Skip to main content

Part of the book series: Vector Optimization ((VECTOROPT))

  • 1132 Accesses

Abstract

In this chapter linear scalarization functionals are studied and characterization results are provided. With these functionals at hand a vector optimization problem can be replaced by a scalar-valued optimization problem which allows for instance the formulation of optimality conditions or can be used as the base of numerical solution methods.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. M. Adán, V. Novo, Partial and generalized subconvexity in vector optimization problems. J. Convex Anal. 8, 583–594 (2001)

    MATH  MathSciNet  Google Scholar 

  2. M. Adán, V. Novo, Optimality conditions for vector optimization problems with generalized convexity in real linear spaces. Optimization 51, 73–91 (2002)

    MATH  MathSciNet  Google Scholar 

  3. S. Al-Homodan, Q.H. Ansari, S. Schaible, Existence of solutions of systems of generalized implicit vector variational inequalities. J. Optim. Theory Appl. 134, 515–531 (2007)

    MathSciNet  Google Scholar 

  4. R.G.D. Allen, The foundations of a mathematical theory of exchange. Economica 12, 197–226 (1932)

    Google Scholar 

  5. G.B. Allende, C. Tammer, Scalar functions for computing minimizers under variable order structures, in Congreso Latino-Iberoamericano de Investigación Operativa, Rio de Janeiro, 2012

    Google Scholar 

  6. Q.H. Ansari, J.C. Yao, On nondifferentiable and nonconvex vector optimization problems. J. Optim. Theory Appl. 106, 475–488 (2000)

    MATH  MathSciNet  Google Scholar 

  7. J.P. Aubin, I. Ekeland, Applied Nonlinear Analysis (Wiley, New York, 1984)

    MATH  Google Scholar 

  8. J.P. Aubin, H. Frankowska, Set-Valued Analysis (Birkäuser, Boston, 1990)

    MATH  Google Scholar 

  9. D. Baatar, M.M. Wiecek, Advancing equitability in multiobjective programming. Comput. Math. Appl. 52, 225–234 (2006)

    MATH  MathSciNet  Google Scholar 

  10. B. Bank, J. Guddat, D. Klatte, B. Kummer, K. Tammer, Non-Linear Parametric Optimization (Akademie-Verlag, Berlin, 1982)

    Google Scholar 

  11. T.Q. Bao, B.S. Mordukhovich, Necessary nondomination conditions in set and vector optimization with variable ordering structures. J. Optim. Theory Appl. (2013). doi:10.2007/s10957-013-0332-6

    Google Scholar 

  12. E.M. Bednarczuk, Bishop-Phelps cones and convexity: applications to stability of vector optimization problems. INRIA Rapport de Recherche, No. 2806 (1996)

    Google Scholar 

  13. E.M. Bednarczuk, M.J. Przybyla, The vector-valued variational principle in Banach spaces ordered by cones with nonempty interior. SIAM J. Optim. 18, 907–913 (2007)

    MATH  MathSciNet  Google Scholar 

  14. J.Y. Bello Cruz, G. Bouza Allende, A steepest descent-like method for variable order vector optimization problems. J. Optim. Theory Appl. (2013). doi:10.1007/s10957-013-0308-6

    Google Scholar 

  15. J. Benoist, N. Popovici, Characterizations of convex and quasiconvex set-valued maps. Math. Method Oper. Res. 57, 427–435 (2003)

    MATH  MathSciNet  Google Scholar 

  16. H.P. Benson, An improved definition of proper efficiency for vector maximization with respect to cones. J. Math. Appl. 71, 232–241 (1979)

    MATH  MathSciNet  Google Scholar 

  17. C. Berge, Topological Spaces (Oliver and Boyd, Edinburgh, 1963)

    MATH  Google Scholar 

  18. C. Berge, Espaces topologiques, Fonctions multivoques (Dunod, Paris, 1966)

    MATH  Google Scholar 

  19. K. Bergstresser, A. Charnes, P.L. Yu, Generalization of domination structures and nondominated solutions in multicriteria decision making. J. Optim. Theory Appl. 18, 3–13 (1976)

