基于概念格约简理论的横贯拟阵及其模糊化研究

Matroid has the important application in combinatorial optimization, integer programming and network flow. The main tools of which scholars take advantage to study matroid are vector space, lattice theory, graph theory and fuzzy mathematics at present. Combining transversal matriod and the concept lattice and starting with the point of view of the concept lattice, this project shall apply the concept lattice reduction theory and fuzzy concept lattice theory to the research on transversal matroid and its fuzziness. The aims of this project are to expand the concept lattice’s application scope at the same time, to provide new approaches and efficiency algorithms for the development of matroid. The main content includes: (1)Discuss the relations between concept lattice theory and transversal matroid . (2) We investigate some properties of transversal matroid via the theory of concept lattice reduction.(3) Taking advantage of fuzzy concept lattice, we fuzzify the matroid and transversal matroid. And we take the fuzzy Galois connection as a tool to study the fuzziness of matroid. The accomplishment of this project will release the connections of matroid and the concept lattice, fuzzy matroid and fuzzy concept lattice, and provide new theory evidence for combinatorial optimization. As a result, this project could have broad application prospects.

拟阵在组合优化、整数规划和网络流等方面有着重要的应用，当前主要的研究工具为向量空间、图论、几何格和模糊数学等。本项目将横贯拟阵和概念格相结合，从概念格的角度出发，把概念格约简理论和模糊概念格应用于横贯拟阵，针对横贯拟阵及其模糊化展开研究。旨在扩大概念格应用领域，同时为拟阵的发展提供新的思路和有效算法,主要内容包括:(1)研究概念格与横贯拟阵的关系;(2)基于概念格属性约简理论展开对横贯拟阵若干性质的研究；(3)借鉴模糊概念格的建构思想展开对拟阵以及横贯拟阵的模糊化研究,并在总结自身工作基础上大胆设想和创新,以模糊Galois 联络为工具，从一个新的角度研究拟阵的模糊化。本项目的成功完成将揭示横贯拟阵与概念格，模糊拟阵和模糊概念格之间的联系,为组合优化等提供新的理论依据,必将具有广阔的应用前景。

概念格是知识发现和数据挖掘等领域的有力工具。拟阵在图论和离散优化等方面有着重要的应用。将概念格理论应用到拟阵的研究当中并进行模糊化研究无疑可以扩大概念格和拟阵的研究领域，并对相应的学科有着促进作用。本课题以概念格和模糊概念格约简为基础，探索(模糊)概念格约简理论在模糊横贯拟阵和模糊优化中的应用。并将拟阵等离散优化问题作用于(模糊)概念格属性约简本身，丰富和提高(模糊)概念格约简的速度。在区间模糊形式背景下构造了一类多水平区间模糊概念格，运用可辨识矩阵法对此类形式背景进行属性约简和粒约简。首次利用0-1整数规划求解概念格属性约简和粒约简，该方法克服了可辨识矩阵法的缺点，极大提高了概念格约简的运算速度。在此基础上，进行了几类广义凸函数的模糊化工作，用模糊分析的方法给出了区间值和模糊值不变凸优化的最优性条件和对偶。 给出了区间值和模糊弱一致凸优化问题的最优性条件。该项研究有助于丰富模糊数学的内涵，深化模糊优化的理论基础，扩大概念格和拟阵的研究领域，为模糊环境下数据挖掘和知识发现提供理论基础，并促进模糊数学，数据科学，最优化等方面的研究。