Motzkin decomposition of closed convex sets via truncation

A nonempty set F is called Motzkin decomposable when it can be expressed as the Minkowski sum of a compact convex set C with a closed convex cone D. In that case, the sets C and D are called compact and conic components of F. This paper provides new characterizations of the Motzkin decomposable sets involving truncations of F (i.e., intersections of FF with closed halfspaces), when F contains no lines, and truncations of the intersection F̂ of F with the orthogonal complement of the lineality of F, otherwise. In particular, it is shown that a nonempty closed convex set F is Motzkin decomposable if and only if there exists a hyperplane H parallel to the lineality of F such that one of the truncations of F̂ induced by H is compact whereas the other one is a union of closed halflines emanating from H. Thus, any Motzkin decomposable set F can be expressed as F=C+D, where the compact component C is a truncation of F̂. These Motzkin decompositions are said to be of type T when F contains no lines, i.e., when C is a truncation of F. The minimality of this type of decompositions is also discussed.

On the stability of the Motzkin representation of closed convex sets

A set is called Motzkin decomposable when it can be expressed as the Minkowski sum of a compact convex set with a closed convex cone. This paper analyzes the continuity properties of the set-valued mapping associating to each couple (C,D) formed by a compact convex set C and a closed convex cone D its Minkowski sum C + D. The continuity properties of other related mappings are also analyzed.

Stability of Supporting and Exposing Elements of Convex Sets in Banach Spaces

* This work was supported by the CNR while the author was visiting the University of Milan.

Projections Onto Convex Sets through Particle Swarm Optimization and its application for remote sensing image restoration

Image restoration attempts to enhance images corrupted by noise and blurring effects. Iterative approaches can better control the restoration algorithm in order to find a compromise of restoring high details in smoothed regions without increasing the noise. Techniques based on Projections Onto Convex Sets (POCS) have been extensively used in the context of image restoration by projecting the solution onto hyperspaces until some convergence criteria be reached. It is expected that an enhanced image can be obtained at the final of an unknown number of projections. The number of convex sets and its combinations allow designing several image restoration algorithms based on POCS. Here, we address two convex sets: Row-Action Projections (RAP) and Limited Amplitude (LA). Although RAP and LA have already been used in image restoration domain, the former has a relaxation parameter (A) that strongly depends on the characteristics of the image that will be restored, i.e., wrong values of A can lead to poorly restoration results. In this paper, we proposed a hybrid Particle Swarm Optimization (PS0)-POCS image restoration algorithm, in which the A value is obtained by PSO to be further used to restore images by POCS approach. Results showed that the proposed PSO-based restoration algorithm outperformed the widely used Wiener and Richardson-Lucy image restoration algorithms. (C) 2010 Elsevier B.V. All rights reserved.

A hybrid image restoration algorithm based on Projections onto Convex Sets and Harmony Search

Image restoration is a research field that attempts to recover a blurred and noisy image. Since it can be modeled as a linear system, we propose in this paper to use the meta-heuristics optimization algorithm Harmony Search (HS) to find out near-optimal solutions in a Projections Onto Convex Sets-based formulation to solve this problem. The experiments using HS and four of its variants have shown that we can obtain near-optimal and faster restored images than other evolutionary optimization approach. © 2013 IEEE.

Continued fractions built from convex sets and convex functions

In a partially ordered semigroup with the duality (or polarity) transform, it is pos- sible to define a generalisation of continued fractions. General sufficient conditions for convergence of continued fractions are provided. Two particular applications concern the cases of convex sets with the Minkowski addition and the polarity transform and the family of non-negative convex functions with the Legendre–Fenchel and Artstein-Avidan–Milman transforms.

[Lecture notes on pure mathematics. Convex sets]

Includes bibliographies.

On Typical Compact Convex Sets in Hilbert Spaces

Let E be an infinite dimensional separable space and for e ∈ E and X a nonempty compact convex subset of E, let qX(e) be the metric antiprojection of e on X. Let n ≥ 2 be an arbitrary integer. It is shown that for a typical (in the sence of the Baire category) compact convex set X ⊂ E the metric antiprojection qX(e) has cardinality at least n for every e in a dense subset of E.

Motzkin Decomposable Sets

Theodore Motzkin proved, in 1936, that any polyhedral convex set can be expressed as the (Minkowski) sum of a polytope and a polyhedral convex cone. We have provided several characterizations of the larger class of closed convex sets, Motzkin decomposable, in finite dimensional Euclidean spaces which are the sum of a compact convex set with a closed convex cone. These characterizations involve different types of representations of closed convex sets as the support functions, dual cones and linear systems whose relationships are also analyzed. The obtaining of information about a given closed convex set F and the parametric linear optimization problem with feasible set F from each of its different representations, including the Motzkin decomposition, is also discussed. Another result establishes that a closed convex set is Motzkin decomposable if and only if the set of extreme points of its intersection with the linear subspace orthogonal to its lineality is bounded. We characterize the class of the extended functions whose epigraphs are Motzkin decomposable sets showing, in particular, that these functions attain their global minima when they are bounded from below. Calculus of Motzkin decomposable sets and functions is provided.

