Optimization of Functionals Which Depend on Distribution Functions: 1. Nonlinear Functional and Linear Constraints

NOT FOR QUOTATION WITHOUT PERMISSIOE;

OF THE AUTHOR

OPTIM3ZATION OF FWNCTIONALS WHICH DEPEND ON DISI'RIBUTION FUNCIIONS: 1. NONLINEAR FUNCTIONAL AND

LWEAR

CONSiTAUWS

k Gaivoronski

November 1983 WP-83-114

Working Papers a r e interim reports on work of t h e International Institute for Applied Systems Analysis and have received only limited review. Views or opinions expressed herein do not necessarily represent those of t h e Institute or of its National Member Organiza- tions.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS 2361 Laxenburg. Austria

PREFACE

The development of optimization techniques for solving complex decision problems u n d e r uncertainty is c u r r e n t l y a major topic of research in t h e Sys- t e m a n d Decision Sciences Area a t IIASA, and in t h e Adaptation and Optimiza- tion g r o u p in particular.This paper deals with methods for t h e solution of prob- lems in which t h e objective function depends on probability distributions.

Such problems a r e common in reliability theory and various o t h e r branches of operations r e s e a r c h , but methods for dealing with t h e m have only recently begun t o emerge.

ANDRZEJ WIERZBICKl C h a i r m a n

S y s t e m a n d Decision S c i e n c e s

ABSTRACT

The main purpose of this paper is to discuss numerical optimization pro- cedures for problems in which both t h e objective function and t h e constraints depend on d s t r i b u t i o n functions. The objective function is assumed to be nonlinear a n d t o have directional derivatives, while t h e constraints a r e taken as linear. The proposed algorithm involves linearization of t h e objective func- tion a t t h e c u r r e n t point and solution of a n auxiliary linear subproblem. This last problem is solved using duality relations and cutting-plane techniques.

OPTIMIZATION OF FUNCTIONALS P33ICH DEPEND 04.;' C?TSI?EIBUTZON FLTNCTIONS: 1. NONLINEAR FUNCTIONAL AND LINEAR CQNSTmdTS

A. Gaivoronski

1. INTRODUCTION

The purpose of this paper is to present numerical methods for solving optimization problems in which both t h e objective function and t h e con- straints depend on distribution functions. Such problems occur in stochastic programming [1,2], reliability theory and various branches of operations research ( surveyed in [3] ), and robust statistics [4] , among others

.

If both the objective function and the constraints a r e linear with respect to t h e distribution functions then the problem can be s t a t e d as follows:

min

f

(x ) d ~ ( x )

X

where

X

is a s e t in Euclidean space Rn

.

Some specific cases of problem (1)- (3) arise in the theory of Markovian moments and can be solved analytically

[5]. However, t h e success of analytical methods is very limited even in t h e linear case (1)-(3) and therefore numerical algorithms a r e needed. Such methods began t o appear in the middle of the last decade.

One idea is t o approximate set X by a sequence of finite subsets

X1.X2, . . . , &

,... , where

This sequence of s e t s has t h e property t h a t

a s s t e n d s t o infinity, i.e., t h e g r e a t e s t distance between points in s e t X and s e t

X,

t e n d s to zero. Let G, be t h e s e t of all distribution functions which correspond to probabilistic measures concentrated in points from

4

1 n ns

G,=[(x,. p l )

. ^{. .}

^.

^,

^{. C p i}

⁼

^{1 . P i 2 01}

^.

i =l

If we include one more constraint.

problem (1)-(4) becomes a finite-dimensional linear programming problem;

t h i s raises t h e possibility of approximating t h e original problem (1)-(3) by a s e q u e n c e of finite-dimensional problems (1)-(4). This idea was explored in [3,6], where i t was also used t o solve nonlinear and minimax problems involv- ing distribution functions. However, this approach can be used only for s e t s X of small dimension ( in fact not exceeding t h r e e ) due t o t h e high dimensional- ity of t h e associated linear programming problems.

Another possible way of solving (1)-(3) is based on t h e duality relations between problem (1)-(3) a n d some finite-dimensional minimax problem. First

proposed in [ I ] , this idea was taken further in [ 2 , 7 ] , where methods of t h e cutting-plane type a r e given.

