L. Lovász (auth.), Wilfried Brauer (eds.)'s Automata, Languages and Programming: 12th Colloquium PDF

By L. Lovász (auth.), Wilfried Brauer (eds.)

ISBN-10: 354015650X

ISBN-13: 9783540156505

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Additional info for Automata, Languages and Programming: 12th Colloquium Nafplion, Greece, July 15–19, 1985

Example text

Define Z = X - Y. It is then obvious that Z is a nonempty convex set and 0 ¢ Z. 2 to show that there exists a nonzero vector cERn such that c T z ::;; 0 "Iz E Z. 5) holds with a = sup c T x, XEclX = (p + q)/2. I We next show a strong separation theorem. 3 (Strong Separation Theorem) Let X S;; Rn be a nonempty, 0, then closed convex set, and Y S;; Rn a nonempty, compact convex set. If X n Y = 20 CHAPTER 2 there exists a hyperplane strictly separating X and Yj that is, there exist a nonzero vector c E R" and a scalar O!

Compactness of aj(x). Since X is an open set, we can choose a positive number c {x E Rn Ilix - xII ::;; c} ~ X. If we suppose that 8f(x) is satisfying clB(x;c) not bounded, then there exists a sequence {C k } ~ 8f(x) such that IICkll-> +00. k} and an index j E {1, ... e -+ 00). Define a sequence {xk'} by x~' = Xi, i :f. j, x7' = Xj + c:(;' 11(;'1. Then it follows that xk, E clB( Xj c:). (xk')} is bounded. k' . e. k t E [)f(x). We can then conclude that [)f(x) is bounded. It follows from the definition of the subgradient that [)f(x) is a closed set.

Then it holds that for any x E Rn, sup min aT x tET aEA(t) = min aOE I1tET sup A(t) a* Ea(T) a*T x Proof. Let x E Rn be arbitrarily fixed. 21) a*Tx;;;;: {3 a(·) E I1tET A(t) satisfying a*T x = supa(tf x;;;;: sup min aT x = a tET tET aEA(t) This is a contradiction implying that a = {3. 20) follows. 4 For the set-valued maps A; : T; ...... 2 Rn and Bi : S; ...... 2 Rn such that B;(s;) are nonempty and compact at each S; E S;, let us define the following positively homogeneous functions h;(x) = inf sup t,ET, a,EA,(t,) h;.

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Automata, Languages and Programming: 12th Colloquium Nafplion, Greece, July 15–19, 1985 by L. Lovász (auth.), Wilfried Brauer (eds.)

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