Quasiconcave vector maximization: Connectedness of the sets of Pareto-optimal and weak Pareto-optimal alternatives

We study the sets of Pareto-optimal and weakly Pareto-optimal solutions to a vector maximization problem defined by a continuous vector-valued quasiconcave criterion functionf and a closed convex set of alternativesS. IfS is compact, it is shown that the set of weakly Pareto-optimal alternatives is connected, but that the set of Pareto-optimal alternatives is not necessarily connected. However, the set of Pareto optima is shown to be connected for some important subclasses of quasiconcave functions. We also provide some reasonable conditions under which the compactness assumption onS may be relaxed and connectedness maintained.     

Pruned Pareto-optimal sets for the system redundancy allocation problem based on multiple prioritized objectives

In this paper, a new methodology is presented to solve different versions of multi-objective system redundancy allocation problems with prioritized objectives. Multi-objective problems are often solved by modifying them into equivalent single objective problems using pre-defined weights or utility functions. Then, a multi-objective problem is solved similar to a single objective problem returning a single solution. These methods can be problematic because assigning appropriate numerical values (i.e., weights) to an objective function can be challenging for many practitioners. On the other hand, methods such as genetic algorithms and tabu search often yield numerous non-dominated Pareto optimal solutions, which makes the selection of one single best solution very difficult. In this research, a tabu search meta-heuristic approach is used to initially find the entire Pareto-optimal front, and then, Monte-Carlo simulation provides a decision maker with a pruned and prioritized set of Pareto-optimal solutions based on user-defined objective function preferences. The purpose of this study is to create a bridge between Pareto optimality and single solution approaches.     

The difference between common knowledge of formulas and sets

Common knowledge can be defined in at least two ways: syntactically as the common knowledge of a set of formulas or semantically, as the meet of the knowledge partitions of the agents. In the multi-agent S5 logic with either finitely or countably many agents and primitive propositions, the semantic definition is the finer one. For every subset of formulas that can be held in common knowledge, there is either only one member or uncountably many members of the meet partition with this subset of formulas held in common knowledge. If there are at least two agents, there are uncountably many members of the meet partition where only the tautologies of the multi-agent S5 logic are held in common knowledge. Whether or not a member of the meet partition is the only one corresponding to a set of formulas held in common knowledge has radical implications for its topological and combinatorial structure.     

Ben's consolidation of knowledge structures about infinite sets

According to a recently proposed model for processes of abstraction in context, the construction of a new structure is to be followed by a consolidation phase. In this paper, we develop an empirically based, theoretical analysis of consolidation that emerges from a sequence of interviews about the comparison of infinite sets with a talented student. We take for granted that construction has occurred in the first interview and analyze the second one. Our analysis shows that consolidation can be identified by means of the psychological and cognitive characteristics of self-evidence, confidence, immediacy, flexibility and awareness. We also found three modes of thinking conducive to consolidation, one related to problem solving, one to reflective activity and an intermediate one.     

Perfect difference sets constructed from Sidon sets

A set of positive integers is a perfect difference set if every nonzero integer has a unique representation as the difference of two elements of . We construct dense perfect difference sets from dense Sidon sets. As a consequence of this new approach we prove that there exists a perfect difference set such that

A methodology for knowledge discovery to support product family design

This paper introduces a methodology for knowledge discovery related to product family design that integrates an ontology with data mining techniques. In the proposed methodology, the ontology represents attributes for the components of products in functional hierarchies. Fuzzy clustering is employed for data mining to first partition product functions into subsets for identifying modules in a given product family and then identify the similarity level of components in a module. Module categorization is introduced to support association rule mining for knowledge discovery related to platform design. We apply the proposed methodology to first develop and then utilize design knowledge for a family of power tools. Based on the developed design knowledge, a new platform is suggested to improve commonality in the power tool family.     

Incremental learning optimization on knowledge discovery in dynamic business intelligent systems

As business information quickly varies with time, the extraction of knowledge from the related dynamically changing database is vital for business decision making. For an incremental learning optimization on knowledge discovery, a new incremental matrix describes the changes of the system. An optimization incremental algorithm induces interesting knowledge when the object set varies over time. Experimental results validate the feasibility of the incremental learning optimization.     

Support vector machine classifiers with uncertain knowledge sets via robust optimization

In this article we study support vector machine (SVM) classifiers in the face of uncertain knowledge sets and show how data uncertainty in knowledge sets can be treated in SVM classification by employing robust optimization. We present knowledge-based SVM classifiers with uncertain knowledge sets using convex quadratic optimization duality. We show that the knowledge-based SVM, where prior knowledge is in the form of uncertain linear constraints, results in an uncertain convex optimization problem with a set containment constraint. Using a new extension of Farkas' lemma, we reformulate the robust counterpart of the uncertain convex optimization problem in the case of interval uncertainty as a convex quadratic optimization problem. We then reformulate the resulting convex optimization problems as a simple quadratic optimization problem with non-negativity constraints using the Lagrange duality. We obtain the solution of the converted problem by a fixed point iterative algorithm and establish the convergence of the algorithm. We finally present some preliminary results of our computational experiments of the method.     