The purpose of this paper is t o use this last approach t o develop solution techniques for nonlinear problems of t h e kind :

min +(H) subject to

where G is defined in t h e following way:

We propose an analogue of t h e linearization ( Frank - Wolfe ) method in which we a r e required to solve subproblems of type (1)-(3). It appears t h a t it is not necessary t o solve these subproblems precisely: duality relations make i t pos- sible t o utilize rough solutions of (1)-(3) so t h a t only a limited number of cal- culations a r e needed a t each iteration.

2.

CHARAC-TION OF OPTIMAL DISllUBUTIONS

We shall use t h e s a m e l e t t e r , say H , t o denote both t h e distribution func- tion and t h e underlying probabilistic measure, where this will not cause con- fusion. For a given probabilistic measure H we shall denote by B+ the collec- tion of all subsets of X with positive measure H , a n d by dom H t h e support s e t of H , i.e.,

Let u s first introduce t h e class of functionals considered, + ( H ) . What we actually need i s some analogue of directional differentiability. Suppose t h a t

Q ( H * + ~ ( H - H * ) ) = + ( H * ) + a

f

^{f '(z} ,,Li")d(hr(z)-hr*(s)) +.r(a.H*,hr) _{( 8 )} X

where

r ( a , H * , H ) = o ( a )

In what follows we a s s u m e t h a t functions f O ( X , H ) , f i ( x ) a r e such t h a t expressions (7) a n d (8) a r e meaningful.

The following simple conditions a r e necessary, a n d in t h e convex c a s e also sufficient, for distribution H t o be a solution of problem (5)-(7) :

Lemma 1

If + ( H e ) r +(H) for some H * E G a n d all H E G t h e n

Proof

The proof is of t h e traditional type for n e c e s s a r y conditions. Note t h a t z * g f f O ( z . ~ * ) d ~ * ( z )

X

i s always t r u e . Suppose, on t h e contrary, t h a t t h e r e exists a y>O s u c h t h a t z

*-

f f O ( z , ~ * ) d ~ * ( z ) ^G -27

X

Then t h e r e exists a n

H

^E G s u c h t h a t

f f

O ( ~ . H * ) H ( Z ) - f f O ( ~ . ~ * ) d ~ * ( ~ )

<

7

X X

Consider now distributions Ha :

H a = a H + ( l - a ) H *

.

According t o ( 8 ) ,

*(Ha)

=

.I.(H*)+~

f ^f

O(z . ~ * ) d ( H ( z ) - ~ ' ( z ) ) + T ( ~ , H * . H ) X

a n d for small a we obtain

which contradicts t h e optimelity of H *

.

This completes the prcof.

Remark. I f , additionally, . k ( H ) is convex, i.e.,

then this l e m m a also gives sufficient conditions for a global minimum.

Lemma 1 implies t h a t to check the necessary conditions for problem ( 5 ) - ( 7 ) requires solution of a linear problem of t h e form ( 1 ) - ( 3 ) . where q ^{( 2 )}

=

f ' ( 2 , ~ ) . The solution of problem ( 1 ) - ( 3 ) can be characterized through the duality relations summarized in t h e following theorem, which was proved in [ 7 ] .

Theorem 1

Suppose t h a t t h e following assumptions a r e satisfied:

1. Set X is compact and functions f i ( z ) , i = O,m are continuous on X

.

2. c o Z # $ where

Then

1. A solution of problem ( 1 ) - ( 3 ) exists and t h e optimal value o f f ' ( z ) is equal t o t h e optimal value of t h e following minimax problem:

where I/+

= t

u : u E Rm ,

q

1

Oj .

2. For any solution H* of problem ( 1 ) - ( 3 ) t h e r e exists a u * E I/+ such t h a t V ( u *)

=

max

p ( u )

, dom H *

r

X*(U *)

U E V + where

3. There exists a finite s e t

4 =

[ z l ,

.

. . , s t

1

^, t I m + l , and a solution

H *

of problem (1)-(3) such t h a t dom H *

=

Xt , i.e., t h e probabilistic measure

H * can be expressed as a collection of t < m + l pairs

t ( z l # p l )

^,

. .

^. ,

( z t , p f ) j -

The probabilities jil ,

. . .

,

pf

a r e solutions of t h e following linear program- ming problem:

Combining Theorem 1 a n d Lemma 1 we obtain t h e following result:

Theorem 2

Suppose t h a t + ( H e )

s

+ ( H ) for all H E G and t h e following assumptions are satisfied :

1. Set X i s compact a n d functions f O ( z , H C ) , f i(z) , i

= l,m

a r e continuous o n X .

