The Proximal Subgradient And The Critical Angles Of Convex Cone In Banach Space

Optimization theory is not only the most important part of optimization, but also an important theoretical basis in operations research, which has a wide range of applications in real life. As a fundamental theory of optimization theory, the proximal subgradient is a crucial point for the differentiation property in non-differentiable functions. The first "subdifferential type" of definition was given by F. H. Clarke in the1970’s, with the "generalized gradient". In recent decades, F. H. Clarke and R. M. Redhee had given several classical subdifferentials: Clarke subdifferential, Michel-Penot subdifferential, Dini subdifferential, Frechet subdifferential, m-subdifferential etc, which were used to study for the nonlinear programming problem, minimal time function problem, Augmented Lagrange multiplier problem etc. G. Colombo and P. R. Wolenski studied the minimal time function with constant dynamics in the context of a Hilbert space using the limiting subdifferential which is defined by proximal sungradient in2004. Bernard, Thibault and Zlateva studied proximal subgradient in Banach space under the assumption that the space has an equivalent norm which is both uniformly convex and uniformly smooth in2006. From2007to2009, M. Soleimani-damaneh studied Characterization of nonsmooth quasiconvex and pseudoconvex functions in R", and some characterizations for generalized invexity and generalized monotonicity under separable Hilbert spaces using the proximal subdifferential. Therefore, it is known that proximal subdifferential is an important theoretical basis in variational analysis, and make proximal sungradient to be focus of domestic and foreign scholars. At the same time, as one of the most important tool for optimization-Convex Cone has also becoming the focus of the study of scholars.This article can be summarized in two parts:In the first part, by using of the concepts, properties and application of proximal sungradient in R" or Hilbert space, the paper introduces the concepts and properties of the normalized duality mapping in Banach space, then we conclude the concept of proximal sungradient for the lower semicontinuous function. Then, this paper introduces the necessary condition for proximal subgradient in uniformly Gateaux smooth Banach space.In the second part, by using of the method of searching critical angle of convex cone in R",the paper introduces the corresponding concept in R" is generalized to Banach spaces and some properties of the critical angles in Banach spaces are given. At the same time, redefined the Gramian matrix and discusses some theories of polyhedron cone.