point z 1 which yields value within t h e s e bounds in s o m e probabilistic sense. These bounds should possess t h e following p r o p e r t i e s :

-

b e valid f o r all s, nonasymptotical;

-

b e successive i.e. permit e a s y updates when new observations a r r i v e , this will allow t o compute them in r e a l time;

-

be independent from any p a r t i c u l a r p a r a m e t r i c family of distributions such as normal, lognormal etc.

This a p p r o a c h , if successfully implemented, would allow t o avoid such ques- tions as "where d o you g e t y o u r distribution"? which are often confronted by sys- t e m s analyst who applies model (1) to r e a l systems. I t could also p r e s e n t a "middle road" between a purely s t o c h a s t i c a p p r o a c h , when unknown p a r a m e t e r s o are con- s i d e r e d to possess known probability distributions

H

and deterministic a p p r o a c h when i t is assumed t h a t all what is known about o is t h a t i t belongs t o s o m e set fl ( f o r m o r e details see Kurzhanski [17]).

This p a p e r p r e s e n t s only f i r s t s t e p s towards this direction. In t h e section 2 t h e p r e c i s e formulation of t h e problem is developed using c e r t a i n techniques of e x t r a c t i n g information a b o u t distribution from observations. This leads to minimax problems with a n i n n e r problem which allows a n explicit solution defined in t h e section 3. Finally in t h e s e c t i o n 4 numerical techniques for computing successive u p p e r bounds is described. Results of s o m e numerical experiments are p r e s e n t e d in t h e c o u r s e of exposition.

I t should b e noted t h a t s o m e techniques f o r computing bounds f o r solution of t h e problem (1) in d i f f e r e n t c o n t e x t s w a s described in Birge and W e t s [I], Cipra [Z], Kall [14], Kankova [15].

2. PROBLEM FORMULATION

In,formal s t a t e m e n t . Solve t h e problem

when all t h a t is known about random p a r a m e t e r s o is t h e s e t of observations to1,.

- . .

^us!.

We shall make two basic assumptions:

1 . Distribution H of random p a r a m e t e r s o e x i s t s , but unknown

2. Observations ol,

. . .

, o, a r e mutually independent and form t h e sample from this distribution H.

In o r d e r t o solve t h e problem (2) i t is necessary t o c l a r i f y what is considered a s solution and l e a r n how t o e x t r a c t information about distribution H from obser- vations wi

.

Let us assume t h a t o belongs t o some s e t

n ^c

Rm with Bore1 field B; probabili- ty measure H i s defined on this field, thus we have a probability s p a c e ( n , B, H).

- - -

For e a c h fixed s l e t us consider t h e sample probability s p a c e ( n , B, P) which is a

- - -

Cartesian product of s s p a c e s ( n , B, H). The s p a c e ( n , B, P) is t h e smallest s p a c e which contain all ( n S ,

w ,

p ) . In what follows t h e "convergence with probability 1 " will mean t h e "convergence with probability 1 in t h e s p a c e ( n S ,

p ,

p ) " . With t h e s e t of observations tol,

. . .

, o, j t h e s e t of distribution G, will b e associated in t h e following way.

Let us fix t h e confidence level a : O

<

^a

<

1 . We shall consider events with probability

p

less t h a n a "improbable" events and discard them. Let us consider a r b i t r a r y set A

c

_B. Among s observations lol,

. . .

, o, j t h e r e a r e iA observa- tions which belong t o set A , 0 S iA S s. The random variable iA is distributed bi- nominally and i t s values c a n be used t o estimate H(A) (Mainland [19]). To d o this let us consider t h e following functions

S s!

9 ( s l k , 2 )

= C

^{z i (I}

-

^{z), -i}

= k i ! ( s

-

i ) !

o b s e r v e t h a t

The function

#

( s , k , z ) i s a monotonically increasing function of z on t h e interval [ O , 11,

#

( s , k , 0 )

=

0 ,

#

( s , k , 1 )

=

1, k

#

0 . T h e r e f o r e t h e solution of equation

# ( s , k , z )

=

c e x i s t f o r any 0 5 c 5 1. Let us t a k e

a ( s , k ) : # ( s , k , a ( s , k ) ) = a , k

#

0 (5)

b ( s , k ) : c k ( s , k , b ( s , k ) ) = a , k

#

s

The values a ( s , k ) and b ( s , k ) a r t h e lower and u p p e r bounds f o r t h e probability H ( A ) in t h e following sense.