2. c o Z # $I where

Then 1. W e have

where

f

O ( z . H e ) d H * ( z )

=

max p ( u )

X U E V +

m

p ( u )

=

min (f O ( Z , H * )

+ C

u i f i ( z ) )

z EX i = l

2. There exists a u, , p ( ~ *) = ma>: p ( u ) , u E ' CJ such t h a t U E V +

dom H

c X*(U

*) where

3. NUMERICAL

METHOD FOR

SOL?TNG PROELEES WTH L I h W CON-15

I t is now possible t o construct a method which finds points satisfying t h e necessary conditions of Lemma 1 or, in t h e convex case, global minima

.

This method is of t h e linearization type.

A l g o r i t h m 1

1. Begin with an initial distribution HI.

2. Suppose we have an approximate solution HS before starting iteration number s

.

Then a t t h e s - t h iteration we do t h e following :

(i) Find a distribution

ps

^{E G} such t h a t

J f O ( ~ . ~ s ) d R S ( ~ ) L Z,

+ 5

X

where

z,= inf J f O ( x . ~ s ) ~ ( x ) H E G

a n d E ,

>

0 is t h e accuracy with which problem (9) is solved. I t is n o t necessary to know t h e value of E, , only t h a t E , -, 0

(ii) Check whether functional +(H) decreases in t h e direction

HS -

HS

.

If not, r e t u r n t o s t e p (i) and solve problem (9) with higher accuracy. Other- wise go to s t e p (iii).

(iii) Choose a stepsize p, : O<p,

r

1 and calculate a new approximation t o t h e optimal solution:

HS+1 = _(1-p,)HS

+ psHS

Then go t o s t e p (i).

Remark. The stepsize can be chosen according t o a n u m b e r of different rules:

- ^OD

( a ) p,

. CFs

^{= m}

s ^=O

(b) p, = a r g min \k(HS

+

a ( p - H S ) ) a20

( c ) Take a sequence a, ,where

a n d

p,

=

minlarg min \ k ( ~ ~

+

a ( H S

-

H')), a, j

.

a20 (13)

We now introduce topology on s e t G . We shall use weak ( s t a r ) convergence topology, which by definition i s t h e weakest topology in t h e s p a c e of all proba- bility m e a s u r e s s u c h t h a t t h e m a p

H

-.

j g ( 4 W z )

is continuous wherever g(z) i s bounded a n d continuous. Note t h a t in t h i s topology t h e s e t of all m e a s u r e s with c o m p a c t support X is c o m p a c t . In t h e discussions t h a t follow we shall use t h i s topology when speaking a b o u t t h e con- tinuity of c e r t a i n functionals with r e s p e c t t o probabilistic m e a s u r e s .

Let v ( H ' . H ~ ) denote s o m e distance between distributions ^HI a n d H2 which will i n d u c e weak ( s t a r ) topology on t h e s e t of all distributions - for example, t h e Levy-Prokhorov distance would do.

We shall now prove t h e convergence of t h e a l g o r i t l ~ m given a b o v ~ . Theorem 3

Suppose t h a t t h e following s t a t e m e n t s are t r u e : 1. X cRn is compact s e t .

2. Functional +(H) satisfies (8) where r ( a , H * . H ) ⁺

a uniformly over H* E G , H E G

.

3. Function f ' ( x , ~ ) is continuous with r e s p e c t t o X E X and satisfies t h e Lipschitz condition with respect to HEG :

for

H I

E G ^,

H2

E

G

, x E X

.

Functions f i ( z ) , i

= %

a r e continuous with respect to z E X .

4. E s + O .

5. Stepsize p, is chosen according to one of (11)-(13)

.

Then

lim inf ff o ( z , ~ s ) d ( ~ ( z ) - ~ ( ~ ) )

=

⁰

s+- HEG

and for all limit distributions H* of sequence HS inf

f

^f O ( z , ~ ) d ( ~ ( x ) - ~ s ( z ) )

=

⁰

.

HEG

x

Proof

1. Note t h a t under assumption 2 of t h e t h e o r e m s e t G is compact in weak ( s t a r ) topology a n d therefore function f O(z,H) is bounded on X x G

.

This together with (8) implies t h e continuity of functional \k(H) on G. Therefore

\k(H) h a s a minimum on G.

