Improved scatter search for the global optimization of computationally expensive dynamic models

A new algorithm for global optimization of costly nonlinear continuous problems is presented in this paper. The algorithm is based on the scatter search metaheuristic, which has recently proved to be efficient for solving combinatorial and nonlinear optimization problems. A kriging-based prediction method has been coupled to the main optimization routine in order to discard the evaluation of solutions that are not likely to provide high quality function values. This makes the algorithm suitable for the optimization of computationally costly problems, as is illustrated in its application to two benchmark problems and its comparison with other algorithms.     

Sampling aspects of rough set theory

Rough Set Theory (RST) originated as an approach to approximating a given set, but has found its main applications in the statistical domain of classification problems. It generates classification rules, and can be seen in general terms as a technique for rule induction. Expositions of RST often stress that it is robust in requiring no (explicit) assumptions of a statistical nature. The argument here, however, is that this apparent strength is also a weakness which prevents establishment of general statistical properties and comparison with other methods. A sampling theory is developed for the first time, using both the original RST model and its probabilistic extension, Variable Precision Rough Sets. This is applied in the context of examples, one of which involves Fishers Iris data.Bruce Curry: The author is grateful to two anonymous referees for various helpful suggestions     

Technical diagnostic of a fleet of vehicles using rough set theory

The paper presents a process of technical diagnostic applied to a fleet of vehicles utilized in the delivery system of express mail. It is focused on evaluation of diagnostic capacity of particular characteristics, reduction of a set of initially selected characteristics to a minimal and satisfactory subset, recognition of a technical condition of vehicles resulting in their condition-based classification. In addition, the decision rules facilitating technical diagnostic and management of a fleet of vehicles are generated and utilized. N-fold cross validation is applied to estimate the efficiency of the decision rules. The rough set theory is applied to support the diagnostic process of vehicles. Classical rough set (CRS) theory is compared with the dominance-based rough set (DRS) approach. The results of computational experiments for both approaches are compared.     

A rough set approach to the characterization of transversal matroids

Rough sets are efficient for data pre-processing during data mining. However, some important problems such as attribute reduction in rough sets are NP-hard and the algorithms required to solve them are mostly greedy ones. The transversal matroid is an important part of matroid theory, which provides well-established platforms for greedy algorithms. In this study, we investigate transversal matroids using the rough set approach. First, we construct a covering induced by a family of subsets and we propose the approximation operators and upper approximation number based on this covering. We present a sufficient condition under which a subset is a partial transversal, and also a necessary condition. Furthermore, we characterize the transversal matroid with the covering-based approximation operator and construct some types of circuits. Second, we explore the relationships between closure operators in transversal matroids and upper approximation operators based on the covering induced by a family of subsets. Finally, we study two types of axiomatic characterizations of the covering approximation operators based on the set theory and matroid theory, respectively. These results provide more methods for investigating the combination of transversal matroids with rough sets.     

A special ordered set approach for optimizing a discontinuous separable piecewise linear function

We give a special ordered set (SOS) approach that optimizes a discontinuous separable piecewise linear function, even when a mixed-integer programming (MIP) model is not available for it. When a MIP model is available, our SOS model gives a linear programming relaxation bound that is as good as the MIPs.     

Implementation of scatter search for multi-objective optimization: a comparative study

Interest in the design of efficient meta-heuristics for the application to combinatorial optimization problems is growing rapidly. The optimal design of water distribution networks is an important optimization problem which consists of finding the best way of conveying water from the sources to the users, thus satisfying their requirements. The efficient design of looped networks is a much more complex problem than the design of branched ones, but their greater reliability can compensate for the increase in cost when closing some loops. Mathematically, this is a non-linear optimization problem, constrained to a combinatorial space, since the diameters are discrete and it has a very large number of local solutions. Many works have dealt with the minimization of the cost of the network but few have considered their cost and reliability simultaneously. The aim of this paper is to evaluate the performance of an implementation of Scatter Search in a multi-objective formulation of this problem. Results obtained in three benchmark networks show that the method here proposed performs accurately well in comparison with other multi-objective approaches also implemented.     

A new methodology for sustainability measurement of banks based on rough set theory

Central European Journal of Operations Research - Sustainability measurement of banks is a complicated process, mainly dependent on the intuition of banking and sustainability experts. In this...     

An approach to the constructivization of Cantor's set theory

A new approach is proposed for the construction of constructive analogs of set theory in hyperarithmetic languages , where is a scale of constructive ordinals. For every ordinal in the language , a special relation of equality = is defined for codes of one-parameter formulas (conditions) of the level in a constructive hyperarithmetic hierarchy corresponding to the scale . The membership relation, (also expressible in the language ), is defined by the conditionx y=z(z= x&z y), where the relation is obtained by suitable refinement of the traditional representations of the constructive relation of membership. This results in a hierarchy of constructive analogsM of the theory of sets (in which the sets are represented by codes of conditions of level , identified modulo the relation =, and is taken as the relation of membership). Some properties of this hierarchy are introduced which show that for the limits ,M is sufficiently rich from the traditional set theoretic standpoint.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 68, pp. 38–49, 1977.     

Global optimization of expensive black box problems with a known lower bound

In this paper we propose an algorithm for the global optimization of computationally expensive black–box functions. For this class of problems, no information, like e.g. the gradient, can be obtained and function evaluation is highly expensive. In many applications, however, a lower bound on the objective function is known; in this situation we derive a modified version of the algorithm introduced in Gutmann (J Glob Optim 19:201–227, 2001). Using this information produces a significant improvement in the quality of the resulting method, with only a small increase in the computational cost. Extensive computational results are provided which support this statement.     

