In classical convex optimization theory, the Karush-Kuhn-Tucker (KKT) optimality conditions are necessary and sufficient for optimality if the objective as well as the constraint functions involved is convex. Recently...In classical convex optimization theory, the Karush-Kuhn-Tucker (KKT) optimality conditions are necessary and sufficient for optimality if the objective as well as the constraint functions involved is convex. Recently, Lassere [1] considered a scalar programming problem and showed that if the convexity of the constraint functions is replaced by the convexity of the feasible set, this crucial feature of convex programming can still be preserved. In this paper, we generalize his results by making them applicable to vector optimization problems (VOP) over cones. We consider the minimization of a cone-convex function over a convex feasible set described by cone constraints that are not necessarily cone-convex. We show that if a Slater-type cone constraint qualification holds, then every weak minimizer of (VOP) is a KKT point and conversely every KKT point is a weak minimizer. Further a Mond-Weir type dual is formulated in the modified situation and various duality results are established.展开更多
Cone-convex, cone-monotonic and positively continuous homogeneous operators are used as duality variables and the Lagrange duality of vector maximization problem in Banach space is discussed. The results are the exten...Cone-convex, cone-monotonic and positively continuous homogeneous operators are used as duality variables and the Lagrange duality of vector maximization problem in Banach space is discussed. The results are the extension of Ref[1,3,4] to some extent.The only tool used in the proof of theorem is Eidelheit separated theorem of two convex sets.展开更多
This paper is concerned with the topological structure of efficient sets for optimizationproblem of set-valued mapping. It is proved that these sets are closed or. connected under someconditions on cone-continuity, co...This paper is concerned with the topological structure of efficient sets for optimizationproblem of set-valued mapping. It is proved that these sets are closed or. connected under someconditions on cone-continuity, cone-convexity and cone-quasiconvexity.展开更多
文摘In classical convex optimization theory, the Karush-Kuhn-Tucker (KKT) optimality conditions are necessary and sufficient for optimality if the objective as well as the constraint functions involved is convex. Recently, Lassere [1] considered a scalar programming problem and showed that if the convexity of the constraint functions is replaced by the convexity of the feasible set, this crucial feature of convex programming can still be preserved. In this paper, we generalize his results by making them applicable to vector optimization problems (VOP) over cones. We consider the minimization of a cone-convex function over a convex feasible set described by cone constraints that are not necessarily cone-convex. We show that if a Slater-type cone constraint qualification holds, then every weak minimizer of (VOP) is a KKT point and conversely every KKT point is a weak minimizer. Further a Mond-Weir type dual is formulated in the modified situation and various duality results are established.
文摘Cone-convex, cone-monotonic and positively continuous homogeneous operators are used as duality variables and the Lagrange duality of vector maximization problem in Banach space is discussed. The results are the extension of Ref[1,3,4] to some extent.The only tool used in the proof of theorem is Eidelheit separated theorem of two convex sets.
文摘This paper is concerned with the topological structure of efficient sets for optimizationproblem of set-valued mapping. It is proved that these sets are closed or. connected under someconditions on cone-continuity, cone-convexity and cone-quasiconvexity.
