Stability of Solutions to Convex Problems of Optimization by K. Malanowski

By K. Malanowski

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Extra info for Stability of Solutions to Convex Problems of Optimization

Example text

31b) holds on where [tN_2,tN_1]. 40) JNCJN_1,then [tN_2,tN_1].

27a) DzL(Uh,Zh;Ph,lh;h) = 0. 2. Lipschitz continuity with Respect to Parameter In this section we are going to show Lipschitz continuity with respect to the parameter of primal and dual optimal variables for (Oh). 9) hold then for any compact and convex there exists a constant c such that llu2-ulil,llz2-z111y. Ilx2-hl I ~ c[h2-h11, where c does not depend on scripts (221) h ~ ~{, and on the left-hand side the sub- 1 and 2 are used instead of h I and h 2. 26) it follows that L(Ul,Zl;p2,12;h I) ~ L(u2,z2;Pl,ll;hl).

N o t e that since x s a t i s f y i n g function, t h e n by ous function. 1) absolutely continuous, vec- tor functions. 4) can be e x p r e s s e d in the form - 8 ( x , h ) ~ K 2. 3) can as well be e x p r e s s e d as - 8 ( x , h ) e K I. 4) w e o b t a i n a little bit d i f f e r e n t forms of the L a g r a n g i a n a s s o c i a t e d w i t h We c o n s i d e r b o t h cases, since each of them w i l l be used in the sequel. We shall need the general nals n o n - n e g a t i v e functionals d e p e n d on K 1 and (0Ch).

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