    MATH  MathSciNet  Google Scholar 

  20. K. Bergstresser, P.L. Yu, Domination structures and multicriteria problems in N-person games. Theory Decis. 8, 5–48 (1977)

    MATH  MathSciNet  Google Scholar 

  21. E. Bishop, R.R. Phelps, The support functionals of a convex set. Proc. Symp. Pure Math. 7, 27–35 (1962)

    Google Scholar 

  22. J. Borde, J.P. Crouzeix, Continuity properties of the normal cone to the level sets of a quasiconvex function. J. Optim. Theory Appl. 66, 415–429 (1990)

    MATH  MathSciNet  Google Scholar 

  23. J.M. Borwein, Proper efficient points for maximizations with respect to cones. SIAM J. Control Optim. 15, 57–63 (1977)

    MATH  Google Scholar 

  24. J.M. Borwein, The geometry of Pareto efficiency over cones. Optimization 11, 235–248 (1980)

    MATH  Google Scholar 

  25. R.I. Boţ, S.-M. Grad, G. Wanka, Duality in Vector Optimization (Springer, Heidelberg, 2009)

    MATH  Google Scholar 

  26. L.-C. Ceng, S. Huang, Existence theorems for generalized vector variational inequalities with a variable ordering relation. J. Glob. Optim. 46, 521–535 (2010)

    MATH  MathSciNet  Google Scholar 

  27. L.C. Ceng, B.S. Mordukhovich, J.C. Yao, Hybrid approximate proximal method with auxiliary variational inequalities for vector optimization. J. Optim. Theory Appl. 146, 267–303 (2010)

    MATH  MathSciNet  Google Scholar 

  28. G.Y. Chen, Existence of solutions for a vector variational inequality: an extension of the Hartmann-Stampacchia theorem. J. Optim. Theory Appl. 74, 445–456 (1992)

    MATH  MathSciNet  Google Scholar 

  29. G.Y. Chen, X. Huang, X. Yang, Vector Optimization, Set-Valued and Variational Analysis (Springer, Berlin, 2005)

    MATH  Google Scholar 

  30. G.Y. Chen, X.Q. Yang, Characterizations of variable domination structures via nonlinear scalarization. J. Optim. Theory Appl. 112, 97–110 (2002)

    MATH  MathSciNet  Google Scholar 

  31. G.Y. Chen, X.Q. Yang, H. Yu, A nonlinear scalarization function and generalized quasi-vector equilibrium problems. J. Glob. Optim. 32, 451–466 (2005)

    MATH  MathSciNet  Google Scholar 

  32. K.L. Chew, Domination structures in abstract spaces, in Southeast Asian Bulletin of Mathematics. Proceedings of the First Franco-Southeast Asian Mathematical Conference, 1979, pp. 190–204

    Google Scholar 

  33. F.H. Clarke, Optimal solutions to differential inclusions. J. Optim. Theory Appl. 19, 469–479 (1976)

    MATH  Google Scholar 

  34. F.H. Clarke, Optimization and Nonsmooth Analysis (SIAM, New York, 1990). Reprint, originally published: Wiley, New York, 1983

    Google Scholar 

  35. P. Daniele, A. Maugeri, W. Oettli, Time-dependent variational inequalities. J. Optim. Theory Appl. 103, 543–555 (1999)

    MATH  MathSciNet  Google Scholar 

  36. J.P. Dauer, R.J. Gallagher, Positive proper efficient points and related cone results in vector optimization theory. SIAM J. Control Optim. 28, 158–172 (1990)

    MATH  MathSciNet  Google Scholar 

  37. J.P. Delahaye, J. Denel, Annex 1: the continuities of the point-to-set maps, definitions and equivalences. Math. Program. Study 10, 8–12 (1979)

    Google Scholar 

  38. F.Y. Edgeworth, Mathematical Psychics (Kegan Paul, London, 1881)

    Google Scholar 

  39. M. Ehrgott, Multicriteria Optimization (Springer, Heidelberg, 2005)

    MATH  Google Scholar 

  40. G. Eichfelder, Parametergesteuerte Lösung nichtlinearer multikriterieller Optimierungsprobleme. Dissertation, Universität Erlangen-Nürnberg, 2006