LEMMA 1. For a n y fixed set A C B t h e b o u n d a ( s , k ) d e f i n e d in ( 5 ) p o s s e s s t h e foLLowing p r o p e r t i e s

1 . p [ a ( s , i d ) > H ( A ) ] 4 a f o r a n y m e a s u r e H .

2. If for some f u n c t i o n c ( i ) , i = O : s , c ( i + I ) > c ( i ) w e h a v e p [ c ( i A )

>

H ( A ) ] 4 a f o r a n y H t h e n c ( i ) 4 a ( s , i )

This lemma shows t h a t a ( s , i d ) i s in a c e r t a i n sense t h e b e s t lower bound f o r t h e probability H ( A ) . The similar r e s u l t holds f o r t h e u p p e r bound b ( s , i d ) :

LEMMA 1'. For a n y f i z e d set A C B b ( s , k ) d e f i n e d in (5) p o s s e s s t h e JoLLowing properties:

1. p [ b ( s , i d )

<

f f ( A ) I 5 a

2. for some f u n c t i o n c ( i ) , i = O : s , c ( i + l ) > c ( i ) w e h a v e p 1 c ( i A ) < H ( A ) I 4 a f o r a n y H t h e n c ( i ) r b ( s , i ) .

PROOF of t h e Lemma 1 1. Statements(3)-(5)imply

w h e r e j ( H , A )

=

min

I

j : @ ( s , j , H ( A ) )

<

a j. T h e r e f o r e

PS

la ( s , i A )

>

H ( A ) j 5 a

j f o r a n y H.

2. C o n s i d e r now a r b i t r a r y f u n c t i o n c ( i ) , i

=

0 ,

. . .

, s . W e o b t a i n :

= 2

^{S !} H ( A ) ~ (1

-

H ( A ) ) ~ - i

i = j ( H , A ) i ! ( s

-

^{i ) !}

w h e r e j ( H , A )

=

min I j : c ( j )

>

H ( A ) j.

j

Assumption 2 of t h e lemma now implies

s 3 3 @ ( ~ , j ( H , A ) , H ( A ) ) 5 a ( 6 )

S u p p o s e now t h a t c ( 0 )

>

0 . Taking H ( A ) s u c h t h a t 0 5 H ( A )

<

c ( 0 ) we o b t a i n j ( H , A )

=

0 a n d @ ( s , j ( H , A ) , H ( A ) )

=

1 which c o n t r a d i c t s ( 6 ) . T h e r e f o r e c ( 0 ) = O . In case if c ( i )

>

a ( s , i ) we c a n t a k e a ( s , i )

<

H ( A )

<

c ( i ) . Then j ( H , A )

=

i a n d @ ( s , j ( H , A ) , H ( A ) )

>

a which a g a i n c o n t r a d i c t s ( 6 ) . Thus, c ( i ) 6 a ( s , i ) f o r a n y i a n d t h e p r o o f i s c o m p l e t e d .

The v a l u e s a ( s

,

k ) a n d b ( s , k ) d e f i n e d by ( 5 ) h a v e t h e following u s e f u l p r o - p e r t y .

LEMMA 2. T h e r e e x i s t s

a

s u c h that for aLL a

< a

t h e b o u n d s a ( s , i ) a n d b ( s , i ) satisfi t h e foLLowing p r o p e r t y :

The p r o o f of t h i s lemma i s v e r y t e c h n i c a l a n d i s t h e r e f o r e omitted. T h e v a l u e of

a

c o m p u t e d with f o u r d i g i t s a c c u r a c y i s

a =

0.4681. This p r o p e r t y i s v e r y i m p o r t a n t f o r f u r t h e r c o n s i d e r a t i o n s a n d i t will b e assumed t h a t a

< a.

N o w i t i s possible to specify p r e c i s e l y t h e p r o c e s s of obtaining bounds f o r t h e solution of t h e problem (1):

Precise statement. The solution p r o c e s s evolves in d i s c r e t e time s

=

0 , 1 ,...

.

Before time i n t e r v a l s t h e set lol,

. . .

, w, of o b s e r v a t i o n s i s a v a i l a b l e which define t h e set G, of admissible d i s t r i b u t i o n s . A t t h e time i n t e r - val s t h e solution p r o c e s s c o n s i s t s of t h e following s t e p s :

1. N e w o b s e r v a t i o n o, a r r i v e s . The whole a v a i l a b l e set of o b s e r v a t i o n s lol,

. . .

