Closing the Duality Gap in Linear Vector Optimization

Using a set-valued dual cost function we give a new approach to duality theory for linear vector optimization problems. We develop the theory very close to the scalar case. Especially, in contrast to known results, we avoid the appearance of a duality gap in case of b = 0. Examples are given.

[1]  D. J. White.  Duality, Indifference and Sensitivity Analysis in Multiple Objective Linear Programming , 1975 .

[2]  J. S. H. Kornbluth,et al.  Duality, Indifference and Sensitivity Analysis inr Multiple Objective Linear Programming , 1974 .

[3]  Alejandro Balbás,et al.  Duality theory and slackness conditions in multiobjective linear programming , 1999 .

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

[5]  A. Göpfert.  Variational methods in partially ordered spaces , 2003 .

[6]  M. Ehrgott.  Approximation algorithms for combinatorial multicriteria optimization problems , 2000 .

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

[8]  George B. Dantzig,et al.  Linear programming and extensions , 1965 .

[9]  丸山 徹.  Convex Analysisの二,三の進展について , 1977 .

[10]  R. Nehse,et al.  Some results on dual vector optimization problems 1 , 1985 .

[11]  J. Focke,et al.  Vektormaximumproblem und parametrische optimierung , 1973 .

[12]  K. S. Banerjee.  Generalized Inverse of Matrices and Its Applications , 1973 .

[13]  H. W. Corley,et al.  Existence and Lagrangian duality for maximizations of set-valued functions , 1987 .

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

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

[16]  C. Tammer,et al.  Theory of Vector Optimization , 2003 .

[17]  Matthias Ehrgott,et al.  Multicriteria Optimization , 2005 .

[18]  J. Jahn.  Mathematical vector optimization in partially ordered linear spaces , 1986 .

[19]  H. W. Corley,et al.  Duality theory for the matrix linear programming problem , 1984 .

[20]  Alan J. Mayne,et al.  Generalized Inverse of Matrices and its Applications , 1972 .