    Google Scholar 

  41. G. Eichfelder, \(\varepsilon\)-Constraint method with adaptive parameter control and an application to intensity-modulated radiation therapy, in Multicriteria Decision Making and Fuzzy Systems, Theory, Methods and Applications, ed. by K.-H. Küfer et al. (Shaker, Aachen, 2006), pp. 25–42

    Google Scholar 

  42. G. Eichfelder, Adaptive Scalarization Methods in Multiobjective Optimization (Springer, Heidelberg, 2008)

    MATH  Google Scholar 

  43. G. Eichfelder, An adaptive scalarization method in multi-objective optimization. SIAM J. Optim. 19, 1694–1718 (2009)

    MATH  MathSciNet  Google Scholar 

  44. G. Eichfelder, Scalarizations for adaptively solving multi-objective optimization problems. Comput. Optim. Appl. 44, 249–273 (2009)

    MATH  MathSciNet  Google Scholar 

  45. G. Eichfelder, A constraint method in nonlinear multi-objective optimization, in Multiobjective Programming and Goal Programming, Theoretical Results and Practical Applications, ed. by V. Barichard et al. Lecture Notes in Economics and Mathematical Systems, vol. 618 (Springer, Heidelberg, 2009), pp. 3–12

    Google Scholar 

  46. G. Eichfelder, Optimal elements in vector optimization with a variable ordering structure. J. Optim. Theory Appl. 151(2), 217–240 (2011)

    MATH  MathSciNet  Google Scholar 

  47. G. Eichfelder, Variable ordering structures in vector optimization, in Recent Developments in Vector Optimization, Chap. 4, ed. by Q.H. Ansari, J.-C. Yao (Springer, Heidelberg, 2012), pp. 95–126

    Google Scholar 

  48. G. Eichfelder, Cone-valued maps in optimization. Appl. Anal. 91(10), 1831–1846 (2012)

    MATH  MathSciNet  Google Scholar 

  49. G. Eichfelder, Numerical procedures in multiobjective optimization with variable ordering structures. J. Optim. Theory Appl. (2013). doi:10.1007/s10957-013-0267-y

    Google Scholar 

  50. G. Eichfelder, Ordering structures in vector optimization and applications in medical engineering. Preprint-Series of the Institute of Mathematics, Technische Universität Ilmenau, 2013

    Google Scholar 

  51. G. Eichfelder, M. Gebhardt, Local specific absorption rate control for parallel transmission by virtual observation points. Magn. Reson. Med. 66(5), 1468–1476 (2011)

    Google Scholar 

  52. G. Eichfelder, T.X.D. Ha, Optimality conditions for vector optimization problems with variable ordering structures. Optimization 62(5), 597–627 (2013)

    MATH  MathSciNet  Google Scholar 

  53. G. Eichfelder, J. Jahn, Vector optimization problems and their solution concepts, in Recent Developments in Vector Optimization, Chap. 1, ed. by Q.H. Ansari, J.-C. Yao (Springer, Heidelberg, 2012), pp. 1–27

    Google Scholar 

  54. G. Eichfelder, R. Kasimbeyli, Properly optimal elements in vector optimization with variable ordering structures. J. Glob. Optim. (2013). doi:10.1007/s10898-013-0132-4

    Google Scholar 

  55. G. Eichfelder, T. Gerlach, Characterization of proper optimal elements with variable ordering structures. Preprint-Series of the Institute of Mathematics, Ilmenau University of Technology, Germany, 2014, http://www.db-thueringen.de/servlets/DerivateServlet/Derivate-28793/IfM_Preprint_M_14_01.pdf

  56. B. El Abdouni, L. Thibault, Optimality conditions for problems with set-valued objectives. J. Appl. Anal. 2, 183–201 (1996)

    MATH  MathSciNet  Google Scholar 

  57. A. Engau, Domination and decomposition in multiobjective programming. Dissertation, University of Clemson, 2007

    Google Scholar 

  58. A. Engau, Variable preference modeling with ideal-symmetric convex cones. J. Glob. Optim. 42, 295–311 (2008)

    MATH  MathSciNet  Google Scholar 

  59. K. Fan, Minimax theorems. Proc. Natl. Acad. Sci. USA 39, 42–47 (1953)

    MATH  Google Scholar 

  60. B. Fischer, E. Haber, J. Modersitzki, Mathematics meets medicine - an optimal alignment. SIAG/OPT Views News 19(2), 1–7 (2008)