, w, ] d e f i n e s t h e set of admissible d i s t r i b u t i o n s G, in t h e following way:

f o r a n y measurable A , w h e r e a (s , iA ) a n d b (s , iA ) are defined in (5). Additional information a b o u t t h e a c t u a l distribution H of random p a r a m e t e r s o c a n b e includ- e d in t h e definition of t h e set G,

.

Some ways of doing t h i s will b e discussed l a t e r . 2. The solution of problem (1) at s t e p s i s defined as t h e p a i r

wf.,

^f:) of lower a n d u p p e r bounds

f f

=

min min f f ( z , o)dH(o)

t E X H €12.

p

: =

min max J f ( z , o ) d ~ ( o ) z E X H E G .

a n d t h e optimal point z, i s defined

*

as follows:

max J f (zs*, o ) ti~(o)

=

/,Y. Z,

*

E

x

H €12,

3. The p r o c e s s i s r e p e a t e d n e x t time i n t e r v a l s

+

¹ with a r r i v a l of new o b s e r v a - tion.

The bounds on solution obtained in t h i s fashion are c o n s t r u c t e d involving t h e

"best" a n d t h e " w o r s t " admissible in t h e s e n s e of lemma 1 distribution H and t h e point z, yields t h e value of t h e o b j e c t i v e within t h e s e bounds. In what follows we

*

s h a l l c o n c e n t r a t e o n t h e numerical a s p e c t s of t h e problems (8)-(10).

3.

THE

SOLUTION OF

THE

M N E R PROBLEM

The minimax problems (8)-(9) look v e r y difficult b e c a u s e t h e i n n e r problems involve optimization o v e r t h e s e t of p r o b a b i l i t y m e a s u r e s defined by q u i t e compli- c a t e d c o n s t r a i n t s . I t a p p e a r s , however, t h a t i n n e r problems h a v e e x p l i c i t solu- tion. Let us c o n s i d e r t h i s problem in more d e t a i l a n d d e n o t e f o r simplicity f (x , o )

=

g ( a ) . W e are i n t e r e s t e d in solving t h e following problem:

minimize ( o r maximize) with r e s p e c t t o H

s u b j e c t t o c o n s t r a i n t s

Let us assume t h a t g ( o O )

=

min g ( o ) a n d g ( o S + I )

=

m a x g ( o ) e x i s t a n d ar-

o s n o s n

r a n g e t h e set of o b s e r v a t i o n s lol,

. . .

, os

1

in o r d e r of i n c r e a s i n g values of t h e function g ( o ) :

oO, o l , .

. .

, u s , us + l

H e r e a n d e l s e w h e r e t h e original o r d e r of o b s e r v a t i o n s i s indicated by s u b s c r i p t a n d a r r a n g e m e n t in i n c r e a s i n g o r d e r of t h e values of g i s indicated by s u p e r - s c r i p t . The f i r s t element of new a r r a n g e m e n t will always b e t h e point with t h e minimal value of t h e o b j e c t i v e function on t h e s e t fl a n d t h e l a s t element (with number s

+

1 ) will b e t h e point with maximal value. This a r r a n g e m e n t d e p e n d s on t h e number s of t h e time i n t e r v a l , but t h i s dependence will n o t b e explicitly indi- c a t e d f o r t h e simplicity of notations.

The solution of t h e problem (10)-(11) i s given by t h e following theorem:

THEOREM 1 Suppose t h a t e z i s t p o i n t s o0 a n d u s + ' s u c h t h a t 8 ( o O )

=

min g (o), g ( o S + l )

=

max g ( o ) . m e n

~n w E n

1. The s o l u t i o n of t h e problem W)-(rZ) e z i s t a n d among eztremal m e a s u r e s a l - w a y s e z i s t d i s c r e t e one w h i c h i s concentrated i n s

+

l points:

where

Ag =

max g ( w )

-

min g ( o )

O E ~ O E ~

3 .

w i t h probability 1 a s s

- ^-.

PROOF

1. Let us consider t h e upper bound

g',

and define the s e t s

and functions

E a c h set Ri c o n t a i n s s

-

ⁱ p o i n t s of

.

