The Application Of Uv Decomposition In Semismoothness

Lemarechal,Mifflin,Sagastizabal and Oustry (2000)introduced the uv-theory, which opens a way to defining a suitable restricted second-order derivative of a convex function f at a nondifferentiable point x. The basic idea is to decompose R~n into two orthogonal subspaces u and v depending on x so that f's nonsmoothness near the point is concentrated essentially in V. A certain Lagrangian associated with the convex function was introduced, called u-Lagrangian. When f satisfies certain structural properties, it is possible to find smooth trajectories, via the intermediate function, yielding a second-order expansion for f. It is an important tool to research about nonsmooth optimization, especially about the function's second-order optimality conditions, and design algorithms (by means of local quadratic approximations).The extension of the uv-decomposition theory to nonconvex function, as well as the study on the second-order properties of nonsmooth functions, are unquestionably important. The focus of this thesis is on the application of the uv-decomposition theory to semismooth optimization problems.In this paper, we focus on the study of uv-theory and its application. We briefly state its contents as follows.1. In chapter 1, we recall the historical background about the uv decomposition theory,the origin of u-Lagrangian and the relative knowledge about semismoothness.2. In chapter 2, we mainly review the uv decomposition theory by Mifflin,Sagastizabaland Oustry, and the convex function's NLP with finite constraints.3. In chapter 3, we apply uv-theory to semismoothness, obtain the properties of u-Lagrangian, the set of minimizers and the second-order expansion in u-space. then, in NLP setting, we discuss the maximum function and D.S. programming with semismooth properties, based on the uv decomposition theory, and study their u-Lagrangians, second-order expansions. we propose a space-decomposition algorithm, namely uv-decomposition algorithm, and demonstrate its superlinear convergence is demonstrated.