A unified vector optimization problem: complete scalarizations and applications

References

• Bonnisseau , J-M and Cornet , B . 1988 . Existence of equilibria when firms follows bounded losses pricing rules . J. Math. Econ. , 17 : 119 – 147 .
   Web of Science ®Google Scholar
• Bonnisseau , J-M and Crettez , B . 2007 . On the characterization of efficient production vectors . Econ. Theory , 31 : 213 – 223 .
   Web of Science ®Google Scholar
• Breckner , WW and Kassay , G . 1997 . A systematization of convexity concepts for sets and functions . J. Convex Anal. , 4 : 109 – 127 .
   Google Scholar
• Debreu , G . 1959 . Theory of Value , New York : Wiley .
   Google Scholar
• Drummond , LDG , Maculan , N and Svaiter , BF . 2008 . On the choice of parameter for the weighting method in vector optimization . Math. Program., Ser. B , 111 : 201 – 216 .
   Web of Science ®Google Scholar
• Durea , M and Tammer , C . 2009 . Fuzzy necessary optimality conditions for vector optimization problems . Optimization , 58 : 449 – 467 .
   Web of Science ®Google Scholar
• Engau , A and Wiecek , MM . 2007 . Generating ϵ-efficient solutions in multiobjective programming . Eur. J. Oper. Res. , 177 : 1566 – 1579 .
   Web of Science ®Google Scholar
• Flores-Bazán , F . 2004 . Semistrictly quasiconvex mappings and nonconvex vector optimization . Math. Methods Oper. Res. , 59 : 129 – 145 .
   Web of Science ®Google Scholar
• Flores-Bazán , F , Hadjisavvas , N and Vera , C . 2007 . An optimal alternative theorem and applications to mathematical programming . J. Global Optim. , 37 : 229 – 243 .
   Web of Science ®Google Scholar
• Flores-Bazán , F and Hernández , E . Optimality conditions for a unified vector optimization problem with not necessarily preordering relation . J. Global Optim. , (Accepted)
   Web of Science ®Google Scholar
• Flores-Bazán , F and Jiménez , B . 2009 . Strict efficiency in set-valued optimization . SIAM, J. Control Optim. , 48 : 881 – 908 .
   Web of Science ®Google Scholar
• Gerstewitz (Tammer) , C and Iwanow , E . 1985 . Dualität für nichtkonvexe Vektoroptimierungsprobleme . Wiss. Z. Tech. Hochsch. Ilmenau , 31 : 61 – 81 . (in German)
   Google Scholar
• Gerth (Tammer) , C and Weidner , P . 1990 . Nonconvex separation theorems and some applications in vector optimization . J. Optim. Theory Appl. , 67 : 297 – 320 .
   Web of Science ®Google Scholar
• Göpfert , A , Riahi , H , Tammer , C and Zălinescu , C . 2003 . Variational Methods in Partially Ordered Spaces , New York : CMS Books in Mathematics, Springer-Verlag .
   Google Scholar
• Gutiérrez , C , Jiménez , B and Novo , V . 2006 . A unified approach and optimality conditions for approximate solutions of vector optimization problems . SIAM J. Optim. , 17 : 688 – 710 .
   Web of Science ®Google Scholar
• Gutiérrez , C , Jiménez , B and Novo , V . 2006 . On approximate solutions in vector optimization problems via scalarization . Comput. Optim. Appl. , 35 : 305 – 324 .
   Web of Science ®Google Scholar
• Gutiérrez , C , Jiménez , B and Novo , V . 2010 . Optimality conditions via scalarization for a new ϵ-efficiency concept in vector optimization problems . Eur. J. Oper. Res. , 201 : 11 – 22 .
   Web of Science ®Google Scholar
• Ha , TXD . 2009 . “ Optimality conditions for several types of efficient solutions of set-valued optimization problems ” . In Nonlinear Analysis and Variational Problems , Edited by: Pardalos , P , Rassias , ThM and Khan , AA . 305 – 324 . Heidelberg : Springer .
   Google Scholar
• Hamel , A and Löhne , A . 2006 . Minimal element theorems and Ekeland's principle with set relations . J. Nonlinear Convex Anal. , 7 : 19 – 37 .
   Web of Science ®Google Scholar
• Hernández , E and Rodríguez-Marín , L . 2007 . Nonconvex scalarization in set optimization with set-valued maps . J. Math. Anal. Appl. , 325 : 1 – 18 .
   Web of Science ®Google Scholar
• Jahn , J . 1984 . Scalarization in vector optimization . Math. Program. A , 29 : 203 – 218 .
   Web of Science ®Google Scholar
• Jeyakumar , V , Oettli , W and Natividad , M . 1993 . A solvability theorem for a class of quasiconvex mappings with applications to optimization . J. Math. Anal. Appl. , 179 : 537 – 546 .
   Web of Science ®Google Scholar
• Kutateladze , SS . 1979 . Convex ϵ-programming . Sov. Math. Dokl. , 20 : 391 – 393 .
   Google Scholar
• Luc , DT . 1989 . Theory of Vector Optimization, Lecture Notes in Economics and Mathematical Systems , Vol. 319 , New York, Berlin : Springer-Verlag .
   Google Scholar
• Luenberger , DG . 1992 . New optimality principles for economics efficiency and equilibrium . J. Optim. Theory Appl. , 75 : 221 – 264 .
   Web of Science ®Google Scholar
• Makarov , VL , Levin , MJ and Rubinov , AM . 1995 . Mathematical Economic Theory: Pure and Mixed Types of Economic Mechanisms , Amsterdam : North-Holland .
   Google Scholar
• Pascoletti , A and Serafini , P . 1984 . Scalarizing vector optimization problems . J. Optim. Theory Appl. , 42 : 499 – 524 .
   Web of Science ®Google Scholar
• Qiu , Q and Yang , X . 2010 . Some properties of approximate solutions for vector optimization problems with set-valued functions . J. Global Optim. , 47 : 1 – 12 .
   Web of Science ®Google Scholar
• Rubinov , AM . 1977 . Sublinear operator and theirs applications . Russ. Math. Surveys , 32 : 113 – 175 .
   Google Scholar
• Rubinov , AM . 2000 . Abstract Convexity and Global Optimization , Dordrecht : Kluwer Academic .
   Google Scholar
• Rubinov , AM and Gasimov , RN . 2004 . Scalarization and nonlinear scalar duality for vector optimization with preferences that are not necessarily a preorder relation . J. Global Optim. , 29 : 455 – 477 .
   Web of Science ®Google Scholar
• Tammer , C and Zălinescu , C . 2010 . Lipschitz properties of the scalarization function and applications . Optimization , 59 : 305 – 319 .
   Web of Science ®Google Scholar
• White , DJ . 1986 . Epsilon efficiency . J. Optim. Theory Appl. , 49 : 319 – 337 .
   Web of Science ®Google Scholar