2. The a r g u m e n t given in the proof of Lemma 1 lezds to the follo~ving inequal- ity:

where ~ ~ ( p , ) -' 0 a s p , -, 0 ; E, -, 0 due t o assumption 1 of t h e theorem and y,

=

inf J ~ O ( I . H ~ ) ~ ( H ( ~ ) - H,(Z))

.

H E G

x

The remaining part of t h e proof depends on t h e way in which t h e stepsize p, is chosen.

2(a) Suppose t h a t p, is chosen according to (12). Define

~ ( 8 ) =

SUP

IP

^{: 0 (P) 4}

BPI

^,

6 1

From assumption 1 of t h e theorem we know t h a t U(P)

>

0 if

8 >

0

.

^Taking

Ys

-

^{-s .}

8 =

₂ ^{~f 7,}

^>

^{E, and p,}

⁼

a(8) we now get from (14):

Inequality (15) immediately gives max 17,

-

^{E, , 01 -,} 0, which implies t h a t y, -, 0 because E, -, 0

2(b) Now l e t p, be chosen according t o (11)

.

Suppose t h a t t h e r e exists an

E

such t h a t for s

>

F

where a

>

0. Now from (14) we g e t :

\k(HS+l) I

mint+(^') -

^{ap, ,} \k(HS

)I =

+(HS )

-

^{ps ,} (1 6)

Summing ( 1 6 ) from s

> s

to k we obtain :

i =s

m

which contradicts + ( H )

>

-= because C p i = =. Therefore t h e r e is a subse- i = I

quence nk s u c h t h a t

maxt0 ⁹ ynk - znk

- T ~ ( P ~ ~ ) I

^{+ 0}

.

Now suppose t h a t t h e r e is a subsequence m k s u c h t h a t

We may assume without loss of generality t h a t

Let u s take a sequence Lk s u c h t h a t

7

-

- 7 a for mk < i < l k ,

71k+l -'lk+l - r I ( ~ ~ k + l ) < a

We shall now estimate t h e value of functional + ( H ) on elements of sequence HLk

.

From ( 1 4 ) we can show t h a t

1, -1

To proceed with this estimate f u r t h e r i t is necessary t o estimate

C

^{pj ,}

j =n, which can be done as follows :

where & l k -, 0 a s k -, m

.

Now l e t us e s t i m a t e t h e t e r m being s u m m e d . From t h e definitio of t h e algorithm we have

,

H i )

^{I c p i} _{( 1 9 )}

where C i s some positive c o n s t a n t . Assumption 2 of t h e t h e o r e m t o g e t h e r with (19) yields t h e following e s t i m a t e :

1

^f O(z,IP+')

-

^f O ( z , H i ) h ( H i , H l f 1 ) ^I C l p , leading t o

Here we also u s e d t h e f a c t t h a t t h e function f O ( z , H ) is bounded on s e t X x G

.

Combining ( 1 8 ) a n d ( 2 0 ) we g e t :

which implies

lk -I

a -

^E~~

C

P i >

i =mk K ^I Substituting ( 2 1 ) i n t o ( 1 7 ) yields

which c o n t r a d i c t s

inf + ( H )

> --

H E C since E~~ + 0

.

^{This again gives}

r,

^{-r 0.}

2(c) Proof for t h e case when t h e stepsize is chosen according t c (13) is similar t o 2(a) a n d 2(b).

This completes t h e proof.

Remark. Assumption 2 of t h e t h e o r e m can be easily s t a t e d without introducing t h e notions of weak ( s t a r ) topology and Levy-Prokhorov distance. One possible way is t o a s s u m e t h e following:

If O ( z . ~ r ) - f O(z,HZ)

1 ₁ J

A ( X . H ~ . H ~ ) d(H1(x) - Hz(.))

1

X

where

I

h(x.H1. Hz)

1 <

K

<

= for some positive K a n d H I E G

.

^{H Z E}

^G .

^{z E}

^X

and1 f O(X,H)\

<

C

<

^m for some positive C and H E G , x

E X .

In order t o obtain a practical method from t h e general framework described in t h i s section, we have t o specify ways of performing s t e p 2(i). This is t h e purpose of t h e n e x t section.

4. SOLVING THE

LWEAR

SUBPROBUCM USING CUTTINGPLANE TECHNIQUES

We shall now consider a m e t h o d for solving l i n e a r subproblem (9) which r e d u c e s s t e p 2(i) of algorithm 1 t o t h e solution of one finite-dimensional linear programming problem. This m e t h o d uses some of t h e s a m e ideas a s general- ized l i n e a r programming

[a],

cutting-plane algorithms [9], and h a s m u c h in common with t h e m e t h o d proposed in [?I for solution of linear problem (1)-(3).

