Points Of Convex Sets Research Articles

Denting points of convex sets and weak property ($$\pi $$) of cones in locally convex spaces

Denting points of convex sets and weak property ($$\pi $$) of cones in locally convex spaces

Relative interior and closure of the set of inner points

Internal points were introduced in the literature of topological vector spaces to characterize the finest locally convex vector topology. Very recently this concept was generalized to the one of inner points in the scope of vector spaces, which, among other things, allows to characterize the linear dimension of a vector space and also serves to provide an intrinsic characterization of linear manifolds that was not possible by using internal points. Inner points of convex sets can be seen as the affine internal points, that is, the internal points with respect to the affine hull. In this manuscript we continue this research line in the scope of topological vector spaces to study the topological structure of the inner points. First, we prove that every infinite dimensional vector space has a convex subset free of inner points which is dense in every vector topology the vector space is endowed with. Also, we find the existence of convex sets in which the set of inner points is not open in the relative topology. Following this line, we also characterize the closed convex sets for which the set of inner points is not empty and open in the relative topology. Finally, we find an example of a convex set whose set of inner points is not contained in the set of inner points of its closure.

Inner structure in real vector spaces

Abstract Internal points were introduced in the literature of topological vector spaces to characterize the finest locally convex vector topology. In this manuscript we generalize the concept of internal point in real vector spaces by introducing a type of points, called inner points, that allows us to provide an intrinsic characterization of linear manifolds, which was not possible by using internal points. We also characterize infinite dimensional real vector spaces by means of the inner points of convex sets. Finally, we prove that in convex sets containing internal points, the set of inner points coincides with the one of internal points.

On exposed points and extremal points of convex sets in Rnand Hilbert space

On exposed points and extremal points of convex sets in Rⁿand Hilbert space

Best Simultaneous Approximation of Finite Set in Inner Product Space

In this paper, we find a way to give best simultaneous approximation of n arbitrary points in convex sets. First, we introduce a special hyperplane which is based on those n points. Then by using this hyperplane, we define best approximation of each point and achieve our purpose.

Imbedding and cutting methods for global optimization and their applications

The article addresses some versions of cutting methods for global optimization, the Chebyshev points of convex sets and applications of the Chebyshev points for global optimization and control theory.

On support points and continuous extensions

A selection theorem concerning support points of convex sets in a Banach space is proved. As a corollary we obtain the following result. Denote by \({\mathcal{BCC}(X)}\) the metric space of all nonempty bounded closed convex sets in a Banach space X. Then there exists a continuous mapping \({S : \mathcal{BCC}(X) \rightarrow X}\) such that S(K) is a support point of K for each \({K \in \mathcal{BCC}(X)}\). Moreover, it is possible to prescribe the values of S on a closed discrete subset of \({\mathcal{BCC}(X)}\).

Methods of Chebyshev points of convex sets and their applications

Consideration is given to methods of Chebyshev points in convex programming and their applications in optimal control. These methods belong to a class of cutting-immersion methods whose idea is in the approximation of convex sets by concave polyhedra.

Methods of Chebyshev points of convex sets and their applications

Chebyshev points of bounded convex sets, search algorithms for them, and various applications to convex programming are considered for simple approximations of reachable sets, optimal control, global optimization of additive functions on convex polyhedra, and integer programming. The problem of searching for Chebyshev points in multicriteria models of development and operation of electric power systems is considered.

Norm-preserving extensions of functionals and denting points of convex sets

Let W and Z be Banach spaces, and let \(X\subset W\) and \(Y\subset Z\) be closed subspaces. Let \(\mathcal{L}\) be a subspace of \(\mathcal{L}(W^*, Z^{**})\), the Banach space of bounded linear operators from W* to Z**, containing \(X\otimes Y\). We describe, for \(x^{*}\in X^{*}\) and \(y^*\in Y^*\), all norm-preserving extensions of \(x^{*}\otimes y^{*}\in (X\otimes Y)^\ast\) to the space \(\mathcal{L}\) in terms of convergence of convex combinations. We also characterize denting points of bounded convex subsets of Banach spaces in similar terms. Various applications are presented.

