Generalized-differentiable functions - Springer Link

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In practice generalized-differentiable functions appear, in particular, when the ... The function F : R n -- R ~ is generalized-differentiable if there exists an upper ...
GE NERA LIZ ED-DIFFERE V. I .

NTIABLE

FUNCTIONS

Norkin

UDC 517.51

S o m e o p t i m i z a t i o n p r o b l e m s involve n o n d i f f e r e n t i a b l e o b j e c t i v e f u n c t i o n s and n o n d i f f e r e n t i a b l e c o n s t r a i n t s . V a r i o u s c l a s s e s of s u c h functions a r e c o n s i d e r e d in m a t h e m a t i c a l p r o g r a m m i n g , e . g . , c o n v e x and c o n c a v e , w e a k l y c o n v e x and w e a k l y c o n c a v e [1], s e m i s m o o t h [2], q u a s i d i f f e r e n t i a b l e [3], a l m o s t d i f f e r e n t i a b l e [4], l o c a l l y L i p s c h i t z [5], e t c . In this p a p e r we i n v e s t i g a t e the g e n e r a l i z e d - d i f f e r e n t i a b l e (GD) f u n c t i o n s , w h i c h c o n s t i t u t e a n a t u r a l g e n e r a l i z a t i o n of the c o n t i n u o u s l y d i f f e r e n t i a b l e f u n c t i o n s . The c l a s s of GD f u n c t i o n s c o n rains c o n t i n u o u s l y d i f f e r e n t i a b l e , c o n v e x and c o n c a v e , w e a k l y c o n v e x a n d w e a k l y c o n c a v e [1], and s e m i s m o o t h [2] f u n c t i o n s . It is c l o s e d u n d e r the finite o p e r a t i o n s of m a x i m u m , m i n i m u m , and s u p e r p o s i t i o n . The g e n e r a l i z e d g r a d i e n t s of GD f u n c t i o n s e x i s t and s i m p l e f o r m u l a s a r e a v a i l a b l e f o r t h e i r c a l c u l a t i o n s . GD f u n c t i o n s s a t i s f y the l o c a l L i p s c h i t z c o n d i t i o n and a g e n e r a l i z a t i o n of the L a g r a n g e f i n i t e - i n c r e m e n t t h e o r e m . N e c e s s a r y c o n d i t i o n s for a n e x t r e m u m c a n be d e r i v e d f o r GD f u n c t i o n s . I n p r a c t i c e g e n e r a l i z e d - d i f f e r e n t i a b l e f u n c t i o n s a p p e a r , i n p a r t i c u l a r , w h e n the a c t u a l d e p e n d e n c e is a p p r o x i m a t e d by a p i e c e w i s e - l i n e a r f u n c t i o n o r w h e n s o m e c o m p o s i t e o b j e c t i v e f u n c t i o n c o n t a i n s the o p e r a t i o n s of m a x i m u m , m i n i m u m , and a b s o l u t e v a l u e on s m o o t h f u n c t i o n s . T h e r e f o r e , nonconvex GD f u n c t i o n s o c c u r , e . g . , i n the o p t i m a l p o w e r g r i d p r o b l e m [4, p. 497], in the o p t i m a l g a s - m a i n s flow d i s t r i b u t i o n p r o b l e m [7, 8], and in the r a t i o n a l c o a l i t i o n s t r a t e g y p r o b l e m [9, p. 63]. D e f i n i t i o n [10]. The f u n c t i o n F : R n - - R ~ is g e n e r a l i z e d - d i f f e r e n t i a b l e if t h e r e e x i s t s a n u p p e r s e m i c o n t i n u o u s p o i n t - s e t m a p p i n g G(F) : x ~ R n ~ G ( F , x) c R n s u c h that the s e t s G ( F , x) a r e b o u n d e d , c o n v e x , a n d c l o s e d , and a t e v e r y point x 6 R n the f o l l o w i n g d e c o m p o s i t i o n h o l d s : F (y) -~ F (x) + (g (y), y - - x) + o (x, y, g), w h e r e o(x, y , g ) / I y - x l ~ 0 u n i f o r m l y in y - - x and g e G (F, y). g r a d i e n t s of F a t the point x.

(1)

The v e c t o r s g E G (F, x) a r e the g e n e r a l i z e d

W e have the f o l l o w i n g p r o p o s i t i o n s . 1. G e n e r a l i z e d - d i f f e r e n t i a b l e functions a r e c o n t i n u o u s . 2. U n i q u e n e s s of G(F). L e t F(x) and H(x) b e g e n e r a l i z e d - d i f f e r e n t i a b l e f u n c t i o n s . If G ( F , x) is a g e n e r a l i z e d g r a d i e n t of F ( x ) , then the s e t s

6' (F, x) =

G (F, x), if F (x) =/=H (x), conv {6 (F, x) U 6 (H, x)}, if

F (x) = H (x),

a r e a l s o g e n e r a l i z e d g r a d i e n t of F(x). 3. A n e x a m p l e of a g e n e r a l i z e d - d i f f e r e n t i a b l e f u n c t i o n w h i c h h a s no d i r e c t i o n a l d e r i v a t i v e s a t the p o i n t x=O:

F(x)-----(2k--lnlln]xII)lx],

if

F (x) = (-- 2k -}- In ] In I x ]]) ] x[, if where e = 2.7183...

e-

e2/e

0, y k _ x ( { y k - x ) / I y k - x l - - d ) g k e 0F{yk) s u c h t h a t

Ig~_xl

ly~_xl

~>8>0.

u s i n g the t h e o r e m of the m e a n [ t l ] , we r e w r i t e the l e f t - h a n d s i d e in the f o r m

r here

gkEOF(x+Ok(g~--x)),

0