Ratios of affine functions

Aratio of affine functions is a function which can be expressed as the ratio of a vector valued affine function and a scalar affine functional. The purpose of this note is to examine properties of sets which are preserved under images and inverse images of such functions. Specifically, we show that images and inverse images of convex sets under such functions are convex sets. Also, images of bounded, convex polytopes under such functions are bounded, convex polytopes. In addition, we provide sufficient conditions under which the extreme points of images of convex sets are images of extreme points of the underlying domains. Of course, this result is useful when one wishes to maximize a convex function over a corresponding set. The above assertions are well known for affine functions. Applications of the results include a problem that concerns the control of stochastic eigenvectors of stochastic matrices.     

几种凸模糊集间的转换关系

在引用扎德所定义的凸模糊集、强凸模糊集、严格凸模糊集等概念的基础上，探讨了这三种凸模糊集间的转换条件，得到凸模糊集与强凸模糊集、强凸模糊集与严格凸模糊集间的等价条件。     

A further characteristic of abstract convexity structures on topological spaces

In this paper, we give a characteristic of abstract convexity structures on topological spaces with selection property. We show that if a convexity structure C defined on a topological space has the weak selection property then C satisfies H₀-condition. Moreover, in a compact convex subset of a topological space with convexity structure, the weak selection property implies the fixed point property.     

Convexity is equivalent to midpoint convexity combined with strict quasiconvextiy

It was recently shown by Nikodem that a function defined on an open convex subset of R ⁿ is convex if and only if it is midpoint convex and quasiconvex. It is shown that quasiconvexity can be replaced by strict quasiconvexity and that the openness condition can be removed altogether. The domain can then be taken from a general real linear space. There will also be given some related results of a “local” nature     

A Study of Downward Sets with p-Functions

In this paper, we study downward sets and increasing functions in a topological vector space and their similarities to the convex sets and convex functions. It will be shown that a very special increasing function, namely, the p-function, can give a geometric interpretation for separating downward sets from outside points. Also, this function can be used to approximate topical functions in the framework of abstract convexity.     

Polyhedral approximation and practical convex hull algorithm for certain classes of voxel sets

In this paper we introduce an algorithm for the creation of polyhedral approximations for certain kinds of digital objects in a three-dimensional space. The objects are sets of voxels represented as strongly connected subsets of an abstract cell complex. The proposed algorithm generates the convex hull of a given object and modifies the hull afterwards by recursive repetitions of generating convex hulls of subsets of the given voxel set or subsets of the background voxels. The result of this method is a polyhedron which separates object voxels from background voxels. The objects processed by this algorithm and also the background voxel components inside the convex hull of the objects are restricted to have genus 0. The second aim of this paper is to present some practical improvements to the discussed convex hull algorithm to reduce computation time.     

Extreme point and halving edge search in abstract order types

Many properties of finite point sets only depend on the relative position of the points, e.g., on the order type of the set. However, many fundamental algorithms in computational geometry rely on coordinate representations. This includes the straightforward algorithms for finding a halving line for a given planar point set, as well as finding a point on the convex hull, both in linear time. In his monograph Axioms and Hulls, Knuth asks whether these problems can be solved in linear time in a more abstract setting, given only the orientation of each point triple, i.e., the set?s chirotope, as a source of information. We answer this question in the affirmative. More precisely, we can find a halving line through any given point, as well as the vertices of the convex hull edges that are intersected by the supporting line of any two given points of the set in linear time. We first give a proof for sets realizable in the Euclidean plane and then extend the result to non-realizable abstract order types.     

Reprint of: Extreme point and halving edge search in abstract order types

Many properties of finite point sets only depend on the relative position of the points, e.g., on the order type of the set. However, many fundamental algorithms in computational geometry rely on coordinate representations. This includes the straightforward algorithms for finding a halving line for a given planar point set, as well as finding a point on the convex hull, both in linear time. In his monograph Axioms and Hulls, Knuth asks whether these problems can be solved in linear time in a more abstract setting, given only the orientation of each point triple, i.e., the setʼs chirotope, as a source of information. We answer this question in the affirmative. More precisely, we can find a halving line through any given point, as well as the vertices of the convex hull edges that are intersected by the supporting line of any two given points of the set in linear time. We first give a proof for sets realizable in the Euclidean plane and then extend the result to non-realizable abstract order types.     

Radius of convexity of certain classes of analytic functions

We consider the classes of analytic functions introduced recently by K.I. Noor which are defined by conditions joining ideas of close-to-convex and of bounded boundary rotation functions. We investigate coefficients estimates and radii of convexity.     

Duality in quasi-convex supremization and reverse convex infimization via abstract convex analysis,and applications to approximation **

We give duality theorems and dual characterizations of optimal solutions for abstract quasi-convex supremization problems and infimization problems with abstract reverse convex constraint sets. Our main tools are dualities between families of subsets, conjugations of type Lau associated to them, and subdifferentials with respect to conjugations of type Lau. These tools permit us to give explicitly the relation between.the constraint sets, and the relation between the objective functions, of the primal problem and the dual problem. As applications, we obtain duality theorems for quasi-convex supremization and reverse convex infimization in locally convex spaces and, in particular, for worst and best approximation in normed linear spaces.     