A polynomial time algorithm for Zero-Clairvoyant scheduling

In the design of real-time systems, it is often the case that certain process parameters, such as the execution time of a job are not known precisely. The challenge in real-time system design then, is to develop techniques that efficiently meet the requirements of impreciseness, while simultaneously guaranteeing safety. In a traditional scheduling model, such as the one discussed in [M. Pinedo, Scheduling: Theory, Algorithms, and Systems, Prentice-Hall, Englewood Cliffs, 1995. [23]]; [P. Brucker, Scheduling Algorithms, second ed., Springer, 1998. [3]], the tendency is to either overlook the effects of impreciseness or to simplify the issue of impreciseness by assuming worst-case values. This assumption is unrealistic and at the same time, may cause certain timing constraints to be violated at run-time. Further, depending on the nature of the constraints involved, it is not immediately apparent, what the worst-case value for a given parameter is. Whereas, in traditional scheduling, constraints among jobs are no more complex than those that can be represented by a precedence graph, in case of real-time scheduling, complicated constraints such as relative timing constraints are commonplace. Additionally, the purpose of scheduling is to achieve a schedule that optimizes some performance metric, whereas in real-time scheduling the goal is to ensure that the imposed constraints are met at run-time. In this paper, we study the problem of scheduling a set of ordered, non-preemptive jobs under non-constant execution times. Typical applications for variable execution time scheduling include process scheduling in Real-Time Operating Systems such as Maruti, compiler scheduling, database transaction scheduling and automated machine control. An important feature of application areas such as robotics is the interaction between execution times of various processes. We explicitly model this interaction through the representation of execution time vectors as points in convex sets. This modeling vastly extends previous models of execution times as either single points or range-bound intervals. Our algorithms do not assume any knowledge of the distributions of execution times, i.e. they are Zero-Clairvoyant. We present both sequential and parallel algorithms for determining the existence of a Zero-Clairvoyant schedule. To the best of our knowledge, our techniques are the first of their kind.

Maximal Points of Convex Sets in Locally Convex Topological Vector Spaces: Generalization of the Arrow–Barankin–Blackwell Theorem

In 1953, Arrow, Barankin, and Blackwell proved that, if C is a nonempty compact convex set in Rn with its standard ordering, then the set of points in C maximizing strictly positive linear functionals is dense in the set of maximal points of C. In this paper, we present a generalization of this result. We show that that, if C is a compact convex set in a locally convex topological space X and if K is an ordering cone on X such that the quasi-interiors of K and the dual cone K* are nonempty, then the set of points in C maximizing strictly positive linear functionals is dense in the set of maximal points of C. For example, our work shows that, under the appropriate conditions, the density results hold in the spaces Rn, Lp(Ω, μ), 1≤p≤∞, lp, 1≤p≤∞, and C (Ω), Ω a compact Hausdorff space, when they are partially ordered with their natural ordering cones.

Reconstructing Boundary Points of Convex Sets from X-ray Pictures

A method for the reconstruction of boundary points of convex sets is provided starting from three X-ray pictures in two orthogonal directions and from a point.

Extreme Points in Convex Sets of Symmetric Matrices

Extreme Points in Convex Sets of Symmetric Matrices

Extreme points in convex sets of symmetric matrices

This paper deals with the following problem: What are the extreme points of a convex set K K of n × n n \times n matrices, which is the intersection of the set S n {S_n} of symmetric matrices of nonnegative type, with another convex subset of symmetric matrices H ? H? ? In the case where the facial structure of H H is known, we expose a general method to determine the extreme points of K K (Theorem 1). Then, we apply this method to the set of correlation matrices, characterizing its extreme points in Theorem 2, which is our main theorem. A corollary describes thoroughly the extreme points of rank 2.

Lattice Points in Convex Sets

Lattice Points in Convex Sets

Extreme points of convex sets of doubly stochastic matrices. II

We prove a conjecture of (5), namely that the convex set of all infinite doubly stochastic matrices whose entries are all strictly less than θ(0 < θ ≤ 1) possesses extreme points if and only if θ is irrational.

Extreme points of convex sets

Extreme points of convex sets

Some topological properties of support points of convex sets

It is shown, as a corollary to Trojanski’s renorming theorem, that the set of support points (of certain closed convex sets) is connected. Some connectedness properties are established for support functionals, and it is also shown that the support points form an Fσ.

Strongly extreme points in Banach spaces

In this paper some properties of a special type of boundary point of convex sets in Banach spaces are studied. Specifically, a strongly extreme point x of a convex set S is a point of S such that for each real number r>0, segments of length 2r and centered x are not uniformly closer to S than some positive number d(x,r). Results are obtained comparing the notion of strongly extreme point to other known types of special boundary points of convex sets. Using the notion of strongly extreme point, a convexity condition is defined on the norm of the space under consideration, and this convexity condition makes possible a unified treatment of some previously studied convexity conditions. In addition, a sufficient condition is given on the norm of a separable conjugate space for every extreme point of the unit ball to be strongly extreme.