On necessary and sufficient conditions in vector optimization

It is shown that some general multiplier rules are necessary conditions for vector optimization in infinite-dimensional spaces. Under additional convexity assumptions, these conditions are sufficient. As an application, the Pontryagin maximum principle for cooperative differential games is examined.The authors are grateful to Professor W. Stadler and the referees of the previous edition of this paper for their valuable remarks and suggestions, which have been very helpful in the preparation of this paper.     

Drybonioside,a new glucoside from the Drynaria bonii H. Christ Rhizomes

Drynaria bonii H. Christ, a Vietnam traditional medicinal plant, is used for the treatment of osteoporosis, bone fractures, tinnitus, etc. (Ho PH. 2002. Vietnamese plants. Hanoi: Publisher of Young; Loi DT. 2004. Medicinal plants and remedy of Vietnam. Hanoi: Publisher of Medicine). Based on column chromatography, a new glucoside named drybonioside (5) and four known compounds α-tocopherol (1), 24-methylencycloartan-3β-ol (2), triphyllol (3) and ethyl β-d-fructopyranoside (4) were isolated and identified from hexane and methanol extracts of D. bonii. The structures of new compound were elucidated on the basis of NMR and MS spectroscopic analysis.     

On necessary optimality conditions in vector optimization problems

Necessary conditions of the multiplier rule type for vector optimization problems in Banach spaces are proved by using separation theorems and Ljusternik's theorem. The Pontryagin maximum principle for multiobjective control problems with state constraints is derived from these general conditions. The paper extends to vector optimization results established in the scalar case by Ioffe and Tihomirov.     

On Duality in Nonconvex Vector Optimization in Banach Spaces Using Augmented Lagrangians

This paper shows how the use of penalty functions in terms of projections on the constraint cones, which are orthogonal in the sense of Birkhoff, permits to establish augmented Lagrangians and to define a dual problem of a given nonconvex vector optimization problem. Then the weak duality always holds. Using the quadratic growth condition together with the inf-stability or a kind of Rockafellar's stability called stability of degree two, we derive strong duality results between the properly efficient solutions of the two problems. A strict converse duality result is proved under an additional convexity assumption, which is shown to be essential.