On finding multiple Pareto-optimal solutions using classical and evolutionary generating methods

In solving multi-objective optimization problems, evolutionary algorithms have been adequately applied to demonstrate that multiple and well-spread Pareto-optimal solutions can be found in a single simulation run. In this paper, we discuss and put together various different classical generating methods which are either quite well-known or are in oblivion due to publication in less accessible journals and some of which were even suggested before the inception of evolutionary methodologies. These generating methods specialize either in finding multiple Pareto-optimal solutions in a single simulation run or specialize in maintaining a good diversity by systematically solving a number of scalarizing problems. Most classical generating methodologies are classified into four groups mainly based on their working principles and one representative method from each group is chosen in the present study for a detailed discussion and for its performance comparison with a state-of-the-art evolutionary method. On visual comparisons of the efficient frontiers obtained for a number of two and three-objective test problems, the results bring out interesting insights about the strengths and weaknesses of these approaches. The results should motivate researchers to design hybrid multi-objective optimization algorithms which may be better than each of the individual methods.     

Generating boxes from ordered sets and graphs

LetE be a set and letL be a family of subsets ofE. A subsets ofE is called atransversal ofL ifs intersects each member ofL in exactly one element, that is |sl|=1, for everyl inL. We denoteT(L) the set of all transversals ofL. A pairB=(L, C) of families of subsets ofE is abox onE if it satisfies the following conditions:

Codebooks from almost difference sets

In direct spread CDMA systems, codebooks meeting the Welch bounds are used to distinguish among the signals of different users. Recently, constructions of codebooks with difference sets meeting Welch’s bound on the maximum cross-correlation amplitude were developed. In this paper, a generic construction of codebooks using almost difference sets is considered and several classes of codebooks nearly meeting the Welch bound are obtained. The parameters of the codebooks constructed in this paper are new.      

Reconstructing plane sets from projections

We give some uniqueness results for the problem of determining a finite set in the plane knowing its projections alongm directions. We apply the results to the problem of the reconstruction of a homogeneous convex body with a finite set of spherical disjoint holes. Ifm X-ray pictures with directions in some plane are given, then the problem is well posed provided the number of the holes is less than or equal tom and the set of the directions satisfies a suitable condition.     

Homogeneous sets from several ultrafilters

We consider Ramsey-style partition theorems in which homogeneity is asserted not for subsets of a single infinite homogeneous set but for subsets whose elements are chosen, in a specified pattern, from several sets in prescribed ultrafilters. We completely characterize the sequences of ultrafilters satisfying such partition theorems. (Non-isomorphic selective ultrafilters always work, but, depending on the specified pattern, weaker hypotheses on the ultrafilters may suffice.) We also obtain similar results for analytic partitions of the infinite sets of natural numbers. Finally, we show that the two P-points obtained by applying the maximum and minimum functions to a union ultrafilter are never nearly coherent.     

Higher generation subgroup sets and the $ \Sigma $-invariants of graph groups

We present a general condition, based on the idea of n-generating subgroup sets, which implies that a given character represents a point in the homotopical or homological -invariants of the group G. Let be a finite simplicial graph, the flag complex induced by , and the graph group, or 'right angled Artin group', defined by . We use our result on n-generating subgroup sets to describe the homotopical and homological -invariants of in terms of the topology of subcomplexes of . In particular, this work determines the finiteness properties of kernels of maps from graph groups to abelian groups. This is the first complete computation of the -invariants for a family of groups whose higher invariants are not determined - either implicitly or explicitly - by ¹. Received: October 18, 1996     

A note on knowledge discovery using neural networks and its application to credit card screening

We address an important issue in knowledge discovery using neural networks that has been left out in a recent article “Knowledge discovery using a neural network simultaneous optimization algorithm on a real world classification problem” by Sexton et al. [R.S. Sexton, S. McMurtrey, D.J. Cleavenger, Knowledge discovery using a neural network simultaneous optimization algorithm on a real world classification problem, European Journal of Operational Research 168 (2006) 1009–1018]. This important issue is the generation of comprehensible rule sets from trained neural networks. In this note, we present our neural network rule extraction algorithm that is very effective in discovering knowledge embedded in a neural network. This algorithm is particularly appropriate in applications where comprehensibility as well as accuracy are required. For the same data sets used by Sexton et al. our algorithm produces accurate rule sets that are concise and comprehensible, and hence helps validate the claim that neural networks could be viable alternatives to other data mining tools for knowledge discovery.     

Global investing risk: a case study of knowledge assessment via rough sets

This paper presents an application of knowledge discovery via rough sets to a real life case study of global investing risk in 52 countries using 27 indicator variables. The aim is explanation of the classification of the countries according to financial risks assessed by Wall Street Journal international experts and knowledge discovery from data via decision rule mining, rather than prediction; i.e. to capture the explicit or implicit knowledge or policy of international financial experts, rather than to predict the actual classifications. Suggestions are made about the most significant attributes for each risk class and country, as well as the minimal set of decision rules needed. Our results compared favorably with those from discriminant analysis and several variations of preference disaggregation MCDA procedures. The same approach could be adapted to other problems with missing data in data mining, knowledge extraction, and different multi-criteria decision problems, like sorting, choice and ranking.