Vector Optimization With Cone Constraints In Banach Space

In this paper, we will discuss a class of vector optimization problems with coneconstraints on a Banach space by using a class of augmented Lagrangian function.Firstly, we give the augmented Lagrangian function and dual function definitions for(VP) problem, and some of the other definitions, such as conjugate cone, an infimumpoint, function is assumed to have a valley at zero. We give penalty function andperturbed problem of (VP), and prove weak duality of (VP) problem. We prove thatseek efcient solutions of vector optimization problems with cone constraints into op-timal solutions for a class of cone constrained optimization problems. Under certainconditions, the zero duality gap have been obtained between the cone constrained op-timization problem and its augmented Lagrangian dual problem. At the same time,we discuss the relationship of saddle points, augmented Lagrange multipliers, and zeroduality gap property between the cone optimization problem and its augmented La-grangian dual problem. Finally, under the condition of some compactness assumption,we obtain two sufcient conditions for the existence of augmented Lagrange multipliersof a cone constrained optimization problem in Banach spaces.