Finding reducts is one of the key problems in the increasing applications of rough set theory, which is also one of the bottlenecks of the rough set methodology. The population-based reduction approaches are attractive to find multiple reducts in the decision systems, which could be applied to generate multi-knowledge and to improve decision accuracy. In this paper, we design a multi-swarm synergetic optimization algorithm (MSSO) for rough set reduction and multi-knowledge extraction. It is a multi-swarm based search approach, in which different individual trends to be encoded to different reduct. The approach discovers the best feature combinations in an efficient way to observe the change of positive region as the particles proceed throughout the search space. The performance of our approach is evaluated and compared with Standard Particle Swarm Optimization (SPSO) and Genetic Algorithms (GA). Empirical results illustrate that the approach can be applied for multiple reduct problems and multi-knowledge extraction effectively.
It is shown that if g is of bounded variation in the sense of Hardy-Krause on ∏m i=1 [ai , bi ], then gχ ∏m i=1 (ai ,bi ) is of bounded variation there. As a result, we obtain a simple proof of Kurzweil’s multidimensional integration by parts formula.
Based on the randomness and fuzziness of the cloud model during the transformation from the qualitative concept to the quantitative numerical value, with the theory that any data distribution can be decompounded into several normal distributions, this paper puts forward a method of multi-classification based on the cloud model. By this method, multiple classification is transformed to a superposed cloud model with training samples as the cloud expectation, while the test samples are regarded as the `cloud droplets', and their classifications of membership degree in a cloud model can be calculated. Considering the effect of the number of training samples on the membership degree, the cloud model is weighted by the ratio of the total number of training samples to the number of training samples in a single class so that the data distribution of the samples can be balanced. The formula of multiple classification based on the cloud model has the structure identical to that of Support Vector Machines, and the hyper entropy in cloud models exerts similar punishment on the noise samples just like the loose coefficients in Support Vector Machines; therefore, the reasonability of the method is theoretically proved. Compared with Support Vector Machine, the method discussed in this paper does not require any large-scale quadratic programming, thus the algorithm of the method is simpler. Last but not the least, five types of data distribution samples are selected for the comparative experiment, and comparison is made with four other classification methods; the result shows that the accuracy and stability of the algorithm is high, and its implementations on the high dimensional multiple classifications are especially satisfactory.
We examine primitive roots modulo the Fermat number Fm = 2 2m + 1. We show that an odd integer n ≥ 3 is a Fermat prime if and only if the set of primitive roots modulo n is equal to the set of quadratic non-residues modulo n. This result is extended to primitive roots modulo twice a Fermat number.
In this paper, a necessity measure optimization model of linear programming problems with fuzzy oblique vectors is discussed. It is shown that the problems are reduced to linear fractional programming problems. Utilizing a special structure of the reduced problem, we propose a solution algorithm based on Bender's decomposition. A numerical example is given.