Updating Credal Networks is Approximable in Polynomial Time

Credal networks relax the precise probability requirement of Bayesian networks, enabling a richer representation of uncertainty in the form of closed convex sets of probability measures. The increase in expressiveness comes at the expense of higher computational costs. In this paper, we present a new variable elimination algorithm for exactly computing posterior inferences in extensively specified credal networks, which is empirically shown to outperform a state-of-the-art algorithm. The algorithm is then turned into a provably good approximation scheme, that is, a procedure that for any input is guaranteed to return a solution not worse than the optimum by a given factor. Remarkably, we show that when the networks have bounded treewidth and bounded number of states per variable the approximation algorithm runs in time polynomial in the input size and in the inverse of the error factor, thus being the first known fully polynomial-time approximation scheme for inference in credal networks.

Convex-transitive Banach spaces and their hyperplanes

The topic of this dissertation is the geometric and isometric theory of Banach spaces. This work is motivated by the known Banach-Mazur rotation problem, which asks whether each transitive separable Banach space is isometrically a Hilbert space. A Banach space X is said to be transitive if the isometry group of X acts transitively on the unit sphere of X. In fact, some weaker symmetry conditions than transitivity are studied in the dissertation. One such condition is an almost isometric version of transitivity. Another investigated condition is convex-transitivity, which requires that the closed convex hull of the orbit of any point of the unit sphere under the rotation group is the whole unit ball. Following the tradition developed around the rotation problem, some contemporary problems are studied. Namely, we attempt to characterize Hilbert spaces by using convex-transitivity together with the existence of a 1-dimensional bicontractive projection on the space, and some mild geometric assumptions. The convex-transitivity of some vector-valued function spaces is studied as well. The thesis also touches convex-transitivity of Banach lattices and resembling geometric cases.

Minimization Problems Based on Relative alpha-Entropy I: Forward Projection

Minimization problems with respect to a one-parameter family of generalized relative entropies are studied. These relative entropies, which we term relative alpha-entropies (denoted I-alpha), arise as redundancies under mismatched compression when cumulants of compressed lengths are considered instead of expected compressed lengths. These parametric relative entropies are a generalization of the usual relative entropy (Kullback-Leibler divergence). Just like relative entropy, these relative alpha-entropies behave like squared Euclidean distance and satisfy the Pythagorean property. Minimizers of these relative alpha-entropies on closed and convex sets are shown to exist. Such minimizations generalize the maximum Renyi or Tsallis entropy principle. The minimizing probability distribution (termed forward I-alpha-projection) for a linear family is shown to obey a power-law. Other results in connection with statistical inference, namely subspace transitivity and iterated projections, are also established. In a companion paper, a related minimization problem of interest in robust statistics that leads to a reverse I-alpha-projection is studied.

On Some Infinite Convex Invariants

The present study on some infinite convex invariants. The origin of convexity can be traced back to the period of Archimedes and Euclid. At the turn of the nineteenth centaury , convexicity became an independent branch of mathematics with its own problems, methods and theories. The convexity can be sorted out into two kinds, the first type deals with generalization of particular problems such as separation of convex sets[EL], extremality[FA], [DAV] or continuous selection Michael[M1] and the second type involved with a multi- purpose system of axioms. The theory of convex invariants has grown out of the classical results of Helly, Radon and Caratheodory in Euclidean spaces. Levi gave the first general definition of the invariants Helly number and Radon number. The notation of a convex structure was introduced by Jamison[JA4] and that of generating degree was introduced by Van de Vel[VAD8]. We also prove that for a non-coarse convex structure, rank is less than or equal to the generating degree, and also generalize Tverberg’s theorem using infinite partition numbers. Compare the transfinite topological and transfinite convex dimensions

Inclusões Dinâmicas em Escalas Temporais: Existência de Soluções sob a Hipótese de Semicontinuidade Inferior

In this paper we consider the problem of differential inclusion in time scales whose vector field is a multifunction, that is, a function that maps points to sets. It is provided conditions of existence without requiring compactness of the vector field; it is required that the vector field is closed, convex, and lower semicontinuous. In previous work in literature, it is required that the field is either scalar or compact, convex, and has closed graph.