Reinhard Nehse

Zur theorie der polarfunktionale

In this paper a generalization of FENCHELs conjugate functions is given. We introduce both the upper and lower polar functionals, respectively, using the upper addition and lower addition (as done by MOREAU). Then we give a few general results. Moreover, a weak duality theorem for certain dual optimization problems is proveci. Finally me generalize FALKs investigations and obtain some properties of the upper and lower convex envelope of a functional.

Konjugierte operatoren und subdifferentiale

In this paper we give separation theorems for convex sets of a product space. Separation is carried out by linear operators. With these theorems we prove several assertions on conjugate operators and subdifferentials of operators (which map a vector space into an order complete vector lattice), where we use the definition of conjugate operators as done by Zowe. Simultaneously we generalize some of his results to such operators. Moreover we prove that an order complete vector lattice is a vector lattice with certain separation properties.

Recent results on separation of convex sets 1

In this paper a survey about the following directions of generalizations for separation theorems involving some new results are given: Separation of finite families of convex sets; separation of sets in product spaces; separation of convex sets in projective spaces; separation of convex sets in convexity spaces. As a basis for all these considerations vector spaces without topology are used.

Relations between generalized concepts of convexity and conjugacy

In this paper some connections between the following generalizations of convex conjugate functions and corresponding sets are given: quasi-conjugate functions, φ-conjugate functionsK-functions, convexity properties of set-families, Especially, a K-function is constructed for quasi-convex optimization problems.

Strong pseudo-convex mappings in dual problems

In this paper some relations between strong pseudo-convexity and other generalized convexity-properties of mappings (convex-likeness, quasi -convexity) into Banach-spaces are described. Furthermore necessary and sufficient conditions for the strong pseudo-convexity of a mapping are presented. Especially, sufficient conditions for the strong pseudo-convexity of composite mappings are proved. Applying these results a correct ion of a direct duality theorem given by Chandra and Lata is formulated.

On duality-separability-relations 1

In this paper relations between duality theorems for nonconvex optimization problems and separation-theorems of certain sets are given. The sets are separated by nonlinear manifolds, Moreover, relation to papers of Rockafellar and Smith/Vande-Linde are pointed out.

Bemerkungen zu einem ergebnis von J. ZOWE

In this note a generalization of a result of ZOWE on FENCHELs duality theorem is given, where, in addition, the assumptions are weakened.