| Optimization theory is an important part of applied mathematics, which has a wide range of applications in real life.Many kinds of approaches have been used for researching Optimization theory.In 17th century, there emerged variational principle, which was based on the calculus established by such scientists as Newton and Leibniz c. In 1947 Danzig presented the conception of the famous simplex algorithm. in 1979, the mathematician Khachiyan in the former Soviet Union put forward the first so-called polynomial-time algorithm--ellipsoid method, which is better than variational principle in theory.At the same time, form the linear programming represented by the Soviet UnionЛ.B. Cantor Petrovich and the United States George Bernard Dantzig; the nonlinear programming represented by by Kuhn and Tucker; the dynamic programming represented by American R. Bellman; the maximum principle represented by the Soviet UnionЛ.C Pontryagin, etc. As one of the most important tool for optimization-Convex Cone has becoming the focus of the study of scholars.This thesis can be summarized in two parts:Firstly, the critical angles of convex cones are investigated. The corresponding concept in Rn" is generalized to Banach spaces and some properties of the critical angles in Banach spaces are given. Especially, a sufficient and necessary condition for critical angles in Hilbert spaces is proposed.Secondly, the concept of P cone in vector spaces is introduced. The relationships between Bishop-Phelps cone, nuclear cone and the P cone are discussed. Some properties of P cone in vector spaces and locally convex spaces are givenThis thesis extends the concept of critical angle and gets some corresponding conclusions in Banach space, which provides an effective tool for measuring the critical angle. At the same time, the properties of P cone in vector space and locally convex space are discussed to provide theories support for researching the efficiency of the Pareto and Ekeland variational principle.
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