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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

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The p r e c e d i n g i n c l u s i o n follows from ( 3 . 1 5 ) will and ( 3 . 3 3 ) be sin- 46 ce they imply f ( x ) < ( e p i - l i m f v ) ( x ) ~< lim i n f f (x ~) yEN yEN < l i m sup f (x ~) yEN < l i m sup ( i n f f +e) < i n f yEN v To prove the second a s s e r t i o n , l e t us f i r s t f + c . assume t h a t lim ( i n f fv) = i n f f . 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.

XER n For an a r b i t r a r y subset D of inf D f : = inf Rn , we w r i t e f(x) . xCDCR n The infinimum of f on bounded below) o r even D , inf D f (if , may be a real number, o r -~ (if f i s not D N dom f = @). The s e t o f p o i n t s t h a t minimize denoted by argmin f : = {x E R n l f ( x ) < i n f f <~} ; f is 41 thus in p a r t i c u l a r argmin f = ~ if dom f = ~ . This convention i s d i c t a t e d by the d e s i r e t o have s o l u t i o n s in the m i n i m i z a t i o n f o r which dom f argmin f is the s e t o f feasible s o l u t i o n .

When the m u l t i f u n c t i o n o t h e r r e l a t e d cases, the d e f i n i t i o n what i s e x p e c t e d . However, i f w e l l not be a l l function on which i t i s u~iformly F of multifunctions b e f o r e we is u n i f o r m l y bounded, or in a number o f of continuity i n t r o d u c e d here i s c o n s i s t e n t w i t h is not u n i f o r m l y bounded then c o n t i n u i t y may v e r y what we e x p e c t from such a concept. Consider f o r example the m u l t i - F : R ~ R with Y F(u) = ~lu-ll} L It is easy t o v e r i f y if u # 0 , if u=O .

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