L e t u s c o n s i d e r t h e p r o b l e m

w h e r e

W e h a v e

t h e r e f o r e

s +1

= x

g(oi)(b(s, s

-

ⁱ

+

¹⁾

-

^{b(s, s}

-

ⁱ⁾⁾

= &

i =1

w h e r e we t o o k b (N,

-

¹⁾

=

0. H e n c e

6

i s t h e s o l u t i o n of p r o b l e m (15) with

&

b e - ing i t ' s optimal v a l u e . W e h a v e

a n d t h e r e f o r e

I t i s l e f t to p r o v e t h a t

is

E G,. To d o t h i s let u s c o n s i d e r a r b i t r a r y m e a s u r a b l e s u b s e t A of R which c o n t a i n s k f r o m s

+

¹ p o i n t s lo1,

. . .

^{, oS}

+'I,

^{s a y}

i l ik

lo , . . . , ⁰

1.

According t o t h e lemma 2

b ( s , s

-

i j

+

1 )

-

^{b ( s , s}

-

^{i j ) 5} b ( s , k

-

^{j )}

-

b ( s , k

-

j - 1 ) T h e r e f o r e

On t h e o t h e r hand

b ( s , s

-

^{i j}

+

1 )

-

b ( s , s

-

^{i j )}

2 b ( s , s

-

j + 1 ) - b ( s , s - j )

=

a ( s , j )

-

a ( s , j

-

^{1 )}

T h e r e f o r e

Inequalities ( 1 6 ) a n d ( 1 7 ) p r o v e t h a t

4

^{( A ) E G,}

.

The proof of t h e f i r s t s t a t e m e n t of t h e theorem i s completed.

2. We h a v e

Consider now t h e d i f f e r e n c e p:

-

q:

The f i r s t term is monotonically i n c r e a s i n g , t h e second i s monotonically d e c r e a s i n g a n d f o r odd s we h a v e

s + l

p,i

-q,i < O f o r 1 5 i

<-

2

s + l

pf

-

^qf

= o

^{f o r i}

^{= -}

2

+ l < i s s pf

-

q,i

> o

^{f o r -} ₂

T h e r e f o r e f o r odd s w e c a n continue ( 1 8 ) as follows:

According t o t h e definition of pf and qf w e h a v e

Taking i n t o a c c o u n t t h a t b ( s , 0 ) = 1

-

a ( s , s ) l a s t e q u a l i t i e s t o g e t h e r with (18) will give

To p r o c e e d f u r t h e r w e h a v e t o obtain t h e lower bound f o r a ( s , ( s

+

1 ) / 2 ) a n d a ( s , ( s

-

I ) / 2 ) . To d o t h i s r e m e m b e r t h a t a ( s , ( s

+

I ) / 2 ) i s t h e solution of equation

with r e s p e c t t o p r o b a b i l i t y z in t h e binomial distribution, by P r we denoted t h e bi- nomial distribution with s t r i a l s . This c a n b e r e w r i t t e n as follows:

On t h e o t h e r h a n d w e h a v e t h e following Okamoto i n e q u a l i t y [ 2 0 ]

f o r a n y c r 0 . Due to 1 / 2

+

1 / 2 s

-

a ( s , ( s

+

I ) / 2 )

>

0 f o r a l l s w e o b t a i n f r o m p r e v i o u s i n e q u a l i t y

f o r z

=

a ( s , ( s

+

1 ) / 2 ) .

T h e r e f o r e t h e s o l u t i o n of t h e e q u a t i o n

will g i v e t h e l o w e r b o u n d f o r a ( s , ( s

+

1 ) / 2 ) This g i v e s a ( s . ( s

+

I ) / 2 )

a

1 / 2

+

1 / 2 s

- ,/%.

In t h e s a m e way we o b t a i n

1 1

a

[ ]

s , 2- ²

- - -

^{2 s}

s u b s t i t u t i n g t h i s i n ( 2 0 ) w e o b t a i n

In case of t h e e v e n s w e o b t a i n

= , i - q , i < O f o r 1 ~ S i ~ - 2

= , i - q , i > O f o r

-

S + 1 5 i 4 s 2

This g i v e s

a n d f i n a l l y

A f t e r applying Okamoto unequality w e again o b t a i n from (21) t h e d e s i r e d unequali- t Y .

3. O b s e r v e t h a t t h e e m p i r i c a l d i s t r i b u t i o n H:

=

[ ( o l , l / s),

. . .

, (o, , l / s )

1

al- ways belongs to G,. T h e r e f o r e t h e last s t a t e m e n t of t h e t h e o r e m follows f r o m state- ment 2 , boundedness of ^g ( o ) o n

R

a n d t h e l a w of l a r g e numbers.

The proof i s completed.

