Optimality Conditions Of A (Weak) Sharp Minimum In Optimization Problems

In this thesis, the optimality conditions of a (weak) sharp minimum for scalar-valued optimization problems are presented. The optimality conditions of generalized weak sharp minima for variational inequality problems are established. The optimality conditions of weak?-sharp minima for vector and set-valued optimization problems are investigated, respectively. This thesis is divided into eight chapters. It is organized as follows.In Chapter 1, we describe the development and researches on the topic of a (weak) sharp minimum for scalar-valued optimization problems, a (weak) sharp minimum for vector-valued optimization problems, a (weak)?-sharp minimum for set-valued optimization problems and a weak sharp solution for variational inequality problems, respectively. We also give the motivation and the main research works.In Chapter 2, we study sufficient optimality conditions for nonsmooth programming problems with inequality constraints. We present a first-order sufficient optimality condition for a local sharp minimum of order 1 when the objective and constraint functions are l-stable at some point. Moreover, we obtain a second-order sufficient optimality condition for a local sharp minimum of order 2 in terms of the lower Dini second-order directional derivative of the Lagrange function. Finally, we present an example to compare our result with others.In Chapter 3, we study weak sharp minima of higher order for optimization problems with cone-constraints. Some necessary conditions are established when the constrained function is Hadamard and Dini directional derivatives, respectively. In addition, a sufficient condition for weak sharp minima of order 1 is also presented when the objective function is a lower semicontinuous convex one and the constrained function is continuously differentiable.In Chapter 4, we investigate a variational inequality problem with functional constraints in finite-dimensional spaces. By virtue of a dual gap function, we convert the variational inequality problem into a convex programming problem with functional constraints. On this basis, the notion of generalized weak sharp minima is introduced. Some characterizations of the generalized weak sharp minima are obtained, and sufficient and necessary conditions for the solution set of the variational inequality problem are discussed. In Chapter 5, we study optimality conditions of weak sharp minima for vector optimization problems. First, we develop the characterization of a weak?-sharp Pareto minimizer by means of an oriented distance function. We investigate the optimality conditions of a weak?-sharp Pareto minimizer for the composition of two functions. Then, we establish a necessary condition of a weak?1-sharp Pareto minimizer in a cone-constrained optimization problem. We get a sufficient and necessary condition of a?1-sharp Pareto minimizer by using a variant of Mordukhovich normal cone when the objective function is strictly differentiable. We also present a sufficient condition of a?2-sharp Pareto minimizer by using a supported function when the objective function is strictly differentiable. Finally, we transfer the weak?-sharp Pareto minimizer of vector optimization problems in infinite-dimensional spaces into the weak?-sharp minimizer for a series of scalar optimization problems and investigate their equivalent relations. By virtue of a nonlinear scalarization function, we also develop the characterization of a weak?-sharp Pareto minimizer.In Chapter 6, we introduce the concept of?-sharp minima in set-valued optimization problems. First, we present some sufficient and necessary conditions in terms of outer limit of the set-valued map. Then, we establish necessary and sufficient conditions by the variant of the Mordukhovich normal cone and the lop-sided paratingent derivative, respectively. Finally, we develop the characterization of weak?-sharp minima by virtue of a generalized nonlinear scalarization function.In Chapter 7, we summarize the results of this thesis and make some discussions.