Applied Mathematics and Optimization 4, 139-142 (1978) Applied Mathematics and Optimization

Optimal Selectors for Stochastic Linear Programs

Wolf-Rtidiger H e i l m a n n

Institut ffir Mathematische Stochastik der Universitat Hamburg Bundesstrasse 55, 2000 Hamburg 13

Communicated by J. Stoer

Abstract. The "distribution problem" of stochastic linear programming consists in answering the following two questions: Is the optimal value of a given stochastic linear program--regarded as a function--measurable, and if so, what is its distribution?

In the present note we make use of a general selection theorem to answer the first question positively. By this approach, an answer to the second question is obtained--at least theoretically--at the same time.

1. Introduction

If in a linear program

cx = Max! A x >1 b x >10

the coefficients A, b, and c are no constants but depend on a parameter 0~, it is meaningful to introduce the conception of selector in this connexion: a selector is a mapping appointing to every possible realization ( A , b , c ) an x which is feasible for the linear program constituted by it.

Probabilistic considerations arise if, moreover, the parameter space f2 is supplied with some o-algebra on which a probability measure is defined.

In the present note, we first make this concept precise and then define an obvious notion of optimality for selectors. Finally, we use a well known selection theorem to show that optimal selectors exist under some assumption. Thus, we obtain a direct approach to the so-called distribution problem of stochastic linear programming.

0095-4616/78/0004-0139 $01.00 ©1978 by Springer-Verlag New York, Inc.

140 W.-R. Heilmann

2. Preliminaries

Let (£, ~ , P ) be a probability space, A:~-->RmX',b:~-->R m, and c:~--*R" be (measurable) random vectors. Thus, for fixed ~0 E £ A(w) can be regarded as an m x n-matrix, b(w) as an m-column vector, and c(~0) as an n-row vector. By this, we obtain a family of linear programs

c(¢o)x = Max! A(~o)x >i b(~o) x >i 0

parametrized by o~ ~ £. Let the range S of the random vector s := (A ,b , c ) be an element of the

o-algebra ~mx,+,~+, of Borel sets in R mx"÷m÷", and let it be supplied with the trace o-algebra $ : = S n 6~mZn+m+n. If s=(A,b,c) is a realization of the ran- dom vector s, let

D ( s ) : = i x ~ R n : A x >/ b , x >/O},

and

r ( s ,x ) :=cx , x ~ R ' .

A measurable map f : S---~R n with f ( s ) ~ D (s) for all s E S is called selector. A selector f * is optimal, if

r (s , f*(s ) )= sup r(s,x) f o r a l l s E S x@D(s)

holds. We shall need the following corollary of a well known selection theorem.

Theorem 1. Let E and H be metric spaces, and let @ or ~ be the o-algebras of Borel sets in E or H, respectively. For every e @ E let G (e) be a nonempO~ compact subset of H, and let u :EXH--~R be continuous. I f the mapping e---~G(e) is separable, i.e. the conditions

(i) ( ( e , h ) : h ~ G ( e ) ) c @ ® ~ , and

(ii) H contains a denumerable dense subset H ' such that H' N G (e) is dense in G(e) for all eEE ,

are fulfilled, then there exists an optimal selector, i.e. a measurable map f : E---> H with f ( e ) ~ G (e) for all e E E and u(e,f(e))= suph ~C(e)U(e, h)for all e E E.

Proof All conditions of Theorem 2 in [2] are fulfilled. [ ] If an optimal selector f * for our stochastic linear program exists, it is--at

least theoretically--possible to solve the "distribution problem" (cf. [1], e.g.), i.e. to determine the distribution of the random vector

f * o ( A , b , c )

Optimal Selectors For Stochastic Linear Programs 141

and thus the distribution of the random variable

cf*o(A,b,c)

and its moments, quantiles etc. For the rest of the paper, the following assumption is made.

VD (s) is bounded and n-dimensional for all s E S.

3. Results

We shall make use of the following

Lemma 2. Let the set Y ' be dense in Y : = R p. Furthermore, p-dimensional compact convex set. Then Y' fq K is dense in K.

Proof The proof is trivial and hence omitted. [] As a consequence of Lemma 2 and V we now obtain

Lemma 3, The set-valued mapping s---~D (s) is separable.

Proof We have to prove the properties (i) and (ii) formulated in Theorem 1. (i) The mappings

• i: ( (A ,b ,c ) , x ) - -~(Ax) i , i= 1 . . . . ,m,

qJ,: ((A,b,c),x)---~b i, i= 1 . . . . . m,

• j : ( (A,b,c) ,x)---~xj , j = l . . . . . n,

where by i or j the ith component of the respective vector is characterized, are measurable. Hence the set

{ ( S , X ) : f ~ i ( ( S , X ) ) ) I ~ i ( ( S , X ) ) , i~---1 . . . . . m, "rj(x)>~O, j = l , . . . , n }

is measurable, too. (ii) Choose Q" (the set of n-tuples of rationals) as separating set, and use

Lemma 2 and V. The following theorem is now immediate.

Theorem 4 (cf. [1], p. 22). There exists an optimal selector.

Proof Obviously, all conditions of Theorem 1 are fulfilled. [] From Theorem 4, we directly conclude

Theorem 5. The mappings

6: 0~---~sup(e(o~)x : A(~o)x i> b(~o), x >i 0)

and

7 : (A ,b , c ) - -+sup(cx :Ax>~b , x>~O)

are measurable.

let K C Y be a

142

Proof. Let f * be an optimal selector. Then we have

~=cf*os,

7((A,b,c))--cf*((A,b,c)), (A,b,c)~S.

References

1. P. KALL, Stochastic Linear Programming, Springer-Verlag, Berlin, 1976. 2.

W.-R. Heilmann

[]

M. SCHOOL, A Selection Theorem for Optimization Problems. Archiv der Mathematik, XXV (1974) 219-224.

Received October 3, 1977