    Google Scholar 

  61. C. Gebhardt, Skalarisierungen für Vektoroptimierungsprobleme mit variablen Ordnungsstrukturen. Diploma thesis, Universität Erlangen-Nürnberg, 2011

    Google Scholar 

  62. N. Georgescu, The pure theory of consumer’s behaviour. Q. J. Econ. 50, 545–593 (1936)

    Google Scholar 

  63. N. Georgescu, Choice and revealed preference. South. Econ. J. 21, 119–130 (1954)

    Google Scholar 

  64. C. Gerstewitz (Tammer), Nichtkonvexe Dualität in der Vektoroptimierung. Wiss. Z. TH Leuna-Merseburg 25, 357–364 (1983)

    Google Scholar 

  65. C. Gerstewitz (Tammer), E. Iwanow, Dualität für nichtkonvexe Vektoroptimierungsprobleme. Wiss. Z. Techn. Hochschule Ilmenau 31, 61–81 (1985)

    Google Scholar 

  66. C. Gerth (Tammer), P. Weidner, Nonconvex separation theorems and some applications in vector optimization. J. Optim. Theory Appl. 67, 297–320 (1990)

    Google Scholar 

  67. F. Giannessi, Theorems of the alternative, quadratic programs and complementarity problems, in Variational Inequalities and Complementarity Problems, ed. by R.W. Cottle et al. (Wiley, New York, 1980)

    Google Scholar 

  68. F. Giannessi, G. Mastroeni, X.Q. Yang, Survey on vector complementarity problems. J. Optim. Theory Appl. 53(1), 53–67 (2012)

    MATH  MathSciNet  Google Scholar 

  69. G. Godini, Set-valued Cauchy functional equation. Rev. Roumaine Math. Pures Appl. 20, 1113–1121 (1975)

    MATH  MathSciNet  Google Scholar 

  70. A. Göpfert, R. Nehse, Vektoroptimierung: Theorie, Verfahren und Anwendungen (Teubner, Leipzig, 1990)

    MATH  Google Scholar 

  71. A. Göpfert, H. Riahi, C. Tammer, C. Zălinescu, Variational Methods in Partially Ordered Spaces (Springer, New York, 2003)

    MATH  Google Scholar 

  72. D. Gourion, D.T. Luc, Generating the weakly efficient set of nonconvex multiobjective problems. J. Glob. Optim. 41, 517–538 (2008)

    MATH  MathSciNet  Google Scholar 

  73. P.J. Guerra, M.A. Melguizo, M.J. Muñoz-Bouzo, Conic set-valued maps in vector optimization. Set-Valued Anal. 15, 47–59 (2007)

    MATH  MathSciNet  Google Scholar 

  74. P.J. Guerra, M.A. Melguizo, M.J. Muñoz-Bouzo, Polar conic set-valued map in vector optimization. Continuity and derivability. J. Optim. Theory Appl. 142, 343–354 (2009)

    MATH  Google Scholar 

  75. A. Guerraggio, E. Molho, A. Zaffaroni, On the notion of proper efficiency in vector optimization. J. Optim. Theory Appl. 82, 1–21 (1994)

    MATH  MathSciNet  Google Scholar 

  76. C. Gutiérrez, B. Jiménez, V. Novo, On approximate solutions in vector optimization problems via scalarization. Comput. Optim. Appl. 35, 305–324 (2006)

    MATH  MathSciNet  Google Scholar 

  77. T.X.D. Ha, Optimality conditions for several types of efficient solutions of set-valued optimization problems, in Nonlinear Analysis and Variational Problems, Chap. 21, ed. by P. Pardalos, Th.M. Rassias, A.A. Khan (Springer, Heidelberg, 2009), pp. 305–324

    Google Scholar 

  78. T.X.D. Ha, Optimality conditions for various efficient solutions involving coderivatives: from set-valued optimization problems to set-valued equilibrium problems. Nonlinear Anal. 75(3), 1305–1323 (2012)

