Generalized Conveity With Applications In Optimization Problems

In this thesis, the generalized convexity and their applications in optimization problems such as extremum problems, dual problems, Hahn-Banach theorems and system of vector quasi-equilibrium problems etc. are researched. The main research works are as follows:In Chapter 2, two properties of strictly preinvex functions are obtained. These properties include a sufficient condition in terms of intermediate-point strict prein-vexity and preinvexity and a necessary and sufficient condition in terms of the semistrict preinvexity and intermediate-point strict preinvexity. we also show that the ratio of preinvex functions is invex. Hence, we give a positive answer to the open question which was proposed by Yang, Yang and Teo in [1].In Chapter 3, a sufficient condition of the strictly B-preinvex function is firstly obtained. Then some properties of the strictly B-preinvex function are shown. Finally, some results for the extremum problem which the objective function is strictly B-preinvex are presented.In Chapter 4, the errors of Theorem 4.6 and Theorem 4.7 in [2] will be correted. And some necessary conditions of (strictly) pseudo-invex monotonicity and quasi-invex monotonicity are established with the condition that is affine in the first argument and skew instead of Condition C.In Chapter 5, the definitions of D-preinvexity, D-semistrict preinvexity and D-strict preinvexity for vector-valued maps are firstly introduced. Then, under the conditions of *-upper semi-continuity and *-lower semi-continuity, some properties of D-preinvexity are given and the interrelations among D-preinvexity, D-semistrict preinvexity and D-strict preinvexity are discussed.In Chapter 6, the definitions such as D-proper prequasiinvexity, D-properly strict prequasiinvexity, D-properly semistrict prequasiinvexity for vector-valued maps are introduced, and under the lower D-Semi-continuous condition or the upper D-Semi-continuous condition, respectively, some equivalent propositions of D-proper prequasiinvexity for vector-valued maps are obtained, and the relationships of D-proper prequasiinvexity, D-properly semistrict prequasiinvexity and D-properly strict prequasiinvexity are discussed, under some conditions the local weak efficient solution of (VP) must be the global weak efficient solution of (VP) is proved.In Chapter 7, two new dual models of the nonsmooth multiobjective programming are constructed and two weak duality theorems for these models are derived.In Chapter 8, Some new results which generalize the Hahn-Banach theorems from scalar or vector-valued case to set-valued case are firstly obtained. Then, the existence of the Borwein-strong subgradient and the Yang-weak subgradient for set-valued maps are also proven. Finally, A new Lagrange multiplier theorem and a new Sandwich theorem for set-valued maps are also presented.In Chapter 9, the notion of affinelike set-valued maps is introduced and some properties of these maps are presented. Then a new Hahn-Banach extension theorem with a D-convex set-valued map dominated by an affinelike set-valued map whichcontains the main results in charter 8 as special cases is obtained.In Chapter 10, we introduce the definition of Di-0-partiaHy diagonal quasicon-vexity which is a generalization of D-quasiconvexity. We also introduce a new system of vector quasi-equilibrium problems and prove its existence of a solution. As applications, some existence results of weak pareto equilibrium for both constrained multicriteria games and multicriteria games without constrained correspondences are also shown.