The m e t h o d is based on t h e duality relations for problem (1)-(3), which were studied in [?I.

Let u s a s s u m e t h a t t h e assumptions of-Theorems 1 and 3 a r e fulfilled.

Then, according t o Theorem 1,

where

Suppose t h a t d s t r i b u t i o n

F

is fixed. Then i t is possible to solve t h e problem max p S ( u )

U E V +

with t h e help of t h e following cutting-plane method.

Algorithm 2

1. First select m + l points x l , x 2 ,

. . .

^, x m + l and set u = m + l . These points a r e used t o approximate function pS ( u ) by t h e function

m

p S ( u , O ) = min (f O ( X ~ , H ~ )

+ C

u i f i ( ~ j ) )

l s j s v i = l

The initial approximation u0 to t h e solution of problem ( 2 2 ) maximizes t h e function pS ( u ,0 ) :

u0

=

arg max pS ( u . 0 ) U E V +

so t h a t we have t o solve a linear programming problem.

2. Suppose t h a t before beginning iteration number k we have v points z 1 , z 2 ,

. . .

, z v and t h e c u r r e n t estimate of t h e minimum uk-'

.

Then itera- tion n u m b e r k involves t h e following stages:

(i) Take v

=

v + l ( i i ) Find

rn

z v

=

arg min ( f O ( Z , H S ) +

C

u i k - 1 f i ( Z ) )

t € X i = l

( i i i ) Calculate the next approximation t o t h e optimal solution u k + l :

uk

=

arg rnax p S ( u , k ) U E V +

where p S ( u , k ) is t h e c u r r e n t approximation of function p S ( u ) :

I t should be realized t h a t t h i s is only a general framev~~ork for so!uiion - m u c h h a s already been done t o avoid increasing t h e n u m b e r of points x i stored and t o implement approximate solutions of problem (23) (for details see [7], [ l o ] ) . The advantage of t h i s m e t h o d is t h a t it becomes possible t o obtain approxi- m a t e solutions of t h e initial problem (9) during t h e solution of problem (22).

These approximations a r e discrete distributions containing no more t h a n m + l points with positive probabilities:

where t h e

Fi

a r e nonzero solutions of t h e following linear programming prob- lem :

v

m i n

C

^{pi f *(xi,}

^HS

^{) ,}

?' i = ]

a n d t h e x i a r e t h e corresponding points. Note t h a t t h e above problem is a c t u - ally dual t o t h e linear program equivalent t o (24), a n d therefore both prob- l e m s c a n b e solved simultaneously.

What we actually n e e d while implementing algorithm 1 i s n o t a precise solution of problem (9) at e a c h s t e p , b u t r a t h e r t o t r a c k i t s changing e x t r e m u m value. The approximate solutions of (9) m a y be very rough for t h e first few iterations, gradually increasing in accuracy. I t appears t h a t algo- r i t h m 2 c a n be u s e d t o follow t h e e x t r e m u m value by tracking t h e changing optimal solution of dual problem (22). It i s only necessary t o m a k e one i t e r a - tion of algorithm 2 for e a c h i t e r a t i o n of algorithm 1.

In what follows we shall simplify the notation, writing f ' ( 2 , ~ ~ )

=

f:(z) .

We now want an algorithm which allows u s t o follo~-; t h e optimal solution of problem ( 2 2 ) a s t h e c u r r e n t distribution H S changes.

A l g o r i t h m 2a

1. First select m + l points z l , z 2 ,

. . .

^, zm+l and s e t v = m + l . The initial approx- imation u0 t o t h e solution of problem ( 2 2 ) maximizes t h e function pO(u,O) :

u 0

=

arg max cpO(u,~)

U E U + m

po(u,O) = min ( f : ( z j )

+ C

u i f i ( z j ) )

lsjsv i = l

2. Suppose t h a t before beginning iteration number s we have v points z 1 . z 2 ,

. . .

^, z Y and t h e c u r r e n t estimate of t h e minimum us-'

.

