We show that the operators E^G (conditional expectation given a tau-field G) and partial (subdifferentiation), when applied to a normal convex integrand f, commute if the effective domain multifunction omega -> {x E R^n | f(omega , x) < +infinity } is G-measurable

Rockafellar, R.T.

Wets, R.J.-B.

International Institute for Applied Systems Analysis (IIASA)

On the Interchange of Subdifferentiation and Conditional Expectation for Convex FunctionalsRockafellar, R.T. and Wets, R.J.-B.IIASA Working PaperWP-81-089July 1981 Rockafellar, R.T. and Wets, R.J.-B. (1981) On the Interchange of Subdifferentiation and Conditional Expectation for Convex Functionals. IIASA Working Paper. WP-81-089 Copyright © 1981 by the author(s). http://pure.iiasa.ac.at/1672/ Working Papers on work of the International Institute for Applied Systems Analysis receive only limited review. Views or opinions expressed herein do not necessarily represent those of the Institute, its National Member Organizations, or other organizations supporting the work. All rights reserved. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage. All copies must bear this notice and the full citation on the first page. For other purposes, to republish, to post on servers or to redistribute to lists, permission must be sought by contacting repository@iiasa.ac.at NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR ON THE INTERCHANGE OF SUBDIFFERENTIATION AND CONDITIONAL EXPECTATION FOR CONVEX FUNCTIONALS R.T. Rockafellar R. J-B. Wets July 1981 WP-81-89 V o r k i n g  Papers are interim reports on work of the International Institute for Applied Systems Analysis and have received only limited review. Views or opinions expressed herein do not necessarily repre- sent those of the Institute or of its National Member Organizations. INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS A-2361 Laxenburg, Austria R.T .  R o c k a f e l l a r ' s  r e s e a r c h  was s u p p o r t e d  i n  p a r t  by  t h e  A i r  F o r c e  O f f i c e  o f  S c i e n t i f i c  Resea rch ,  A i r  F o r c e  Sys tems Command, USAF u n d e r  g r a n t  no. 77-3204.  R. J - B .  W e t s '  r e s e a r c h  w a s  s u p p o r t e d  i n  p a r t  by a g r a n t  o f  t h e  N a t i o n a l  S c i e n c e  F o u n d a t i o n .  ABSTRACT We show that the operators E~ (conditional expectation given a T-field G) and 3 (subdifferentiation), when applied to a normal convex integrand f, commute if the effective domain multifunction o + {x E R ~ (  f o,x) < +a1 is G-measurable. ON THE INTERCHANGE OF SUBDIFFERENTIATION AND CONDITIONAL EXPECTATION FOR CONVEX FUNCTIONALS R.T. Rockafellar and R. J-B. Wets We deal with interchange of conditional expectation and subdifferentiation in the context of stochastic convex analysis. The purpose is to give a condition that allows the commuting of these two operators when applied to convex integral functionals. Let (R,A,P) be a probability space, G a T-field contained in A, and f an A-normal convex integrand defined on R x R" with values in R ~ { w ) .  The latter means that the map is a closed-convex-valued A-measurable multifunction. See [2] and [9] for more on normal integrands and their properties. In particular recall that for any A-measurable function x: R + R ~ ,  the function is a A-measurable and the integral functional associated with f is defined by To bypass some trivialities we impose the following summability conditions: (1) there exists a G-measurable x:R + Rn such that If(x) is finite, 1 1 (2) there exists V E  Ln(G) = L (R,G,P;R~) such that If, (v) is finite, * where f is the (A-normal) conjugate