Convexificators And Approximate Generalized Hessian Matrices For The Directionally Differentiable Functions

The notion of convexificator is originally given for the positively homogeneous functions, and it is defined as a convex and compact set. It can describe the upper convex approximation and lower concave approximation of the positively homogeneous functions. Because directional derivative is the positively homogeneous function, we apply the convexificator to the directional derivative of the directionally differentiable function in the application. With the deeper realization the notion of convexificator is extended for a continuous function and it is defined as a closed set and it is not necessarily convex or compact set. As the smaller convexificator can describe the function better, so the notion of minimal convexificator is introduced. But the question of finding conditions for minimal convexificators of a continuous function and also of guaranteeing uniqueness of minimal convexificators has reminded so far open. The main results, obtained in this dissertation, may be summarized as follows:1. Chapter 2, because the quasidifferentiable function is an important class of the non-smooth analysis. This dissertation introduces the convexificator to the quasidifferentiable function. The conclusion we draw is that the convexificators of quasidifferentiable functions is closed about linear operation; we construct two convexificators for quasidifferentiable functions(one is smaller than the other); minimizing and maximizing quasidifferentiable functions still admit a convexificator, and their operational formulas are given.2. Chapter 3, in terms of the generalized convexificator, we present the characterization of pseudoconvexity and quasiconvexity; K-T sufficient condition with equality and inequality constraints is presented, various calculus rules and extremality about convexificators are given.3. Chapter 4, the definitions of the generalized second-order directional derivative and the upper and lower approximate generalized Hessian matrices are introduced; generalized Taylor's expansions for directionally differentiable functions are presented by using approximate generalized Hessian; we present conditions in terms of the set of extreme points for minimal and the unique minimal approximate generalized Hessians; Several calculus rules are given for directionally differentiable functions in terms of approximate generalized Hessian.