The Knaster-Kuratowski-Mazurkiewicz theorem and abstract convexities

The Knaster-Kuratowski-Mazurkiewicz covering theorem (KKM), is the basic ingredient in the proofs of many so-called “intersection” theorems and related fixed point theorems (including the famous Brouwer fixed point theorem). The KKM theorem was extended from Rn to Hausdorff linear spaces by Ky Fan. There has subsequently been a plethora of attempts at extending the KKM type results to arbitrary topological spaces. Virtually all these involve the introduction of some sort of abstract convexity structure for a topological space, among others we could mention H-spaces and G-spaces. We have introduced a new abstract convexity structure that generalizes the concept of a metric space with a convex structure, introduced by E. Michael in [E. Michael, Convex structures and continuous selections, Canad. J. Math. 11 (1959) 556-575] and called a topological space endowed with this structure an M-space. In an article by Shie Park and Hoonjoo Kim [S. Park, H. Kim, Coincidence theorems for admissible multifunctions on generalized convex spaces, J. Math. Anal. Appl. 197 (1996) 173-187], the concepts of G-spaces and metric spaces with Michael's convex structure, were mentioned together but no kind of relationship was shown. In this article, we prove that G-spaces and M-spaces are close related. We also introduce here the concept of an L-space, which is inspired in the MC-spaces of J.V. Llinares [J.V. Llinares, Unified treatment of the problem of existence of maximal elements in binary relations: A characterization, J. Math. Econom. 29 (1998) 285-302], and establish relationships between the convexities of these spaces with the spaces previously mentioned.     

Global saddle-point duality for quasi-concave programs,II

A saddle-point duality is proposed for the quasi-concave non-differentiable case of the maximization of the minimum between a finite number of functions. This class of problems contains quasi-concave (convex) programs that are known to be irreducible to convex ones. With the aid of the saddle-point duality involving conjugate-like operators, a Lagrangian is presented, the saddle-points of which give the exact global solutions. A few particular cases are discussed, among them the Von Neumann economic model and discrete rational approximation.     

Convexifactors,generalized convexity and vector optimization

In this article we study a recently introduced notion of non-smooth analysis, namely convexifactors. We study some properties of the convexifactors and introduce two new chain rules. A new notion of non-smooth pseudoconvex function is introduced and its properties are studied in terms of convexifactors. We also present some optimality conditions for vector minimization in terms of convexifactors.     

A new algorithm for minimizing convex functions over convex sets

Let be a convex set for which there is an oracle with the following property. Given any pointz∈ℝ ⁿ the oracle returns a “Yes” ifz∈S; whereas ifz∉S then the oracle returns a “No” together with a hyperplane that separatesz fromS. The feasibility problem is the problem of finding a point inS; the convex optimization problem is the problem of minimizing a convex function overS. We present a new algorithm for the feasibility problem. The notion of a volumetric center of a polytope and a related ellipsoid of maximum volume inscribable in the polytope are central to the algorithm. Our algorithm has a significantly better global convergence rate and time complexity than the ellipsoid algorithm. The algorithm for the feasibility problem easily adapts to the convex optimization problem.     

Some Geometric Properties of Julia Sets and filled-in Julia Sets of Polynomials

In this paper, some geometric properties of connected Julia sets and filled-in Julia sets of polynomials are given.     

Objective function and logarithmic barrier function properties in convex programming: level sets,solution attainment and strict convexity *

We give several results, some new and some old, but apparently overlooked, that provide useful characterizations of barrier functions and their relationship to problem function properties. In particular, we show that level sets of a barrier function are bounded if and only if feasible level sets of the objective function are bounded and we obtain conditions that imply solution existence, strict convexity or a positive definite Hessian of a barrier function. Attention is focused on convex programs and the logarithmic barrier function. Such results suggest that it would seem possible to extend many recent complexity results by relaxing feasible set compactness to the feasible objective function level set boundedness assumption.     

Optimality conditions and duality in multiple objective programming involving semilocally convex and related functions

A nonlinear multiple objective programming problem is considered where the functions involved are nondifferentiable. By considering the concept of weak minima, the Fritz John type and Karush-Kuhn- Tucker type necessary optimality conditions and Wolfe and Mond-Weir type duality results are given in terms of the right differentials of the functions. The duality results are stated by using the concepts of generalized semilocally convex functions     

Comments on abstract convexity structures on topological spaces

All results in “Some properties of abstract convexity structures on topological spaces” by S.-w. Xiang and H. Yang [S.-w. Xiang, H. Yang, Some properties of abstract convexity structures on topological spaces, Nonlinear Analysis 67 (2007) 803-808] and “A further characteristic of abstract convexity structures on topological spaces” by S.-w. Xiang and S. Xia [S.-w. Xiang, S. Xia, A further characteristic of abstract convexity structures on topological spaces, J. Math. Anal. Appl. 335 (2007) 716-723] are shown to be consequences of known ones or can be stated in more general forms.