Multifunctions and Integrands: Stochastic Analysis, by R. T. Rockafellar, Roger J -B. Wets (auth.), Gabriella

By R. T. Rockafellar, Roger J -B. Wets (auth.), Gabriella Salinetti (eds.)

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Additional resources for Multifunctions and Integrands: Stochastic Analysis, Approximation and Optimization Proceedings of a Conference held in Catania, Italy, June 7–16, 1983

Example text

YEN In view o f the above, f o r a l l c >0 l i m i n f (e-argmin fv) c lim sup (c-argmin fv) c e-argmin f . 13) with ( 3 . 1 2 ) - - t h a t there e x i s t s N' c ~ and of epi-convergence --combi{x ~ ,v E ( N ' , ~ ) } such that x = l i m x~ ~EN' If, and f o r some ( f i l t e r e d ) lim f (x v) = f ( x ) yEN' collection done. Otherwise, there e x i s t s f (x v) > i n f Taking l i m i t s on H' f . ~ ~0 , we have t h a t H'E~ x~ E ~ -argmin fv ' we are such t h a t f o r some c' > 0 + e' (with ~ r e s t r i c t e d to H' ), we obtain and a l l ~ E H' , 47 f(x) = l i m f (x ~) > E ' ~EH' + l i m ( i n f f ) = e' + min f > f ( x ) ~@H' a clear contradiction.

But nonetheclose to and at F(-u') 0 0 , are is concer- 29 r,y / r'(u) = {u - I } Figure 6. Continuous multifunctions. ned between F and F' . In many a p p l i c a t i o n s , between the d i r e c t i o n s o f r e c e s s i o n , i . e . lues of the m u l t i f u n c t i o n . it is i m p e r a t i v e to d i s t i n g u i s h the asymptotic behavior a t ~ o f the va- This leads us to a more s t r i n g e n t n o t i o n o f c o n t i n u i t y f o r m u l t i f u n c t i o n s r e q u i r i n g in a d d i t i o n " c o n t i n u i t y " o f the d i r e c t i o n s o f r e c e s s i o n ; t h i s i s e l a b o r a t e d in R o c k a f e l l a r and Wets, 1985.

COROLLARY. u "min f " in place , and speak o f mi~Lmwm in place o f infimum, as an i n d i c a t i o n t h a t the infimum i s a c t u a l l y a t t a i n e d a t some p o i n t nuo~ at . x . We s h a l l a l s o have recourse to t h i s on the e x i s t e n c e o f a minimum. : Rd # Rn . Then Consider a v ~ t i a t i o n a l system F i s epiconti- i f and only i f l i m sup ( i n f G fu, ) < i n f G fu u' ~ u for a t l open lim inf for a l l compact K c Rn . sc. 27) 42 Indeed, in the space of epigraphs {E • E these open sets correspond to E IE n (G X(-~, a ' ) ) # @ , G open, a' E R} and {E E E IE cl (K X ( - ~ , a ] ) = ~ , K compact, a E R} .

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