Then itera- tion n u m b e r s involves t h e following stages:

( i ) Take v

=

v+l ( i i ) Find

n

zY = arg min ( f : ( z ) + C uis-If i ( z ) ) z EX i =l

( i i i ) Calculate the next approximation t o t h e optimal solution us+' : us

=

arg max p S ( u , s )

U E V +

The following theorem proves t h e convergence of this method.

Theorem 4 Assume that:

1. S e t X is compact and functions f i(z) , i

=

T r n a r e continuous.

2. Functions f :(x) a r e conkinuous for x E X uniformly on s and

a s s + m .

3. There exists a y

>

0 such t h a t for any u E

u+,

^IIu

1 1 =

¹ t h e r e exists an i E 1,

...,

m + l j for which

Then

max pS ( u ) - p S ( u S ) ⁺ 0

U E V + Proof

1. We shall first prove t h a t sequence us is bounded. Take any point ZLE U + and estimate t h e value of pS ( u , s ) a t this point. Select i Ir rn +1 such t h a t

which from assumption 3 of t h e theorem will always be possible. Then

P " ( " , ~ )

"

^f:(xi)-yI

^{l " I I .}

(25)

The uniform continuity of functions f:(x) implies t h e existence of a constant C such t h a t

1

f:(x) ¹⁵ C

.

Combining this with ( 2 5 ) yields

p S ( ' l l * s ) ~ C - y I I

G O

and

These two inequalities lead t o

p S ( 0 , s ) - p ~ ( i i , s ) > y I 1'111

I

- 2 C

which implies t h a t t h e n o r m of a n y point u * which maximizes ~ + ~ ( u . , s ) is bounded, i.e.,

where c o n s t a n t s C a n d y do not depend on s

.

This proves t h a t s e q u e n c e us is bounded, because us maximizes rpS ( u , s )

.

2. Now suppose, arguing by contradiction, t h a t t h e t h e o r e m is not t r u e and t h a t t h e r e exists an a

>

0 a n d a sequence sk s u c h t h a t

m a x p S k ( u ) - p S k ( u S k )

>

a

.

U E V+

From t h e boundedness of sequence u s a n d assumptions 1 a n d 2 we m a y a s s u m e without loss of g e n e r a l i t y t h a t

where

vk =

sk

+

¹

+

m

+

¹

.

We shall now e s t i m a t e t h e difference

We have

for s o m e

K >

0 f r o m assumption 1. We also know t h a t pSk(u *) + p * , which t o g e t h e r with assumptions 1 a n d 2 implies t h a t

p S k * l ( u S k ) ⁺ p* a n d p S k ( u S k )

-

^p*

^.

Hence

p S k + l ( u S k ) ² p S k ( u S k )

-

c l ( k )

where c l ( k ) + 0 a s k ^{+ m.} We have a s s u m e d t h a t r n a ~ I f : + ~ ( z ) - f P ( z ) I ⁺

o

zEX

as s -+ and t h i s gives

sk + I

(pSk(usk) 2 $0 ( u S k ) - c 2 ( k ) where c Z ( k ) ⁺ 0 as k ^{-+ m.} But from t h e algorithm we have

where

vk =

^sk

+

¹

+

^m

+

1

.

From estimate ( 2 9 ) , we obtain:

m m

f ; + , ( z v k )

+ C

u i S k f i ( x v k ) 2 m i n [ f & ( 2 ' )

+ C

^uiSk f i ( x j ) ] 2

i = l l S j l ~ ~ + ~ - l a = I

min

[

f;+l ( z j )

+ 9

u i S k + ' f ( z j ) ] -

l s j s ~ ~ + ~ - l i = l

m a x min

[

f;+, ( x j )

+ 2

y f i ( x j ) ]

-

u ^E U+ I s j s v k + l - l i =l

K ]

I I

^uSk

-

^uSk+l

I I

² m a x ipSk+'(u)

-

K ]

I I

^{u S k}

-

^uSk+l

I I .

U E U + (30)

Combining (26)-(30) gives

m a x p S k + ' ( u ) - p S k + l ( u S k + l ) ^I K

I I

uSk

-

^uSk+l

I I +

U E V+

e l ( k )

+

e Z ( k )

+

^{cq(k )}

+

^K1

1 1

^uSk

-

^usk+l

I 1

^{I e5(k} )

where c 5 ( k ) + 0 as k + m

.

This contradicts t h e initial assumption

a n d t h u s completes t h e proof.

We shall now give an algorithm based on t h e r e s u l t s obtained in Sections 3 a n d 4. I t is a s s u m e d t h a t t h e conditions of Theorem 3 a r e m e t .

