Some New Results On The Uv-Decomposition Method

In nonsmooth optimization, second-order derivatives constitute a significant tool both for deriving optimality (by means of optimality conditions) and developing algorithms (by means of local quadratic approximations). Therefore, the study concerning the theory and application of the second-order behaviour of nonsmooth function has been paid much attention.Lemarechal, Mifflin, Sagastizabal and Oustry [35] (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 Rn 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. But, there are no results on characterizing of the set of minimizers and the smoothness of the u-Lagrangian. Therefore the further study on the higher-order behaviour of convex or nonconvex functions is quite necessary for handling the related nonsmooth optimization problems.In this dissertation, we focus on the study of uv-theory and its application. We briefly state our researches as follows.1. In Chapter 2, the basic characteristic, upper semi-continuity and the continuity at 0 of the set of minimizers are presented and demonstrated. Some operations of the conjugate of u-Lagrangian of a finite convex function and some results on the radial strong convexity of the conjugate of u-Lagrangian are given. These results can help us to deeply understand the first and second-order behaviour of f on the u-space and the characteristic of the fast track.2. In Chapter 3, the uv-decomposition theory and the u-Lagrangian are provided for three special functions. Since the theory of uv-decomposition of spaces and subdifferentials of convex functions can not be directly extended to nonconvex functions, the notions of proximal subdifferential and regular subdifferential are needed for studying nonconvex functions. Based on these concepts, the uv-decomposition theory and the new representations of the u-Lagrangian for D. C. functions and lower semi-continuous functions are constructed, respectively. At the end of this chapter the uv-decomposition theory and the u-Lagrangian for convex functionals on the Hilbert spaces are presented.3. In Chapter 4, the uv-decomposition theory is applied to NLP. The results on the penalty function of constrained minimization with a finite number of constraints [35] are generalized.The uv-decomposition theory to the exact penalty functions in NLP for a semi-infinite minimization problem with a polytope and with a convex compact set are established.4. In Chapter 5, based on the uv-theory for D. C. functions introduced in Chapter 3, new opti-mality conditions for unconstrained D. C. programming and constrained D. C. programming are presented. A space-decomposition algorithm, namely uv-decomposition algorithm, for solving unconstrained D. C. programming problems is given, and its superlinear convergence is demonstrated. In particular, an uv-decomposition algorithm and its convergence theorems for a max-type D. C. programming, the uv-decomposition theory and the properties of the u-Lagrangian for lower-C2 functions are also provided.