Notion Of Convexity Research Articles

Weak quasi-invexity of nonsmooth functions

The notions of quasi-1 convexity, weak quasi-convexity and weak quasi-invexity are introduced. The relations among strict quasi-preinvexity, weak quasi-invexity and pseudo-invexity for a nonsmooth function are studied by means of the properties of the Clarke’s generalized subdifferential. As an application of the main results in this work, the relations among quasi-1 convexity, strict quasi-convexity, weak quasi-convexity and pseudo-convexity are established.

A characterization of convex games by means of bargaining sets

The aim of the paper is to characterize the classical convexity notion for cooperative TU games by means of the Mas-Colell and the Davis–Maschler bargaining sets. A new set of payoff vectors is introduced and analyzed: the max-Weber set. This set is defined as the convex hull of the max-marginal worth vectors. The characterizations of convexity are reached by comparing the classical Weber set, the max-Weber set and a selected bargaining set.

On some abstract convexity notions in real linear spaces

We introduce some abstract convexity notions in a real linear space and investigate which of the results from the convex analysis in topological vector spaces still work in a linear space. The differences between these abstract convexity notions and those established in spaces endowed with a topology are underlined by some examples. Mathematics subject classification (2000): 15A03, 52A01, 52A41.

On convex functions of higher order

Based on J. L. W. V. Jensen’s concept of convex functions as well on its generalization byE. M. Wright and related to T. Popoviciu’s convexity notions, higher-order convexity properties of real functions are introduced and surveyed. Mathematics subject classification (2000): 26A51, 26A48, 39B62.

Convexities of metric spaces

We introduce two kinds of the notion of convexity of a metric space, called k-convexity and L-convexity, as generalizations of the CAT(0)-property and of the nonpositively curved property in the sense of Busemann, respectively. 2-uniformly convex Banach spaces as well as CAT(1)-spaces with small diameters satisfy both these convexities. Among several geometric and analytic results, we prove the solvability of the Dirichlet problem for maps into a wide class of metric spaces.

Convex functions on Carnot groups

We consider the definition and regularity properties of convex functions in Carnot groups. We show that various notions of convexity in the subelliptic setting that have appeared in the literature are equivalent. Our point of view is based on thinking of convex functions as subsolutions of homogeneous elliptic equations.

Holomorphic Generation of Continuous Inverse Algebras

AbstractWe study complex commutative Banach algebras (and, more generally, continuous inverse algebras) in which the holomorphic functions of a fixedn-tuple of elements are dense. In particular, we characterize the compact subsets of ℂnwhich appear as joint spectra of suchn-tuples. The characterization is compared with several established notions of holomorphic convexity by means of approximation conditions.

A note on integral-convexity in Banach spaces

In the present paper we focus on a generalization of the notion of integral convexity. This concept, introduced in [J.Y. Wang, Y.M. Ma, The integral convexity of sets and functionals in Banach spaces, J. Math. Anal. Appl. 295 (2004) 211–224] by replacing, in the definition of classical notion of convexity, the sum by the integral, has interesting applications in optimal control problems. By using, instead of Bochner integral, a more general vector integral, that of Pettis, we obtain some results on integral-extreme points of subsets of a Banach space stronger than those given in [J.Y. Wang, Y.M. Ma, The integral convexity of sets and functionals in Banach spaces, J. Math. Anal. Appl. 295 (2004) 211–224]. Finally, a natural example coming from measure theory is included, in order to reflect the relationships between different kinds of integral convexity.

Optimization in the Heisenberg group

In this article, the local unconstrained and the constrained optimization problems in the Heisenberg group are investigated. The framework on which we work is given by the class of weakly H-convex functions recently introduced in the literature. This geometric notion of convexity, that is strictly related to the stratified structure of the group and has not an analogous in a Euclidean setting, requires a condition of convexity to be satisfied at the points of the horizontal planes. We find second-order sufficient conditions for a local extremum in both the unconstrained and the constrained optimization problems exploiting the weak H-convexity and the geometric behaviour of the horizontal planes in .

Shape preserving approximation by Bernstein-type operators which fix polynomials

In this paper we present a family of sequences of linear Bernstein-type operators, an element of which was recently introduced by J.P. King. As well as showing that they fix certain polynomials and studying some shape preserving and convergence properties, we make a comparison with Bernstein polynomials which shows that for certain usual situations and without too much additional computational cost, the elements of the family represent a good shape preserving approximation process versus that of the referred polynomials. The notion of convexity with respect to a function plays an important role.

Cremona convexity, frame convexity and a theorem of Santaló

Abstract In 1940, Luis Santaló proved a Helly-type theorem for line transversals to boxes in ℝ d . An analysis of his proof reveals a convexity structure for ascending lines in ℝ d that is isomorphic to the ordinary notion of convexity in a convex subset of ℝ2d−2. This isomorphism is through a Cremona transformation on the Grassmannian of lines in ℙ d , which enables a precise description of the convex hull and affine span of up to d ascending lines: the lines in such an affine span turn out to be the rulings of certain classical determinantal varieties. Finally, we relate Cremona convexity to a new convexity structure that we call frame convexity, which extends to arbitrary-dimensional flats in ℝ d .

A note on discrete convexity and local optimality

One of the most important properties of a convex function is that a local optimum is also a global optimum. This paper explores the discrete analogue of this property. We consider arbitrary locality in a discrete space and the corresponding local optimum of a function over the discrete space. We introduce the corresponding notion of discrete convexity and show that the local optimum of a function satisfying the discrete convexity is also a global optimum. The special cases include discretely-convex, integrally-convex, M-convex, M♮-convex, L-convex, and L♮-convex functions.

