The Approximate Karush-Kuhn-Tucker Of Vector Optimization Problems

As we all know, there are lots of vector optimization problems (VOP) arose in the financial management, economic analysis, ecological protection, sustainable development and so on. A main research direction in vector optimization problems is to study the existence theorem for its solutions. In recent years, many authors use AKKT point, SAKKT point and AGP point to study the solvability of scalar optimization problems and related problems. Basing on these previous work, we continue to study the relationship between AKKT point, SAKKT point and AGP point for vector problems. We discuss the solvability of VOP by using approximate KKT point, AGP point and AGP point. This paper is organized as follows:In Chapter 1, we introduce the background of vector optimization problem and AKKT point, and list some common symbols and basic conceptions which used in this dissertation.In Chapter 2, we study the AKKT point for vector optimization problems. First, by changing the vector optimization problem to corresponding scalar optimization problem, we introduce the AKKT point for vector optimization and show that the locally weak effective solution for VOP is its AKKT point. Furthermore, if X is a finite dimensional space, then similar results are obtained without assuming the constrained set being closed and convex. Finally, we show that the SAKKT point is weak effective solution for convex VOP.In Chapter 3, we study the AKKT point for semi-infinite vector optimization problem. First, by changing the semi-infinite vector optimization problem to corresponding scalar semi-infinite optimization problem optimization problem, we introduce the AKKT point for semi-infinite vector optimization problem. Furthermore, we show that the locally weak effective solution for semi-infinite vector optimization problem is its AKKT point.In Chapter 4, we use the convexificator to study the AKKT point of nonsmooth vector optimization problem. First, by using the convexificator, we construct a sequence of scalar optimization problem and prove that its solution sequence converges to the weak effective solution of corresponding nonsmooth vector optimization problem. This enables us to define the AKKT point for nonsmooth vector optimization problem. Then, we discuss the relation-ship between the weak efficient solution and AKKT point for vector optimization problem under the nonsmooth constraint condition. Compared with Chapter 2, we only require that the objective function satisfies locally Lipschitz condition.In Chapter 5, we study the AKKT point for vector variational inequality. First, we change the vector variational inequality into scalar variational inequality. By using the AKKT point for scalar variational inequality, we define the AKKT point for vector variational inequality. Second, we discuss the relationship between AKKT point, SAKKT point and AGP point for vector variational inequality. We show that the weak effective solution for vector variational inequality is its AKKT point, and SAKKT point is its weak effective solution. Finally, we show that SAKKT point is AGP point and AGP point is weak effective solution for vector variational inequality. The results established in this chapter generalize some corresponding results of [16] from finite dimensional spaces to infinite dimensional spaces, the method employed is somewhat different with the one used in [16].