R e s u l t s of numerical e x p e r i m e n t s f o r computing bounds

g',

a n d g,

-

are shown in Figures 1-6. These bounds were computed f o r c o n f i d e n c e l e v e l a

=

0.1. They a l l e x h i b i t similar b e h a v i o r : r a p i d c o n v e r g e n c e f o r a small number of o b s e r v a t i o n s which slowed down as t h e number of o b s e r v a t i o n grows in a c c o r d a n c e with r e s u l t of t h e t h e o r e m . Almost in a l l examples t h e a c t u a l value of

f

g ( o ) d H ( o ) always s t a y e d within bounds, although t h e value of a w a s c h o s e n 0.1. This h a p p e n e d b e c a u s e t h e bounds w e r e computed f o r t h e worst d i s t r i b u t i o n s which are t h o s e c o n c e n t r a t e d in a f i n i t e number of points. When s u c h d i s t r i b u t i o n s were t a k e n t h e b e h a v i o r of bounds worsened (Figures 2-4) a n d in some cases t h e bounds did n o t c o n t a i n a c t u a l value (Figure 4). A t t h e same time f o r s m o o t h e r d i s t r i b u t i o n s c o n v e r g e n c e i s fas- ter (Figure 5, w h e r e d i s t r i b u t i o n i s close to normal).

Within t h e framework of n o n p a r a m e t r i c s t a t i s t i c s [16] t h e bounds p r o p o s e d in t h i s s e c t i o n c a n b e c o n s i d e r e d as a s p e c i a l t y p e of L-estimates.

4. INCORPORATING ADDITIONAL INFORMATION

The method developed in t h e last s e c t i o n d e a l s with t h e case when t h e only a v a i l a b l e information o n t h e d i s t r i b u t i o n H of random p a r a m e t e r s o comes from ob- s e r v a t i o n s oi. In many c a s e s , however, additional information i s a v a i l a b l e which i s d r a w n from o b s e r v a t i o n s o n similar systems. One of t h e ways of using t h i s informa- tion i s c o n s i d e r e d i n t h i s s e c t i o n .

S u p p o s e t h a t additional information comes in t h e form of c o n s t r a i n t s o n t h e v a l u e s of moments l i k e e x p e c t a t i o n , v a r i a n c e e t c . This c a n b e e x p r e s s e d in t h e fol- lowing way:

The problem of g e t t i n g u p p e r a n d lower bounds in t h i s case c a n b e e x p r e s s e d similarly t o (10)-(11). Let u s t a k e t h e problem of g e t t i n g t h e u p p e r bound:

maximize with r e s p c t t o H

s u b j e c t t o c o n s t r a i n t s

The L a g r a n g e multipliers are used t o t a k e a c c o u n t of c o n s t r a i n t (24) a n d r e d u c e t h e problem (22)-(24) to t h e problem (10)-(11).

Let u s c o n s i d e r t h e function

a n d assume t h a t f o r e a c h u 2 0 e x i s t oO(u ) and o S "(u) s u c h t h a t

F o r e a c h fixed u a r r a n g e w i in i n c r e a s i n g o r d e r :

where L ( o f ( u ), u ) S

L

( o f +l(u ), u ). H e r e again w e use s u p e r s c r i p t t o indicate o r d e r e d o b s e r v a t i o n s , t h i s time, however, o r d e r i n g will depend on u

.

c o n s t r u c t

HS

( u )

=

[(oO(u ), p:),

. . .

, (aS +'(u ), p: +I) j a n d t a k e

where p: are defined in (14).

THEOREM 2 S u p p o s e t h a t

1. For u ² 0 e x i s t w O ( u ), wS ( u ) s u c h t h a t (25) i s s a t i s f i e d . 2 . E x i s t s b

>

0 s u c h t h a t

* *

Let U S t a k e u : u r 0,

p

_{( u}

*

)

=

min

p

_{( u} ) (ij t h i s u

*

e s i s t s ) . T h e n t h e u 2 0

m e a s u r e f i S ( u * ) d e f i n e d in (26) i s t h e s o l u t i o n of t h e problem 622)-(24) w i t h p r o b a b i l i t y PS i n t h e space ( R S ,

BS

, H S ) a t least e q u a l to y w h e r e

qS ( u ) i s d e f i n e d in (27) a n d t h e c o r r e s p o n d i n g u p p e r b o u n d i s qS ( u

*).