Algorithm 1 ^a

1. We begin by choosing a n initial distribution

H0

which satisfies assumption 3 of Theorem 4

.

Let x l , x 2 , . . . , x m + l be t h e m t l points which form t h e ini- tial distribution HO. Consider t h e following linear programming problem:

Assumption 3 of Theorem 1 is satisfied if and only if problem (31)-(33) h a s a solution .ELI , iiz ^,

. . .

, - s u c h t h a t urn+]

<

0

.

If t h i s is t h e c a s e , t h e solu- tion is t h e s a m e a s t h a t of t h e dual problem

where t h e optimal value of p m + ~ is less t h e n 0

.

The distribution

I?,

^where

R =

t ( z l , p l ) , ( x 2 , p z ) ,

. . .

^, ( z m + l Frn+l)j

a n d & i s t h e optimal solution of (34)-(36), h a s t h e property

J

^{f i} ( z ) d H ( x )

< o

X ⁽³⁷⁾

a n d t h e r e f o r e condition 2 of Theorem 1 is satisfied. The converse i s also t r u e (see [ 7 ] ) . Thus if condition 2 of Theorem 1 is fulfilled, i t is possible t o find m t l points z l , . . . , z m + l s u c h t h a t problem (34)-(36) h a s a solution jil ,

. . .

, pm+z - with

h+z ^<

⁰

^.

This g u a r a n t e e s t h a t condition 3 of Theorem 4

is fulfilled. These points, together with t h e probabilities

pi,

now form t h e desired initial distribution H O , which is a solution of t h e following problem :

min p H

Jf

~ ( z ) d H ( z ) r p , j =

i;;;;

X

This problem is of t h e form (1)-(3) and may be solved using algorithm 2 (see [7]). We do not need t o solve (38)-(40) exactly - we can stop when t h e c u r r e n t solution satisfies (37). This will occur after a finite number of iterations of algorithm 2 if condition 2 of Theorem 1 is satisfied. The initial s t e p of algo- r i t h m l a therefore involves the following stages:

(i) Take vo

=

m +l , where us is t h e number of points in distribution HS.

(ii) Obtain t h e initial distribution H1, where

by applying algorithm 2 to problem (38)-(40). This algorithm will produce a sequence of distributions

ES

which after substitution in problem (38)- (40) gives corresponding pS

.

Take as H1 t h e first

Bs

with p,

<

0.

(iii) Take t h e initial point u 1 E

U+

for solution of t h e dual problem.

2. Suppose t h a t before beginning iteration number s we have t h e c u r r e n t approximation t o optimal solution

HS

:

and point u s

.

Iteration number s t h e n involves t h e following operations, where steps (i)-(iii) correspond t o steps (i)-(iii) of algorithm 2a and s t e p (i) of algorithm 1, and steps (iv)-(v) correspond t o steps (ii)-(iii) of algorithm 1 :

(i) Take

J

= $

+

1

(ii) Find a new point xVst1 , where

z v s t ~

=

arg min

[f

'(x ,HS)

+ C

m ^uf

^f

^(x)]

z EX i=l

(iii) Solve the following linear programming problem :

a=l together with its dual:

This will give u s t h e next approximation t o t h e solution of t h e dual prob- lem, us+], a n d also vector

pS+'

:

g+l=

(p;+l

^,

.

^,

.

^, pv*+i -s+l )

which will have n o more t h a n m+1 nonzero elements, say ( ^,

. . .

^I _Pkmtl ^{-s + I ).}

(iv) Take t h e family of distributions

H"+l(a)

= I(

zl,p;+' (a))

. ^{. . .}

^{, (} ~ " + ~ . p ~ ~ + ~ ( a ) ) j where

, i f ( # kj for j = l , m + l ( l

-

^{a) +}

.

^otherwise

^.

Then find

a, =

arg rnin \II(HS+l(a))

(ka4 1

a n d take p, = minl a,

,fl,

j , where

(v) Take

HS+1 =

Hs +YP,

⁾

a n d go to s t e p 2(i).

Remark. It is not necessary to solve nonlinear programming problem (41) with g r e a t accuracy. All we need is a point zVs+' such t h a t

m

lim

i[

f O ( z V s + 1 , ~ )

+ C ut

f i ( Z v s + l ) ]

-

s +- i =I

m

min [f O ( Z , H ~ )

+ C

^uf f i ( z ) ] ] = 0 ,

z EX i = l

I t is also possible t o avoid increases in t h e dimension of linear programming problem (42)-(44) by considering only points zi which satisfy some additional inequality (see [7]).