convex integrand, i.e. * f (w,x) = sup [v'x-f(w,x)I . XER" Finally, we assume that A --and hence also G --is countably gen- erated, and that there exists a r e g u l a r  conditional probability (given G )  , pG: A x R + [O, 1 1  . Whenever we refer to the conditional expectation given G, we always - mean the version obtained by in- G tegrating with respect to P . Consequently all conditional expectations will be regular. G In particular the conditional expectation E f of f is the G-normal integrand defined by + n Also given I.:R+R , a closed-valued A-measurable multifunction, its conditional expectation given G is a closed-valued G-measur- able multifunction obtained via a projection-type operation from a set 1 1 onto Ln (G) = L (a, G,P;R") . Valadier has shown that a regular G version E I.:R Rn is given by the expression We refer to [12] and the references given therein for the prop- G erties of E f; in particular to the article of Dynkin and Estigneev [3], which specifically deals with regular conditional expectations of measurable multifunctions. We consider If and I as (integral) functionals on 1, (A) E f and L;(G) respectively. The natural pairings of Lm with L1 and m * (1 ) yield for each functional two different subgradient multi- functions. We shall use aIf and 31 for designating 1 -sub- * * Emf * gradients and a If and a I for ( L ) -subgradients. Rockafellar E f [8, Corollary 1B] shows that when the summability conditions (1) and (2) are satisfied, one has the following representation for m * (1 ) -subgradients: * ( 3 )  a I~ (x) = {v+v~ J V E ~ I ~  (x) .V,ES~ (A) with vs [x-XI] -> O  V X ' E ~ O ~  I 1f where S (A) is the space of singular continuous linear functionals n on L~(A), and m dom If = {x E L, (A) I I~ (x) < * is the effective domain of If. (For the decomposition of (Lm) 4 n consult [2, Chapter VIII I ) . Furthermore the L ' -subgradient set is given by The summability conditions (1 ) and (2) on f imply similar prop- G erties for E f, so the formulas above also apply to I . Thus for X E  L,(G) we get ~~f with us [x-x ' 1 - > 0 . Vx ' E dom I } and ~~f W e  a r e  i n t e r e s t e d  i n  t h e  r e l a t i o n s h i p  be tween  a I f  and  3 1  . R e l y i n g  on  t h e  f o r m u l a s  j u s t  g i v e n ,  C a s t a i n g  and  V a l a d i e r  E f [ 2 ,  Theorem VI I I . 371  show t h a t  i f  i n  p l a c e  o f  t h e  s u m a b i l i t y  c o n d i t i o n s  ( 1 )  and ( 2 )  , o n e  makes t h e  s t r o n g e r  a s s u m p t i o n :  ( 7 )  t h e r e  e x i s t s  x O  E  L ~ ( G )  a t  which  I i s  f i n i t e  and  norm f  c o n t i n u o u s ,  t h e n  f o r  e v e r y  x  E L: ( G )  o n e  g e t s  : where  rc d e n o t e s  t h e  r e c e s s i o n  ( o r  a s y m p t o t i c )  c o n e  [ 2 , 7 ] .  I f  x E i n t  dom I , 3 1  ( x )  i s  weakly  compact  and  t h e n  r c [ a 1  ( x ) ]  = E  f  E  f  E  f  ( 0 1 ,  i n  which  c a s e  T h i s  was a l r e a d y  o b s e r v e d  by  Bismut  [ I ,  Theorem 41. F o r  t h e  s u b s p a c e  o f  L: o f  c o n s t a n t  f u n c t i o n s ,  H i r i a r t - U r r u t y  [ 4 ]  o b t a i n s  a  s i m i l a r  r e s u l t  f o r  t h e  & - s u b d i f f e r e n t i a l s  o f  convex  f u n c t i o n s .  Here w e  s h a l l  g o  one  s t e p  f u r t h e r  and p r o v i d e  a  c o n d i t i o n  u n d e r  which  t h e  rc  t e r m  c a n  be  d r o p p e d  from t h e  i d e n t i t y  w i t h o u t  r e q u i r i n g  t h a t  x ~ i n t  dom I f .  Very s i m p l e  examples  show t h a t  t h e  rc  t e r m  i s  somet imes  i n e s c a p a b l e  i n  ( 8 ) .  