Classical descriptive set theory as a refinement of effective descriptive set theory

The (effective) Suslin-Kleene Theorem is obtained as a corollary of a standard proof of the classical Suslin Theorem, by noticing that it is mostly constructive and applying to it a naive realizability interpretation.     

Classical descriptive set theory as a refinement of effective descriptive set theory

The (effective) Suslin–Kleene Theorem is obtained as a corollary of a standard proof of the classical Suslin Theorem, by noticing that it is mostly constructive and applying to it a naive realizability interpretation.     

A dominance-based rough set approach applied to evaluate the credit risk of sovereign bonds

4OR - Even though sovereign bonds represent low-risk alternatives that give investors a healthy income, the risk assessment process for these bonds is still considered subjective because of the...     

基于熵权的投资评价模型在风险投资中的应用

本着“实用性和现实操作性”原则,本文根据风险投资评价的实际操作,在引入粗糙集信息熵理论,导出基于多指标评价的熵权投资模型的基础上,通过问卷调查的实证研究方法,确定评价指标和权重,并例举实际(经适当简化)案例演算具体运算过程,以验证在实际风险投资中的可操作性。从而试图克服目前相关领域研究文献基本停留在方法研究阶段、所给的证例过于简单、没有实际运用价值的缺陷,也尝试探索粗糙集理论在风险投资管理中的应用。     

Rough set theory: a novel approach for extraction of robust decision rules based on incremental attributes

Rough set theory is a new data mining approach to manage vagueness. It is capable to discover important facts hidden in the data. Literature indicate the current rough set based approaches can’t guarantee that classification of a decision table is credible and it is not able to generate robust decision rules when new attributes are incrementally added in. In this study, an incremental attribute oriented rule-extraction algorithm is proposed to solve this deficiency commonly observed in the literature related to decision rule induction. The proposed approach considers incremental attributes based on the alternative rule extraction algorithm (AREA), which was presented for discovering preference-based rules according to the reducts with the maximum of strength index (SI), specifically the case that the desired reducts are not necessarily unique since several reducts could include the same value of SI. Using the AREA, an alternative rule can be defined as the rule which holds identical preference to the original decision rule and may be more attractive to a decision-maker than the original one. Through implementing the proposed approach, it can be effectively operating with new attributes to be added in the database/information systems. It is not required to re-compute the updated data set similar to the first step at the initial stage. The proposed algorithm also excludes these repetitive rules during the solution search stage since most of the rule induction approaches generate the repetitive rules. The proposed approach is capable to efficiently and effectively generate the complete, robust and non-repetitive decision rules. The rules derived from the data set provide an indication of how to effectively study this problem in further investigations.     

A comparison of parallel large-scale knowledge acquisition using rough set theory on different MapReduce runtime systems

Nowadays, with the volume of data growing at an unprecedented rate, large-scale data mining and knowledge discovery have become a new challenge. Rough set theory for knowledge acquisition has been successfully applied in data mining. The recently introduced MapReduce technique has received much attention from both scientific community and industry for its applicability in big data analysis. To mine knowledge from big data, we present parallel large-scale rough set based methods for knowledge acquisition using MapReduce in this paper. We implemented them on several representative MapReduce runtime systems: Hadoop, Phoenix and Twister. Performance comparisons on these runtime systems are reported in this paper. The experimental results show that (1) The computational time is mostly minimum on Twister while employing the same cores; (2) Hadoop has the best speedup for larger data sets; (3) Phoenix has the best speedup for smaller data sets. The excellent speedups also demonstrate that the proposed parallel methods can effectively process very large data on different runtime systems. Pitfalls and advantages of these runtime systems are also illustrated through our experiments, which are helpful for users to decide which runtime system should be used in their applications.     

The uncovered set and indifference in spatial models: A fuzzy set approach

The uncovered set was developed in order to predict outcomes when spatial models result in an empty core. In contrast to conventional approaches, fuzzy spatial models induce a substantial degree of individual and collective indifference over alternatives. Hence, existing definitions of the covering relationship return differing results. We develop a definition for a fuzzy covering relation. Our definition results in an uncovered set that is, in most cases, contained within the Pareto set. We conclude by characterizing the exceptions.     

The strength of Mac Lane set theory

Saunders Mac Lane has drawn attention many times, particularly in his book Mathematics: Form and Function, to the system of set theory of which the axioms are Extensionality, Null Set, Pairing, Union, Infinity, Power Set, Restricted Separation, Foundation, and Choice, to which system, afforced by the principle, , of Transitive Containment, we shall refer as . His system is naturally related to systems derived from topos-theoretic notions concerning the category of sets, and is, as Mac Lane emphasises, one that is adequate for much of mathematics. In this paper we show that the consistency strength of Mac Lane's system is not increased by adding the axioms of Kripke–Platek set theory and even the Axiom of Constructibility to Mac Lane's axioms; our method requires a close study of Axiom H, which was proposed by Mitchell; we digress to apply these methods to subsystems of Zermelo set theory , and obtain an apparently new proof that is not finitely axiomatisable; we study Friedman's strengthening of , and the Forster–Kaye subsystem of , and use forcing over ill-founded models and forcing to establish independence results concerning and ; we show, again using ill-founded models, that proves the consistency of ; turning to systems that are type-theoretic in spirit or in fact, we show by arguments of Coret and Boffa that proves a weak form of Stratified Collection, and that is a conservative extension of for stratified sentences, from which we deduce that proves a strong stratified version of ; we analyse the known equiconsistency of with the simple theory of types and give Lake's proof that an instance of Mathematical Induction is unprovable in Mac Lane's system; we study a simple set theoretic assertion—namely that there exists an infinite set of infinite sets, no two of which have the same cardinal—and use it to establish the failure of the full schema of Stratified Collection in ; and we determine the point of failure of various other schemata in . The paper closes with some philosophical remarks.