PROOF O b s e r v e t h a t f o r e m p i r i c a l d i s t r i b u t i o n H: consisting of s p o i n t s con- s t r a i n t s ( 2 3 ) are s a t i s f i e d . Let us e s t i m a t e t h e p r o b a b i l i t y with which J v i ( w ) c W , e ( w )

= ~ / S Z ~ = ~ V ~ ( O ~ )

^{< - E with E > O .}

According to g e n e r a l i z a t i o n of Tchebyshev inequality

T h e r e f o r e

f o r some E

>

0 with p r o b a b i l i t y at l e a s t y. Convexity of t h e s e t , defined by ( 2 3 ) similarly to [9] implies now e q u i v a l e n c e of t h e problem (22)-(24) a n d t h e problem

min max

f

^{L (w. u )} d H ( o )

u 2 O H ( 2 9 )

s u b j e c t t o ( 2 3 ) , which holds f o r a l l o E RS such t h a t ( 2 8 ) is satisfied. The i n n e r problem in ( 2 9 ) h a s e x p l i c i t solution defined by t h e o r e m 1. This solution i s d e s c r i b e d by r e l a t i o n s ( 2 6 ) - ( 2 7 ) .

The proof i s completed.

A similar r e s u l t holds f o r lower bound if w e t a k e L ( o , u )

=

g ( o )

+

E ; ~ . ' ~ U ~ ~ ( G J ) a n d q S ( o )

=

~ f ~ ~ q f ~ ( o i ( u ) , u ) .

The t h e o r e m 2 r e d u c e s t h e problem of g e t t i n g u p p e r bound t o minimization of function q S ( u ) , defined in ( 2 7 ) . This i s a convex function with r e a d i l y available values of s u b g r a d i e n t . T h e r e f o r e s u i t a b l e nondifferentiable optimization t e c h - niques c a n b e a p p l i e d t o g e t i t s minima [18, 221. D i f f e r e n t s i t u a t i o n s c a n o c c u r d u r - ing s u c h computations.

1 . I t was found t h a t q S ( u ) i s not bounded from below. This means t h a t problem ( 2 2 ) - ( 2 4 ) i s infeasible. To g e t a n u p p e r bound in t h i s case i t is n e c e s s a r y t o d r o p c o n s t r a i n t ( 2 4 ) a n d solve t h e problems ( 1 0 ) - ( 1 1 ) instead. According t o t h e o r e m 2 t h e p r o b a b i l i t y of t h i s case t e n d s t o 0 as s ^--,

=.

2 . The point u * w a s found s u c h t h a t within p r e s c r i b e d a c c u r a c y

ckS

( u * ) will b e t h e optimal value of t h e problem ( 2 2 ) - ( 2 4 ) , i.e. t h e d e s i r e d u p p e r bound with p r o - bability t h a t t e n d s t o 1 when s ^--,

=.

F o r finite s , however, i t is possible t h a t q s ( u *) e x c e e d s t h e optimal value of ( 2 2 ) - ( 2 4 ) . In both cases i t will b e t h e u p p e r bound, but in t h e s e c o n d case not t h e b e s t one.

5. NUMERICAL TECHNIQUES FOR FRYDING THE UPPER BOUNDS FOR THE SOLUTION OF OPTIMIZATION PROBLEMS

Let u s r e t u r n t o t h e problems ( 8 ) - ( 9 ) now. Define

_F(s, x ) = min f j ' ( x , o)dH(o)

H E C ,

I t is now possible t o compute t h e values of t h e s e functions using r e s u l t s of t h e sec- tion 3 . The problem of finding u p p e r a n d lower bounds f o r t h e solution of ( 1 ) c a n b e formulated as follows:

4; =

₂ min F ( s _{E X}

.

^{I ) ; j'?}

⁼

₂ ^min _EX- ^{F ( S , z)}

S u c c e s s i v e bounds f o r simple problem are given in F i g u r e 7 to g i v e a feeling how t h e y evolve. F o r m o r e complicated problems i t i s n e c e s s a r y to d e v e l o p s p e c i a l nu- m e r i c a l techniques. The problem of finding lower bound :'j i s n o t c o n v e x b e c a u s e t h e function _F(s, z ) i s n o t convex with r e s p e c t to z . T h e r e f o r e i t n e e d s s p e c i a l t r e a t m e n t in e a c h p a r t i c u l a r case a n d g e n e r a l e f f i c i e n t t e c h n i q u e s are o u t of r e a c h so f a r . In t h i s s e c t i o n w e s h a l l c o n c e n t r a t e o n t h e problem of finding t h e u p p e r bound j': which i s much e a s i e r b e c a u s e t h e function F ( s , z ) i s convex. This allows a p p l i c a t i o n of c o n v e x programming algorithms to find jt

.

