Karl-Heinz Elster

Abstract cone approximations and generalized differentiability in nonsmooth optimization

In the present paper a connection between cone approximations of sets and generalized differentiability notions will be given. Using both conceptions we present an approach to derive necessary optimality conditions for optimization problems with inequality constraints. Moreover, several constraint qualifications are proposed to get Kuhn-Tucker-type-conditions.

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.

Generalized conjugate functions

Conjugate functions introduced in nonlinear programming by Fenchel are closely connected with polarity with respect to a special hypersurface of the order two. In the paper a wider class of conjugate functions is considered, basing on the polarity with respect to a nondegenerate hypersurface φ of order two. Important properties of so-called φ-conjugate sets and φ-conjugate functions are given.

Duality theorems and optimality conditions for nonconvex optimization problems

Continuing former papers of the authors the present paper gives assertions about φ-conjugate in connection with duality theorems and optimality conditions for a rather broad class of nonconex optimization problems. A geometrical interpretation of the statements follows in a natural way by the projective extension of the Euclideanspace.

Duality theorems for nonconvex optimization problems

The paper is dealt with duality theorems for a special class of nonconvex optimization problems. Thereby the Φ- conjugate functions (introduced by the authors in earlier papers) are used. The primal objective function contains a pair of functions f g - but in contrast to the problem of Fenchel not in terms of a difference. In proving duality theorems the separation of the sets epif and hypog is used essentially

Konvexe mengen in projektiven räumen

In projective space three notions of convexity (weak convexity, strong convexity, p-convexity) are regarded systematically. Since these notions are defined only by incidence relations, there can be introduced dual notions. We consider relations.between the introduced notions and the most essential properties of convex sets. To all assertions. can be formulated dual assertions, too. The most important theorems given by Fenchel can be generalised. The property of a point set (a set of hyperplanes) to be strongly convex or p-convex, respectively, is invariant with respect to correlations.

Differentialrechnung für Funktionen mit mehreren unabhängigen Variablen

Okonomische und technische Erscheinungen werden im allgemeinen nicht durch einen, sondern durch mehrere Faktoren bestimmt. Eine Beschrankung auf Funktionen mit nur einer unabhangigen Variablen gibt daher in vielen Fallen nur eine unzureichende Widerspiegelung der objektiven Realitat. Besonders in mathematisch-okonomischen und statistisch-okonomischen Modellen ist die Verwendung von Funktionen mehrerer Variablen zur Beschreibung der vielfach komplizierten Zusammenhange und Abhangigkeiten zwischen okonomischen Erscheinungen und Prozessen unumganglich.

Grundbegriffe der Mengenlehre

In diesem Abschnitt soll eine kurze Zusammenstellung einiger Grundbegriffe der Mengenlehre erfolgen. Fur eine ausfuhrlichere und axiomatisch strengere Behandlung dieses Themas verweisen wir auf die im Literaturverzeichnis aufgefuhrten Lehrbucher.