凸锥运算及其性质的讨论毕业论文.doc

本文首先给岀凸集、锥、凸锥的定义及其基本性质。其次，研究凸锥与子 空间的关系，通过对凸锥的具体分析得出“子空间必是凸锥，反Z不一定成立” 这一结果，接着就能否由一个凸锥扩充成一个子空间这个问题进行了讨论。第 三，本文讨论了凸集与凸锥Z间的关系，将屮的凸集与/?切屮的凸锥Z间建 立起一一对应关系，并由凸锥的定理推导出儿个凸集的定理。第四，研究了凸 锥的代数运算。最后，讨论了闭凸锥的相关性质，并将结果应用到关于向量子 空间的研究中，得到了线性代数中的一些结果，同时也得到了在线性规划和非 线性规划的最优性条件导出中起着重要作用的Fakas引理。 关键词凸集锥凸锥极锥 Abstract In this paper, firstly, the author gives definitions of convex sets, cone and convex cone as well as their basic qualities. Secondly, the author discusses relations between convex cone and sub-space. By the specific analysis about convex cone, the author reaches a conclusion that sub-space must be convex cone. On the contrary, the conclusion does not necessarily set up. And the author discusses the issue that whether convex cone can be expanded into sub-space. Thirdly, the author also discusses the relationships between convex sets and convex cone. In this section, the author sets up one-to-one relationship between the convex set in Rn and convex cone in Rn^ . At the same time , the author deduces some theorems of convex sets from that of convex cone. Following , the author studies the algebra operations of convex cone. At last, the author discusses some qualities of closed convex cone and applies the results to the studying about vector subspaces. Besides, the author obtains some results of linear algebra and the Farkas lemma which plays an important role in deducing the optimality conditions of linear programming and nonlinear programming. Key words : convex sets ； cone ； convex cone ； polar cone 目录 TOC \o 1-5 \h \z 摘 要 I ABSTRACT I I HYPERLINK K2 C2 A (4) /Cp(^,x)g= 1,^ 0,^ 0}, 由凸集的相关性质易证：Kd是一个凸集. 由(人/) G K],(人，兀)G K2 O XG /IjCpXG /UC2 = x g AjC] n/t)c2 O U (GMCJ， 人 +/^2 —1, U (AC.n^G )可定义为C,#C2? 人 1 = h 入?空0 下面证明G#c?是一个凸集： 证明：因对Vx.y ^Cj#C2及a w (0」)， 3xpx2,/11, A2, 使 x =人再 =A2x2,xl g C^x2 g C2,+ Z = 1,人 0 及为L 儿，“，“2，使)=“)i = 〃2)d )1 G G*2 G Cl\ + 仏=1，丄 0 ? 这样[\-a^x + ay (1—Q) (1—Q)人 (―)入+谢无+匸亦亦？ [(iM+谢 (1 -4) +Q“]C| . 乂(1 _q) 乂(1 _q)兀+ ay= (1-7)/?2x2 + ?//2 y2 (l-a)入 6