F o r  i n s t a n c e ,  G s u p p o s e  G = {+, i l l  ( s o  E  = E) and  c o n s i d e r  £ ( a , * )  = $ ( - m , E ( w ) l ,  . . - .  . -  t h e  i n d i c a t o r  o f  t h e  unbounded i n t e r v a l  ( -m ,E(w) ] ,  where  5 i s  a  random v a r i a b l e  u n i f o r m l y  d i s t r i b u t e d  o n  [ 0 , 1 ] .  I n  t h i s  c a s e  - b = E f  = E f = I  G $ ( - , I  01 SO t h a t  a 1  ( 0 )  = R+ b u t  E ( a I f ( o ) )  = EGf E f  -E { O )  = ( 0 ) .  Thus ( 8 )  would f a i l  w i t h o u t  t h e  rc  t e r m .  THEOREM. Suppose f  i s  a n  A-normal convex integrand such t h a t  t h e  closure of its effective d o m a i n  muZtifunction ( 1  0 )  u ~ D ( u ) :  = c l  dam f ( w , . )  = c l  { X E R " ~ ~ ( U , X )  < + . . I  i s  G-measurable. Assume t h a t  I f ( x )  < +m f o r  e v e r y  x  E L ~ ( G )  such t h a t  x ( w )  Edom f  ( w , . )  a . s . ,  and  t h a t  t h e r e  e x i s t s  x0  E L ~ ( G )  a t  which If i s  f i n i t e  and norm c o n t i n u o u s .  Then f o r  eve ry  x  E  L ~ ( G )  one has  o r  i n  o t h e r  words, t h e  c l o s e d - v a l u e d  G-measurable m u l t i -  f u n c t i o n s  and G W E  [ a f c . , x c - ) )  I a r e  a l m o s t  s u r e l y  e q u a l .  Proof .  From ( 8 )  it f o l l o w s  t h a t  G a I G  ( X I C E  ( a I f ( x ) )  . E f  I n  v iew of  ( 6 )  and ( 4 )  t h i s  h o l d s  i f  and o n l y  i f  I t  t h u s  s u f f i c e s  t o  p r o v e  t h e  r e v e r s e  i n c l u s i o n .  L e t  u s  suppose  G t h a t  U E ~ E  f ( - , x ( * ) ) .  For  e v e r y  ~ E R " ,  d e f i n e  T h i s  i s  a n  A-normal convex i n t e g r a n d  which i n h e r i t s  a l l  t h e  p r o p e r t i e s  assumed f o r  f  i n  t h e  Theorem ( r e c a l l  t h a t  u E  L ; ( G ) ) .  G G Moreover OEaE g ( * , x ( * ) ) .  W e  s h a l l  show t h a t  OEE a g ( * , x ( = ) ) ,  G which i n  t u r n  w i l l  imply t h a t  u  E  E af ( -  , x  ( ) ) and t h e r e b y  com- p l e t e  t h e  p roof  o f  t h e  Theorem. C S i n c e  a l m o s t  s u r e l y  0  E aE ' g  ( u , x  (u) ) , w e  know t h a t  * 0  E  a 1  ( x )  C a I I: ( X I .  Hence x  min imizes  I on  L ~ ( G ) .  ~ e t  E Gg E g  E Gg inj denote the natural injection of L ~ ( G )  into L,(A) with Now note that inj ; = x also minimizes I on W C L;(A), or equi- E g valently I on W, since the two integral functionals coincide on W (by the definition of conditional expectation.) Thus where OW is the indicator function of W, or equivalently: since g is (norm) continuous at some x0 = inj x0 E W. By (3) , this 1 means that there exist V E  Ln(A), vsESn(A), such that (13) vS[x -x11 - > 0 for all x' ~ d o m  I t 9 and -(v+vS) is orthogonal to W, i.e. This last relation can also be expressed as (v+vs) [inj yl = 0 for alyEL,(G) t or still for all y E 1, (GI * inj (v + vs) [yl = 0 , * * * where inj : A + ( L: (G) ) is the adjoint of inj. Thus the * continuous linear functional inj ( v + v  ) must be identically 0 S on L,(G), i.e. on L,(G) one has G * The last equality follows from the observation that E = inj * 1 when inj is restricted to Ln(A), cf. [2, p.2651 for example. We shall complete the proof by showing that the assumptions (121, (13) and (15) imply that This will certainly do, since it trivially yields the sought-for relation To obtain (1 6) , it will be sufficient to show that for all y Edom I C L:(A). To see this, recall that the relations g G (17) and ~ € 3 1  (x) (cf. (12)) imply that v - E v E a I  (x), from g g which (1 6) follows via the representation of L' -subgradients given by (4). In fact, because the effective