LINK MONOTONIC ALLOCATION SCHEMES

A network is a graph where the nodes represent players and the links represent bilateral interaction between the players. A reward game assigns a value to every network on a fixed set of players. An allocation scheme specifies how to distribute the worth of every network among the players. This allocation scheme is link monotonic if extending the network does not decrease the payoff of any player. We characterize the class of reward games that have a link monotonic allocation scheme. Two allocation schemes for reward games are studied, the Myerson allocation scheme and the position allocation scheme, which are both based on allocation rules for communication situations. We introduce two notions of convexity in the setting of reward games and with these notions of convexity we characterize the classes of reward games where the Myerson allocation scheme and the position allocation scheme are link monotonic. As a by-product we find a characterization of the Myerson value and the position value on the class of reward games using potentials.

Staffing decisions for heterogeneous workers with turnover

In this paper we consider a firm that employs heterogeneous workers to meet demand for its product or service. Workers differ in their skills, speed, and/or quality, and they randomly leave, or turn over. Each period the firm must decide how many workers of each type to hire or fire in order to meet randomly changing demand forecasts at minimal expense. When the number of workers of each type can by continuously varied, the operational cost is jointly convex in the number of workers of each type, hiring and firing costs are linear, and a random fraction of workers of each type leave in each period, the optimal policy has a simple hire- up-to/fire-down-to structure. However, under the more realistic assumption that the number of workers of each type is discrete, the optimal policy is much more difficult to characterize, and depends on the particular notion of discrete convexity used for the cost function. We explore several different notions of discrete convexity and their impact on structural results for the optimal policy.

Convexity and the well-formedness of musical objects

It is well known that subsets of the two-dimensional space ℤ2 can represent prominent musical and music-theoretical objects such as scales, chords and chord vocabularies. It has been noted that the major and minor diatonic scale form convex subsets in this space. This triggers the question whether convexity is a more widespread concept in music. This article systematically investigates the convexity for a number of musical phenomena including scales, chords and (harmonic) reduction. It is hypothesised that the notion of convexity may be a covering concept of musical phenomena and possibly reflects other mathematical properties of these musical structures. Furthermore, convexity can be used in a pitch-spelling model.

New Generalized Convexity Notion for Set-Valued Maps and Application to Vector Optimization

In this paper, we introduce a new generalized convexity notion for set-valued maps, called ic-cone-convexlikeness, and use it as the main tool to derive an alternative theorem and necessary conditions for efficient, weakly efficient, and Benson properly efficient solutions of the problem of minimizing a set-valued map subject to set-valued constraints. Our results are valid for a class of optimization problems broader than that of the problems considered in Refs. 1--6 and generalize the corresponding results of these references.

Stability of multiagent systems with time-dependent communication links

We study a simple but compelling model of network of agents interacting via time-dependent communication links. The model finds application in a variety of fields including synchronization, swarming and distributed decision making. In the model, each agent updates his current state based upon the current information received from neighboring agents. Necessary and/or sufficient conditions for the convergence of the individual agents' states to a common value are presented, thereby extending recent results reported in the literature. The stability analysis is based upon a blend of graph-theoretic and system-theoretic tools with the notion of convexity playing a central role. The analysis is integrated within a formal framework of set-valued Lyapunov theory, which may be of independent interest. Among others, it is observed that more communication does not necessarily lead to faster convergence and may eventually even lead to a loss of convergence, even for the simple models discussed in the present paper.

Cyclic Hypomonotonicity, Cyclic Submonotonicity, and Integration

Rockafellar has shown that the subdifferentials of convex functions are always cyclically monotone operators. Moreover, maximal cyclically monotone operators are necessarily operators of this type, since one can construct explicitly a convex function, which turns out to be unique up to a constant, whose subdifferential gives back the operator. This result is a cornerstone in convex analysis and relates tightly convexity and monotonicity. In this paper, we establish analogous robust results that relate weak convexity notions to corresponding notions of weak monotonicity, provided one deals with locally Lipschitz functions and locally bounded operators. In particular, the subdifferentials of locally Lipschitz functions that are directionally hypomonotone [respectively, directionally submonotone] enjoy also an additional cyclic strengthening of this notion and in fact are maximal under this new property. Moreover, every maximal cyclically hypomonotone [respectively, maximal cyclically submonotone] operator is always the Clarke subdifferential of some directionally weakly convex [respectively, directionally approximately convex] locally Lipschitz function, unique up to a constant, which in finite dimentions is a lower C2 function [respectively, a lower C1 function].

Fuzzy Φ-convexity and fuzzy decision making

The notion of convexity and its various generalization is important for quantitative and qualitative studies in operations research or applied mathematics. It has also been considered by many authors in fuzzy set theory. In this paper, we study the concept of Φ-convex and Φ-quasiconvex fuzzy sets which was proposed byChen et al. [1], and develop some useful extrema properties of these fuzzy sets. We prove that any local maximizer of a Φ-convex fuzzy set is also a global maximizer, and that any strict local maximizer of a Φ-quasiconvex fuzzy set is also a global maximizer. We also study the class of strictly Φ-convex (respectively, strictly Φ-quasiconvex) fuzzy sets that is more restricted than the class of Φ-convex (respectively, Φ-quasiconvex) fuzzy sets. We prove for both families of strictly Φ-convex and strictly Φ-quasiconvex fuzzy sets that every local maximizer is also the unique global maximizer. In addition, some applications to fuzzy decision making are discussed.

Tropical convexity

The notions of convexity and convex polytopes are introduced in the setting of tropical geometry. Combinatorial types of tropical polytopes are shown to be in bijection with regular triangulations of products of two simplices. Applications to phylogenetic trees are discussed.