Some General Separation Theorems

✍ Scribed by Reinhard Nehse

• Publisher: John Wiley and Sons
• Year: 1978
• Tongue: English
• Weight: 438 KB
• Volume: 84
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19780840119

✦ Synopsis

Separation theorems are known to have fundamental importance for several fields of mathematics. Recently some authors have proved new separation theorems for set.s in real vector spaces and generalizations of such spaces (for instance, convexity spaces), respectively (see

The purpose of this paper is to prove that under certain assumptions two non-void subsets of a product space can be separated by an affine 'honvertical" manifold of that product space. In the last section an application of such theorems t.0 problems of nonconvex optimization is presented. Additional applications to convex optimization problems are to be found in [ 5 ] , [7], [9], [lo], and [12].

1. Preliminaries

Throughout this paper E denotes a real vector space and F an order-complete partially ordered real vector space, that is a vector space where a binary reflexive, transitive and ant.i-symmetrical relation " s'' is defined such that =-Az SAY for all real A E 0 , and every nonempty subset of F which has an upper bound in F has also a least upper bound in F. This last property of F is equivalent t n the validity of the HmN-BANAcH-theorem in F (see [ 1 11).

Further, we apply some abbreviations : F, : = (yCF/O s g ] ; 9(E, l') denotes the real vector space of all linear operators mapping E into F. Now let A be a nonempty subset of E . 'A denotes the affine manifold spanned by A ; 'A denotes 1) This paper is part of the author's dissertation [lo] prepared under the supervision of h f . Dr. K.-H. ELSTEB. The author wishes to express his thanks to Prof. ELYTER for his guidance and encouragement.