Characterizations And Applications Of The Weak Sharp Minima For Non-smooth Optimization Problems

In this paper, we study the primal characterizations and dual characterizations of the weak sharp minima for unconstrained optimization problem (P) min f(x) subject to x ∈ X, where X is a Banach space,f:Xâ†'R is a proper lower semi-continuous extended-real-valued function. To solve the infinite optimization problem in a consistent way, we introduce a more generalized notion of weak sharp minima for unstained optimization problem (P). By using the directional derivative and tangent cone in Banach space and using normal cone and subdifferential in dual space, we study the geometrical characterizations of the weak sharp minima under the case that the objective function f is convex or nonconvex. The main works done in this dissertation are organized as follows.In the first part, we study the objective function f which is convex. For optimization problem with a closed convex constrained set, to decouple the roles of the objective function and constraint region in the most basic first-order optimality conditions, we always assume that the regularity condition MQC holds. Based on this, we consider varies characteriza-tions of weak sharp minima. Primal characterizations involve directional derivatives and tangent cones while dual characterizations involve subgradients and normal cones. The pri-mal characterizations are more elementary in the sense that they are derived directly from the definition whereas the dual characterizations require the application of dual results and properties of the subdifferential calculus. By using the dual skills and the convex analytic techniques in Banach spaces, the necessary and sufficient conditions of global weak sharp minima, local weak sharp minima and boundedly weak sharp minima for corresponding op-timization problem are obtained. In particular, when the constraint set is the whole space, we obtain the equivalent characterizations of weak sharp minima for (P) in general Banach spaces. Our results extend and improve the corresponding conclusions in convex optimiza-tion. As an application, we study the weak sharp minima for convex infinite optimization problem. By the broader definition of weak sharp minima, we obtain the equivalent rela-tionship between the infinite optimization problem and the essential unstained optimization problem. Then a number of variational characterizations of the weak sharp minima are obtained by the properties of the subdifferential of the supremum function.In the second part, we consider the objective function f which is nonconvex. We introduce the definition of D-subsmoothness and D-semi-subsmoothness for a function family, which is an extension of convexity. By the nonsmooth analysis and variational analysis, we provide the necessary and sufficient conditions of the local weak sharp minima for unconstrained optimization problem (P) in Banach spaces and Asplund spaces, respectively. Based on this, by the equivalent relationship with the essential unconstrained optimization problem, we study the infinite optimization problem. To solve this problem, we provide formulas of the subdifferential of the pointwise supremum function and the sum function under the assumption of D-subsmoothness or D-semi-subsmoothness. Then we obtain the Frechet subdifferential, limiting subdifferential and Clarke subdifferential characterizations of local weak sharp minima. Consider the relationship between the infinite optimization problem and the linear regularity in mathematical programming. We provide the dual equivalent characterization of the linear regularity for closed uniformly D-subsmooth or closed convex-composite sets. Most of our results are new and extend and improve the corresponding conclusions in the nonconvex case.