These a l g o r i t h m s however, need a s u b s t a n t i a l number of i t e r a t i o n s to g e t close to a solution a n d with a r r i v a l of e a c h new o b s e r v a t i o n t h e p r o c e s s should b e s t a r t e d anew. The main p o i n t of t h i s s e c t i o n i s t h a t t e c h n i q u e s c a n b e developed t h a t p e r f o r m v e r y limited, p e r h a p s only o n e , i t e r a t i o n of convex programming algorithm with e a c h new o b s e r - vation a n d s t i l l g e t r e a s o n a b l e u p p e r bound. G e n e r a l l y s p e a k i n g t h e p r o c e s s of ob- taining bounds look like t h i s

1. Start from some f i x e d number of o b s e r v a t i o n r a n d p o i n t z T .

2. Suppose t h a t p r i o r to s t e p number s t h e o b s e r v a t i o n s ol,

. . .

, us a r r i v e d , t h e point x S w a s o b t a i n e d a n d t h e c u r r e n t u p p e r bound i s t a k e n e q u a l to F ( s

-

^{1, z S -I).} The following c a l c u l a t i o n s are p e r f o r m e d at t h e s t e p number

S .

Za. The o b s e r v a t i o n os a r r i v e s . The function F ( s , x ) a n d i t s s u b g r a d i e n t i s computed at t h e p o i n t x S a n d possibly at some additional points.

2 b . These v a l u e s are used to p e r f o r m o n e s t e p of minimization of t h e function F ( s , x ) which g i v e s t h e new point z S

.

The p r o c e d u r e g o e s to t h e n e x t s t e p .

W e s h a l l c o n s i d e r a p a r t i c u l a r method b a s e d on g e n e r a l i z e d l i n e a r p r o g r a m - ming [3]. This t e c h n i q u e p r o v i d e s n a t u r a l bounds f o r t h e solution of t h e problem min,F(s, x ) which e n a b l e s to c o n t r o l a c c u r a c y . With e a c h new o b s e r v a t i o n new s u p p o r t i n g h y p e r p l a n e i s i n t r o d u c e d .

1. Take a n initial c o l l e c t i o n of p o i n t s

F o r e a c h y E

?

compute u ( y ) :

Take F(0, y )

=

^'j(y , o(y )) a n d compute

Solve t h e problem

min max [F(o, y )

+

<z

-

^{y ,}

F,

(0, y ) ] x y € 9

This i s a l i n e a r programming problem which solution i s zl. This will b e t h e initial solution of t h e original problem. Take t h e set of o b s e r v a t i o n s R0

= 4.

2 . A t t h e beginning of s t e p number s we h a v e t h e set of o b s e r v a t i o n s

RS

t h e set of approximating points YS c u r r e n t approximation t o t h e minimum of t h e u p p e r bound z S a n d f o r e a c h y E YS w e h a v e a l r e a d y computed j'(y, o) o E RS j

' ( v , o(y )),

f,

(y, o),

f,

( y , o(y )), a n d estimates

t h e n w e d o t h e following:

( a ) Obtain new o b s e r v a t i o n ^os a n d t a k e RS

=

RS

u

^{to, j.} The set RS consists now of s points ol,

. . . .

^OS

(b) F o r y

=

z S compute o(y ):

a n d compute ^'j(y , o), j', _(y , o) f o r a l l o E CIS

.

A r r a n g e t h e set RS U io(y) j in t h e o r d e r of i n c r e a s i n g values of 'j(y , w):

a n d oS +l(y)

=

o(y) Assign ps (y)

=

s a n d t a k e

F o r y E YS t a k e ps ( y )

=

pS - l ( y ) . Update t h e set YS

( c ) Find

min max [ F ( p s (Y ), Y ) + <z

-

^{Y , Fz ( ~ s ( Y ) I Y}

)>I

E X y € r

This i s l i n e a r programming problem and s u p p o s e t h a t is i t s solution. Consider t h e s e t

=

min max [ F ( p s ( y ) , Y )

+

<z

-

y , F , ( p S ( y ) , Y ) > ]

E X y E r

and p S ( Y ) < s j

If Z S

= #

t h e n t a k e z S + I

= 5

and g o to t h e s t e p number s

+

1, o t h e r w i s e p r o c e e d t o (d).