Remark. Algorithm 2a adds one additional point to t h e c u r r e n t approximation of optimal solution

HS

a t each iteration, which may not be convenient if we have restrictions on t h e amount of memory available for storing t h e distribu- tion. In this case measures should be taken t o avoid this expansion, perhaps a t the expense of accuracy. Some possible ways of achieving this are discussed below.

1. Suppose we want t o find t h e best possible approximation, in no more than

N

points, t o t h e optimal solution of (5)-(7). (It is assumed t h a t some additional memory i s available for storing N further points.) We then proceed as follows:

(i) Run algorithm 2a until t h e c u r r e n t distribution HS contains 2N points.

Arrange t h e s e points in order of decreasing probabilities :

(ii) S t a r t algorithm 2a again from the distribution

(iii) Continue this process as long as t h e new ZN-point distribution has a b e t t e r value of + ( H ) t h a n t h e previous one.

2. Suppose t h a t we want t o find an approximation t o t h e optimal solution using a t m o s t N points.

(i) Run algorithm 2a until t h e c u r r e n t distribution contains N points. Let

- m a x p t . Divide t h e s e t 1 ,

. . .

, ZN i n t o two subsets:

=

^lsisN

where

x >

0 should be chosen previously.

(ii) S t a r t algorithm 2a again from distribution HI :

(iii) Continue t h i s process a s long as t h e value of *(H) in consecutive 2 N - point distributions improves a n d s e t

I2

is not empty.

3. Another possibility is t o u s e approximation t e c h n i q u e s t o fit d i s c r e t e distri-

butions by continuous ones. For example, splines c a n be used when t h e dimen- sions a r e small or when t h e distributions

HS

have independent components.

This approach needs f u r t h e r study.

The convergence of algorithm 2 follows directly from Theorems 3 a n d 4.

The n e x t r e s u l t m a y b e derived using t h e r e m a r k following Theorem 3.

Theorem 5

Let t h e following conditions be satisfied:

1. X is a compact set.

2. Functions f i ( z ) a r e continuous for z ^E A' , i

=

r m . 3. 9 ( H ) is convex and satisfies (8) and

If O ( z . ~ l ) - f O ( ~ , ~ Z ) I ~ I ~ X ( ~ . H I . H Z ) d ( H 1 - H Z ) \ X

f o r some summable ~ ( z ,-Y1,H2) such t h a t

I

A(z,H,,H,)

1 <

K

<

^m

for any H I , H Z E G

.

4. There exists an H ( x ) such t h a t

Then

lim

[+(PI -

min + ( H ) ]

=

0 s + = H E C

It is interesting to compare this algorithm with t h e methods for solving sto- chastic problems with recourse recently proposed by Wets [ l l ] . Although applied to quite different problems, they both use generalized linear program- ming techniques and on each iteration require solution of one linear program- ming problem and one nonlinear optimization problem.

More complex problems with nonlinear constraints can be t r e a t e d in t h e same way. If t h e definition of s e t G of constraints is changed to t h e following:

and functions \ k i ( H ) have directional derivatives of form ( 8 ) then analogues of Lemma 1 and Theorem 2 will hold

.

In t h i s case we can construct a feasible- direction type algorithm, which inherits all of t h e important characteristics of algorithm 1. The same ideas can be used t o solve minimax problems which

d e p e n d on d i s t r i b u t i o n f u n c t i o n s . This, t o g e t h e r x i t h t h e r e s u l t s of s o m e n u m e r i c a l e x p e r i m e n t s , will provide t h e s u b j e c t of a s u b s e q u e n t p a p e r .

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3. A. Golodnikov, "Optimization problems with distribution functions", Disser- tation, V. Glushkov Institute for Cybernetics, Ukrainian Academy of Sci- ences, Kiev (1979).

4. P.J. Huber, Robust S a t i s t i c s , John Wiley and Sons (1981).

5. M.G. Krein, "The ideas of P.L. Tchebysheff and A A Markov in t h e theory of limiting values of integrals and their further developments", Am. Math.

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8. G.B. Dantzig, L i n e a r P r o g r a m m i n g a n d E x t e n s i o n s , P r i n c e t o n University P r e s s (1963).

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