"Variational analysis and applications, Springer monographs in mathematics." Optimization, 68(5), pp. 1073–1074

Having a function being a difference of sublinear functions defined on a plane, we present a formula for effective calculation of sublinear functions such that their difference is equal to the given one. Moreover, these newly calculated sublinear functions are minimal and as such unique-up-to-linear-summand. We also provide examples of such functions.

In this paper, we study properties of a special class of ordered median functions, called block-functions. These are ordered median functions which belong to a generating binary (row)-vector of the form lambda = 0, 0,..., 0, 1, 1,..., 1, 0, 0,..., 0 is an element of R-n, called a block vector. The aim of this paper is to explicitly determine the simplicial complexes and all steepest descent and ascent directions of descent and ascent cones of ordered median block-function.

In this paper we give a survey of the influence of the quasidifferential calculus of V. F. Demyanov and A. M. Rubinov to the field of generalized convexity. In particular, we will show the strong relations between the order cancellation property of bounded closed convex set and the separation property of bounded closed convex sets by sets. Moreover, a generalization of the Demyanov difference of compact convex sets infinite dimension and its role in the context of convex sets by sets is discussed. Refs 15. Figs 2.

It is a great pleasure for us to praise Rosalind Elster’s long-standing professional and courageous commitment to improve the journal Optimization to a high-level international journal. Our journal...

The paper examines different kinds of p-convexity of a function g which are sufficient for the existence of a linear functional such that in Theorem 1.13 of Simons in his monograph ‘From Hahn–Banach to monotonicity’, published in Springer lecture notes (2008). We replace sublinearity of p with convexity, the field with Dedekind vector lattice and present -convexity which is also necessary. In Theorem 4.7 we also generalize a result of MM. Neumann from 1991 published in Czech. Mathem. Journal Vol 41 on the Mazur–Orlicz theorem.

Within this paper we treat a version of the Shapley value (see [13]) for countably many players. However, our approach differs from the one favored by most authors, in particular by Shapley [14] and Artstein [2] , who attacked the problem of constructing a Shapley value on a suitable class of games by a basically measure–theoretic approach (see also [I], [15], [9]). Our version is rather based on group-theoretical considerations, more precisely, we use the structure of the symmetric group of ℕ. This group, viewed as a subset of the orderings of ℕ, has measure 0. However, if we introduce a suitable metric, it admits nevertheless a normalized, nonatomic additive measure, which, on a sufficiently large subalgebra of the clopen subsets is invariant under left and right multiplication (and a fortiori on all group automorphisms). Integration of the marginal worth of a player with respect to this measure yields the Shapley value.

Order unit normed linear spaces are a special type of regularly ordered normed linear spaces and therefore the first section is a short collection of the fundamental results on this type of normed linear spaces. The connection between order unit normed linear spaces and base normed linear spaces within the category of regularly ordered normed linear spaces is described in Section 2, and Section 3 at last, contains the results on Banach limits in an arbitrary order unit normed linear space. It is shown that the original results on Banach limits are valid for a greater range.

In this paper Banach Limits are studied for locally bounded Lipschitz functions on a Hausdorff space.

We prove that in finite dimensional spaces every ordered median function is the Minkowski dual of a reduced pair of polytopes. This implies a very general theorem on the representation of an ordered median function as a uniquely determined difference of two sublinear functions up to adding and subtracting one and the same arbitrary sublinear function.

In this paper we study Minkowski duality, i.e. the correspondence between sublinear functions and closed convex sets in the context of dual pairs of vector spaces.

This is an excellent and clearly written book which provides an easy access to the most fundamental parts of convex analysis and its applications to optimization. Convex analysis and optimization a...

In this paper the universal properties of spaces of Lipschitz functions, defined over metric spaces, are investigated.

An ordered median functions is a continuous piecewise-linear function. It is well known that in finite dimensional spaces every continuous piecewise-linear function admits a max-min representation in terms of its linear functions. An explicit representation of an ordered median function in max-min form is given by the authors and will appear in a forthcoming issue of this journal. Based on this representation, we give a topological classification of ordered median functions through their simplicial complex of ascent (resp. descent) cones.

In this paper we consider a generalization of the separation technique proposed in [10,4,7] for the separation of finitely many compact convex sets \(A_i,~i \in I \) by another compact convex set \(S\) in a locally convex vector space to arbitrary sets in real vector spaces. Then we investigate the notation of shadowing set which is a generalization of the notion of separating set and construct separating sets by means of a generalized Demyanov-difference in locally convex vector spaces.

An ordered median function is a continuous piecewise linear function. It is well known, that in finite dimensional spaces every continuous piecewise linear function admits a max-min representation in terms of its linear functions. We give an explicit representation of an ordered median function in max-min form using a purely combinatorial approach.