(d) F o r e a c h y ^E Z S compute f ( y , w i ) a n d f , ( y , w i ) , p S ( y )

<

i 5 S . Take

Assign p s ( y )

=

s a n d g o t o (c). The following t h e o r e m d e a l s with t h e c o n v e r g e n c e of t h i s method.

THEOREM 3. S u p p o s e that t h e following c o n d i t i o n s a r e satisj'ied 1. S e t s

X c

Rn a n d D

c

_Rm a r e compact sets.

2 . The f u n c t i o n f ( z , w ) i s c o n v e z o n z a n d c o n t i n u o u s o n O,

I l

^o

- ^I

⁵

^{L I Z ~}

^{-z21 for z l , z 2 EX, o E}

n

^{m e n}

F ( s , z ^{' )}

-

^min

F ( s

^, z ) ⁴ 0 as s ^{4 m} w i t h p r o b a b i l i t y 1.

z E X PROOF Let us d e n o t e

F S ( s , z )

=

max [ F ( p s ( y ) , y )

+

<z - y , F z ( p S ( v ) , y ) > l

Y E r

Then

Let u s p r o v e t h a t F ( s , z S )

- F S

^{( s ,} z S ) 0 with probability 1. Note t h a t

F ( s , z S ) ²

FS

^{( s ,} z S ) i s always satisfied. Suppose that f o r some A

>

⁰ e x i s t subse- quence sk such that

Due t o compactness of the s e t X w e may assume without loss of generality that

Z s k 4 X

*

,

where s k S p ( x S k ) S sk + l . According t o condition 2 w e have

S k

I F ( s ,

z l ) - F ( s , x 2 ) 1 S L i z l - z z l , I F S ( s , x l ) - F S ( s , x 2 ) d L Ix l

-

^x21

f o r all s and x l , x E X . Therefore

l F ( S k , X S k ) - F ( s t , z * ) d L I x S k

-

^{z *}

1

-st +I

I F

^{( s k + I ,} x s k + l ) -FSk+l(sk

s

L ^IzSk+'

I f

( p S k ( x s k ) , x

-

F . ( p S k ( ~ s k ) , * )

1 s

L

I X

^sk

-

^I

* I

The theorem 1 gives

1F(sk, x * ) - F ( X * ) I 4 0 / F ( p s k ( x s k ) , x * )

-

F ( z * ) ~ 4 0 with probability 1.

Definition of t h e a l g o r i t h m implies

Compactness of t h e sets X a n d fl with continuity of f ( z

.

^o) implies boundedness of F ( s , z ) uniformly on s . T h e r e f o r e w e may assume without loss of g e n e r a l i t y t h a t

combining (32)-(36) we o b t a i n

with p r o b a b i l i t y 1.

This c o n t r a d i c t s initial t h e assumption a n d t h e r e f o r e

with p r o b a b i i t y 1.

Thus

F ( s , z S )

-

2 m i n F ( s , z ) EX

- ^o

b e c a u s e

The proof i s completed.

The t h e o r e m s u g g e s t s t h a t p r o p o s e d t e c h n i q u e s could b e viable f o r computing u p p e r bounds. The i m p o r t a n t question now i s w h e t h e r t h e s p e e d of c o n v e r g e n c e to u p p e r bound i s f a s t e r t h a n c o n v e r g e n c e of t h e bounds themselves. To find condi- tions which g u a r a n t e e t h i s i s t h e o b j e c t i v e of f u r t h e r s t u d y .

.I L? . m m -

17. VIB -

=--*--

-a-- ----..2---;1- + 3----sit--p- -s---e 2 ----i?? 3

-

^3--1-- -+--*----f--P--*--f . -F.--f- 3 -... - 3 _f f --.--

*

^-.f --.-+-.I-.-- *--*

---,-/*-

^{- ----}

.-'-

/'

9-

.. u-

- 3

-

^{El rn ---}

1 I ?

.

Elm

number of observations

FIGIJRK 5. Bounds f o r / o d ~ ( o ) w h e r e H i s d i s t r i b u t i o n of t h e sum of 10 independent random v a r i a b l e s , e a c h d i s t r i b u t e d uniformly on 1-1, 11.

ACKNOWLEDGMENT

The a u t h o r i s g r a t e f u l to P r o f e s s o r s A. Kurzhanski a n d R. Wets f o r useful dis- cussions.

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