Geometric consideration of duality in vector optimization

Recently, duality in vector optimization has been attracting the interest of many researchers. In order to derive duality in vector optimization, it seems natural to introduce some vector-valued Lagrangian functions with matrix (or linear operator, in some cases) multipliers. This paper gives an insight into the geometry of vector-valued Lagrangian functions and duality in vector optimization. It is observed that supporting cones for convex sets play a key role, as well as supporting hyperplanes, traditionally used in single-objective optimization.

[1]  W. Rödder.  A generalized saddlepoint theory , 1977 .

[2]  Johannes Jahn,et al.  Duality in vector optimization , 1983, Math. Program..

[3]  Shelby Brumelle,et al.  Duality for Multiple Objective Convex Programs , 1981, Math. Oper. Res..

[4]  Heinz Isermann.  On some relations between a dual pair of multiple objective linear programs , 1978, Z. Oper. Research.

[5]  Y. Sawaragi,et al.  Duality theory in multiobjective programming , 1979 .

[6]  J. Nieuwenhuis.  Supremal points and generalized duality , 1980 .

[7]  P. Naccache,et al.  Connectedness of the set of nondominated outcomes in multicriteria optimization , 1978 .

[8]  R. Hartley.  On Cone-Efficiency, Cone-Convexity and Cone-Compactness , 1978 .

[9]  Y. Sawaragi,et al.  Conjugate maps and duality in multiobjective optimization , 1980 .

[10]  Klaus Ritter,et al.  Optimization theory in linear spaces. I , 1969 .

[11]  Hirotaka Nakayama,et al.  On subjective programming in multi-criterion decision making , 1978, 1978 IEEE Conference on Decision and Control including the 17th Symposium on Adaptive Processes.

[12]  P. Yu.  Cone convexity, cone extreme points, and nondominated solutions in decision problems with multiobjectives , 1974 .

[13]  K. Ritter,et al.  Optimization theory in linear spaces , 1970 .

[14]  H. W. Corley,et al.  Duality theory for maximizations with respect to cones , 1981 .

[15]  H. Isermann,et al.  Duality in Multiple Objective Linear Programming , 1978 .

[16]  Hidefumi Kawasaki,et al.  A Duality Theorem in Multiobjective Nonlinear Programming , 1982, Math. Oper. Res..

[17]  G. R. Bitran,et al.  Duality for nonlinear multiple-criteria optimization problems , 1981 .