In recent years, a large number of evolutionary and other population-based heuristics were proposed in the literature. In 2009, we suggested to combine the very efficient bacterial evolutionary algorithm with local search as a new Discrete Bacterial Memetic Evolutionary Algorithm (DBMEA) (Farkas et al., In: Towards intelligent engineering & information technology, Studies in Computational Intelligence, Vol 243. Berlin, Germany: Springer-Verlag; 2009. pp 607–625). The method was tested on one of Traveling Salesman Problem (TSP) benchmark problems, and a difference was found between the real optimum calculated by the new and the published result because the Concorde and the Lin–Kernighan algorithm use an approximation substituting distances of points by the closest integer values. We modified the Concorde algorithm using real cost values to compare with our results. In this paper, we systematically investigate TSPLIB benchmark problems and other VLSI benchmark problems (http://www.math.uwaterloo.ca/tsp/vlsi/index.html) and compare the following values: optima found by the DBMEA heuristic and by the modified Concorde algorithm with real cost values, run times of DBMEA, modified Concorde, and Lin–Kernighan heuristic. In this paper, for the evaluation of metaheuristic techniques, we suggest the usage of predictability of the successful run in addition to the accuracy of the result and the computational cost as third property. We will show that in the case of DBMEA, the run time is more predictable than in the case of Concorde algorithm, so we suggest the use of DBMEA heuristic as very efficient for the solution of TSP and other nondeterministic polynomial-time hard optimization problems.

Preface by the Editors of the Special Issue on Computational Intelligence and MathematicsMathematics is the foundation of almost all areas of sciences, especially it is so with Engineering, Computer Science, Physics, Chemistry, and Business, and new mathematical models and approaches continuously improve the efficiency of current methodologies and provide solutions for new challenges.An important and emerging field of Computer Science is Computational Intelligence (CI), whose aim is to provide methods to be able to deal with complex real-world problems for which traditional approaches are not feasible.Some of the methods that CI encompasses are, among others, fuzzy logic, evolutionary computation, neural networks, as well as probabilistic and statistical approaches, such as Bayesian networks or kernel methods.Approaches based on CI often provide a compromise between resource intensity and accuracy or precision of the solution.So, for example, NP-hard problems, which are known to be intractable (at least as far no polynomial complexity algorithm has been ever found for any of the-mathematically equivalent-NP-hard problems), may be rather well solved for limited size and limited problem structure by various heuristics.The classical Traveling Salesman Problem (TSP) may be rather well tackled by the Lin-Kernighan heuristics for smaller size graphs (up to a few 100), while the CONCORDE approach delivers good results up to sizes 1000-1500.However, for problems with sizes over 2000 often there is no known solution as far.This is an excellent training field for CI approaches, and reference data sets are available in abundance.It is clear that both areas, CI and Mathematics, are closely related since the latter is the fundamental base of the former, and continuous interactions between them will bring more and more robust and efficient CI models and approaches-while sometimes the new CI methods deliver unexpected results useful even for getting closer to the solution of mathematically unsolved problems.This special issue includes a small selection of papers written by scientists and engineers working in the field of CI and applied mathematics and proposes some new results that might interest fellow scientists in both fields.The first paper by Rodriguez-Lorenzo et al. defines a sound and complete inference system for triadic conditional attribute implication (CAI) generated from a formal triadic context and expressed as a set of axioms "a la Armstrong."Moreover, it proposes a method to compute CAIs from Biedermann's implications and introduces an algorithm to compute the closure of an attribute set X with respect to a set of CAIs given a set of conditions.

In our previous papers, fuzzy model identification methods were discussed. The bacterial evolutionary algorithm for extracting fuzzy rule base from a training set was presented. The Levenberg–Marquardt method was also proposed for determining membership functions in fuzzy systems. The combination of the evolutionary and the gradient-based learning techniques is usually called memetic algorithm. In this paper, a new kind of memetic algorithm, the bacterial memetic algorithm, is introduced for fuzzy rule extraction. The paper presents how the bacterial evolutionary algorithm can be improved with the Levenberg–Marquardt technique. © 2009 Wiley Periodicals, Inc.

One of the main types of information retrieval systems produces a word frequency measure estimated by some important parts of the document using neural network approaches. This paper reports a general neural network for this task. It is specialised considering the main difficulties of these kinds of applications, namely, the calculation time complexity. It will be pointed out that the calculation, hence, the learning time could be much reduced applying the reduction algorithm proposed here. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.