domain multifunc- tion, or more precisely its closure wbD(w), is G-measurable, it is sufficient to show that (17) holds for every yEdom I nu. 9 Suppose to the contrary that (17) holds for every yEdom I nW -- 9 or equivalently because of the - < inequality that (17) holds for 1 every y E c l  dom I nW --but there exists $ E  Ln(A) such that 9 Ig(Y) < +m and for which (17) fails, i.e. we have G Because -E v and x are G-measurable, this inequality implies that Moreover, since I ( < +m, it follows that almost surely g Taking conditional expectation on both sides, we see that because D is a closed-valued G-measurable multifunction. Natur- ally E W .  Because I is by assumption finite on {z E L; (G) ( 9 z(w) Edom g(w,.) a.s.1, and D(w) = cl dom g(w,-), it follows from G (19) that E yEcl dom I . Hence (17) cannot hold for every 9 y Edom I nW since E ~ $  belongs to (cl dorn I ) nW and satisfies (1 8) . g 9 There remains only to show that (1 7) holds for every y E L:(G) such that inj y = yEdom I . But now from (13) we have that for 9 each such y vs[x-y] = vS[inj x-inj yl > 0 t - or again equivalently: for each y Edom I ~L;(G), 9 * (inj vs) [x-y] - > 0 . But this is precisely (1 71, since we know from (1 5) that on L;(G), * G inj v s =  -Ev. COROLLARY. Suppose  f i s  a  A-normal c o n v e x  i n t e g r a n d  s u c h  t h a t  F(x) c +m whenever  x Edom f(w,*) a.s., where Suppose  m o r e o v e r  t h a t  t h e r e  e x i s t s  x0 ERn a t  t ' h i c h  F i s  f i n i t e  and c o n t i n u o u s ,  and t h a t  t h e  m u l t i f u n c t i o n  W ~ ~ ( ~ )  = c1 dom f(w,*) i s  a l m o s t  s u r e l y  c o n s t a n t .  Then f o r  a l l  XER", where t h e  e x p e c t a t i o n  o f  t h e  c l o s e d - v a l u e d  m e a s u r a b l e  m u l t i -  f u n c t i o n  I' i s  d e f i n e d  by PROOF. Just apply the Theorem with G = {$,R), and identify the class of constant functions --the G-measurable functions -- with R". This Corollary was first derived by Ioffe and Tikhomirov [5] and later generalized by Levin [6]. Note that our definition of the expectation of a closed-valued measurable multifunction is at variance with the definition now in vogue for the integral of a measurable multifunction, which does not involve the closure operation. (Otherwise the definition of the integral of a multi- function would be inconsistent with that of its conditional ex- pectation, in particular with respect to G = {@,R)r and also when r+Er is viewed as an integral on a space of closed sets it could generate an element that it is not an element of that space.) APPLICATION Consider the s t o c h a s t i c  o p t i m i z a t i o n  p r o b l e m :  (21) find inf E[f(wIxl (w) .x2 (w) ) 1 over all x  EL^ (G) , x  EL^ (A) , nl "2 where A and G are as before, and f is an A-normal convex inte- grand which satisfies the norm-continuity condition: (22) 0 0 there exists (xl ,x2) E L: (G) x (A) 1 "2 at which If is finite and norm continuous. Suppose also that the effective domain multifunction n "2 w +dom f(w,*.-) = { (xl ,x2) E R x R I f(w.xl ,x2) < +rn) is uniformly bounded and that there exists a summable function 1 h E L (A) such that (x, ,x2) E dom f (w. - )  implies that I f (u1,x1 ,x2)I 2 h(l~). Finally suppose that the multifunction "1 n2 w * Dl ( w )  = cl {xl E R (3x2 E R such that f (w,xl ,x2) < +m} is G-measurable. For a justification and discussion of these assumptions cf. ['I 1, Section 21. From Theorem 1 of [1 1 1 ,  it follows that the problem ( 2 3 )  find inf E lg (w,xl (w) ) I over all xl E 1; (G) 1 where is equivalent to (21) in the sense that if (XI ,E2) solves (21), then xl solves (23), and similarly any solution xl of (23) can be "extended" to a solution (xl 'X2 ) of (21 ) . Both problems also have the same optimal value. The hypotheses imply that is an A-normal convex integrand, since the multifunction wbepi(inf f(w,xl,x2)) is closed-convex-valued and A-measurable. X Its effecti4e domain multifunction, or more precisely is G-measurable. Combining (11) with the representation for the subgradients of infimal functions [13, VIII.41, we have that for every xl E l m  (G) "1 q 1  ( 1  = ~?v(w) I (v(w) ,o) af(w,xl (w) ,x2) for some x2 E Rn2 I ( ) , from which Theorem 2, the main result of [11], follows directly. REMARK. If the underlying probability measure P has finite w *  and (1 1) and (20) are satisfied with- support, then (Ln) = Lnt out any other restriction. On the other hand, if P is nonatomic, and the effective domain multifunction (or its closure) is not G-measurable, then the identities (1 1) and (20) do not apply. More precisely, suppose that there exists a subset C of R" such that the A- measurable set has (strictly) positive mass and is not G-measurable. Then the term rc[aI (x)] can never be dropped from the representation E f of a1 given by ( a ) ,  as can be seen from an adaptation of the E f arguments in Section 4 of [lo]. In those cases the inclusion G G E af C aE f will be strict for at least some x E L;(G). REFERENCES Bismut, J.-M. Int4grales convexes et probabilitgs. J .  M a t h e m a t i c a l  A n a l y s i s  and A p p l i c a t i o n s ,  42(1973): 639-673. Castaing, C. and M. Valadier. Convex A n a l y s i s  and Measure-  a b l e  M u l t i f u n c t i o n s .  Springer-Verlag Lecture Notes in Mathematics 580. Berlin, 1977. Dynkin, E.B. and I.V. Estigneev. Regular conditional expectations of correspondences. T h e o r y  o f  Prob- a b i l i t y  and i t s  A p p l i c a t i o n s .  21(1976):325-338. Hiriart-Urruty, J.-B. About properties of the mean value functional and the continuous inf-convolution in stochastic convex analysis. In O p t i m i z a t i o n  T ech -  n i q u e s  Mode l ing  and O p t i m i z a t i o n  i n  t h e  S e r v i c e  o f  Man. Ed. J. Cea, Springer-Verlag Lecture Notes in Computer Science. Berlin, 1976, 763-789. Ioffe, A.D. and V.M. Tikhomirov. On the minimization of integral functionals. Fun.?. A n a l i z .  3 (1 969) :61-70. Levin, V.L. On the subdifferentiability of convex func- tional~. U s p e k h i  Mat. Nauk. 25 (1970) : 183-184. Rockafellar, R.T. Convex A n a l y s i s .  Princeton University Press, Princeton, 1970. Rockafellar, R.T. Integrals which are convex functionals, 11, P a c i f i c  J. M a t h e m a t i c s ,  39 (1 971) :439-469. (91 Rockafellar, R.T. Integral functionals, normal integrands and measurable selections. In Nonlinear Operators and the Calculus of Variations. Springer-Verlag Lecture Notes in Mathematics 543, Berlin, 1976, 157-207. [lo] Rockafellar, R.T. and R. Wets. Stochastic convex pro- gramming: relatively complete recourse and induced feasibility. SIAM J. Control Optimization, 14 (1 976), 574-589. 1 [11] Rockafellar, R.T. and R. Wets. Nonanticipativity and L - martingales in stochastic optimization problems. Mathematical Programming Study, 6 (1 976) , 170-1 87. [I21 Valadier, M. Sur l'espgrance conditionnelle multivoque non convexe. Afin. Inst. Henri PoincarG, 16 (198O), 109-116. [13] Wets, R. J-B. Grundlage konvexer Optirnierung. Springer- Verlag Lecture Notes in Economics and Mathematical Systems 137, Berlin, 1976. 

On the Interchange of Subdifferentiation and Conditional Expectation for Convex Functionals

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