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.

Stationarity and regularity of infinite collections of sets. Applications to infinitely constrained optimization

This article continues the investigation of stationarity and regularity properties of infinite collections of sets in a Banach space started in Kruger and López (J. Optim. Theory Appl. 154(2), 2012), and is mainly focused on the application of the stationarity criteria to infinitely constrained optimization problems. We consider several settings of optimization problems which involve (explicitly or implicitly) infinite collections of sets and deduce for them necessary conditions characterizing stationarity in terms of dual space elements—normals and/or subdifferentials.

Pitfalls in the Evaluation of the Thermodynamic Consistency of Experimental VLE Data Sets

The thermodynamic consistency of almost 90 VLE data series, including isothermal and isobaric conditions for systems of both total and partial miscibility in the liquid phase, has been examined by means of the area and point-to-point tests. In addition, the Gibbs energy of mixing function calculated from these experimental data has been inspected, with some rather surprising results: certain data sets exhibiting high dispersion or leading to Gibbs energy of mixing curves inconsistent with the total or partial miscibility of the liquid phase, surprisingly, pass the tests. Several possible inconsistencies in the tests themselves or in their application are discussed. Related to this is a very interesting and ambitious initiative that arose within the NIST organization: the development of an algorithm to assess the quality of experimental VLE data. The present paper questions the applicability of two of the five tests that are combined in the algorithm. It further shows that the deviation of the experimental VLE data from the correlation obtained by a given model, the basis of some point-to-point tests, should not be used to evaluate the quality of these data.

Algorithm for the detection of outliers based on the theory of rough sets

Outliers are objects that show abnormal behavior with respect to their context or that have unexpected values in some of their parameters. In decision-making processes, information quality is of the utmost importance. In specific applications, an outlying data element may represent an important deviation in a production process or a damaged sensor. Therefore, the ability to detect these elements could make the difference between making a correct and an incorrect decision. This task is complicated by the large sizes of typical databases. Due to their importance in search processes in large volumes of data, researchers pay special attention to the development of efficient outlier detection techniques. This article presents a computationally efficient algorithm for the detection of outliers in large volumes of information. This proposal is based on an extension of the mathematical framework upon which the basic theory of detection of outliers, founded on Rough Set Theory, has been constructed. From this starting point, current problems are analyzed; a detection method is proposed, along with a computational algorithm that allows the performance of outlier detection tasks with an almost-linear complexity. To illustrate its viability, the results of the application of the outlier-detection algorithm to the concrete example of a large database are presented.

With or without you...judging politically in the field of Area of Freedom, Security and Justice? Research Papers in Law, 3/2007

From the Introduction. The study of the European Court of Justice’s (ECJ) case law of the regarding the Area of Freedom Security and Justice (AFSJ) is fascinating in many ways.1 First, almost the totality of the relevant case law is extremely recent, thereby marking the first ‘foundational’ steps in this field of law. This is the result of the fact that the AFSJ was set up by the Treaty of Amsterdam in 1997 and only entered into force in May 1999.2 Second, as the AFSJ is a new field of EU competence, it sets afresh all the fundamental questions – both political and legal – triggered by European integration, namely in terms of: a) distribution of powers between the Union and its member states, b) attribution of competences between the various EU Institutions, c) direct effect and supremacy of EU rules, d) scope of competence of the ECJ, and e) measure of the protection given to fundamental rights. The above questions beg for answers which should take into account both the extremely sensible fields of law upon which the AFSJ is anchored, and the EU’s highly inconvenient three-pillar institutional framework.3 Third, and as a consequence of the above, the vast majority of the ECJ’s judgments relating to the AFSJ are a) delivered by the Full Court or, at least, the Grand Chamber, b) with the intervention of great many member states and c) often obscure in content. This is due to the fact that the Court is called upon to set the foundational rules in a new field of EU law, often trying to accommodate divergent considerations, not all of which are strictly legal.4 Fourth, the case law of the Court relating to the AFSJ, touches upon a vast variety of topics which are not necessarily related to one another. This is why it is essential to limit the scope of this study. The content of, and steering for, the AFSJ were given by the Tampere European Council, in October 1999. According to the Tampere Conclusions, the AFSJ should consist of four key elements: a) a common immigration and asylum policy, b) judicial cooperation in both civil and penal matters, c) action against criminality and d) external action of the EU in all the above fields. Moreover, the AFSJ is to a large extent based on the Schengen acquis. The latter has been ‘communautarised’5 by the Treaty of Amsterdam and further ‘ventilated’ between the first and third pillars by decisions 1999/435 and 1999/436.6 Judicial cooperation in civil matters, mainly by means of international conventions (such as the Rome Convention of 1981 on the law applicable to contractual obligations) and regulations (such as (EC) 44/20017 and (EC) 1348/20008) also form part of the AFSJ. However, the relevant case law of the ECJ will not be examined in the present contribution.9 Similarly, the judgments of the Court delivered in the course of Article 226 EC proceedings against member states, will be omitted.10 Even after setting aside the above case law and notwithstanding the fact that the AFSJ only dates as far back as May 1999, the judgments of the ECJ are numerous. A simple (if not simplistic) categorisation may be between, on the one hand, judgments which concern the institutional setting of the AFSJ (para. 2) and, on the other, judgments which are related to some substantive AFSJ policy (para. 3).

Rounding out a satisfactory Trio Presidency: Greece sets the stage for its Italian successor. CEPS Commentary, 3 July 2014

Expectations of the Greek presidency were not high: the budget was limited, the legislative term was drawing to a close and the European Parliament dissolved in mid-April for the elections. However, Greece made the most of its resources to progress on some very important dossiers and brought about a satisfactory close to the Trio presidency previously held by Ireland and Lithuania. The Greek presidency managed to finalise work on the Trio priorities, mainly in relation to banking union, the Digital Agenda, the competitiveness of EU enterprises and the Compact for Growth and Jobs. It also advanced legislation to tackle tax evasion as a necessary complement to spending cuts, and set the agenda for migration and maritime affairs, in close cooperation with Italy.