    MATH  MathSciNet  Google Scholar 

  79. T.X.D. Ha, J. Jahn, Properties of Bishop-Phelps cones. Preprint series of the Institute of Applied Mathematics, Universität Erlangen-Nürnberg, No. 343, 2010

    Google Scholar 

  80. A. Hamel, Translative sets and functions and their applications to risk measure theory and nonlinear separation. Preprint series of IMPA, Rio de Janeiro, 21, 2006

    Google Scholar 

  81. S. Helbig, An interactive algorithm for nonlinear vector optimization. Appl. Math. Optim. 22(2), 147–151 (1990)

    MATH  MathSciNet  Google Scholar 

  82. S. Helbig, Approximation of the efficient point set by perturbation of the ordering cone. Z. Oper. Res. 35(3), 197–220 (1991)

    MATH  MathSciNet  Google Scholar 

  83. I. Henig, Proper efficiency with respect to cones. J. Optim. Theory Appl. 36, 387–407 (1982)

    MATH  MathSciNet  Google Scholar 

  84. J.-B. Hiriart-Urruty, New concepts in nondifferentiable programming. Bull. Soc. Math. France 60, 57–85 (1979)

    MATH  MathSciNet  Google Scholar 

  85. J.-B. Hiriart-Urruty, Tangent cones, generalized gradients and mathematical programming in Banach spaces. Math. Oper. Res. 4, 79–97 (1979)

    MATH  MathSciNet  Google Scholar 

  86. C. Hirsch, P.K. Shukla, H. Schmeck, Variable preference modeling using multi-objective evolutionary algorithms, in Evolutionary Multi-criterion Optimization - 6th International Conference, ed. by R.H.C. Takahashi et al. Lecture Notes in Computer Science, vol. 6576 (Springer, Heidelberg, 2011)

    Google Scholar 

  87. N.J. Huang, X.Q. Yang, W.K. Chan, Vector complementarity problems with a variable ordering relation. Eur. J. Oper. Res. 176, 15–26 (2007)

    MATH  MathSciNet  Google Scholar 

  88. A.D. Ioffe, Approximate subdifferentials and applications 3: the metric theory. Mathematika 36, 1–38 (1989)

    MATH  MathSciNet  Google Scholar 

  89. A.D. Ioffe, J.-P. Penot, Subdifferentials of performance functions and calculus of coderivatives of set-valued mappings. Well-posedness and stability of variational problems. Serdica Math. J. 22, 257–282 (1996)

    MathSciNet  Google Scholar 

  90. G. Isac, Sur l’existence de l’optimum de Pareto. Riv. Mat. Univ. Parma, IV. Ser. 9, 303–325 (1983)

    Google Scholar 

  91. G. Isac, A.O. Bahya, Full nuclear cones associated to a normal cone. Application to Pareto efficiency. Appl. Math. Lett. 15, 633–639 (2002)

    MATH  MathSciNet  Google Scholar 

  92. J. Jahn, Introduction to the Theory of Nonlinear Optimization, 3rd edn. (Springer, Heidelberg, 2007)

    MATH  Google Scholar 

  93. J. Jahn, Bishop-Phelps cones in optimization. Int. J. Optim. Theory Methods Appl. 1, 123–139 (2009)

    MATH  MathSciNet  Google Scholar 

  94. J. Jahn, Vector Optimization - Theory, Applications, and Extensions, 2nd edn. (Springer, Heidelberg, 2011)

    MATH  Google Scholar 

  95. J. Jahn, T.X.D. Ha, New order relations in set optimization. J. Optim. Theory Appl. 148, 209–236 (2011)

    MATH  MathSciNet  Google Scholar 

  96. J. Jahn, U. Rathje, Graef-Younes method with backward iteration, in Multicriteria Decision Making and Fuzzy Systems - Theory, Methods and Applications, ed. by K.-H. Küfer et al. (Shaker, Aachen, 2006), pp. 75–81

    Google Scholar 

  97. R. John, The concave nontransitive consumer. J. Glob. Optim. 20, 297–308 (2001)

    MATH  Google Scholar 

  98. R. John, Local and global consumer preferences, in Generalized Convexity and Related Topics, ed. by I. Konnov, D.T. Luc, A. Rubinov (Springer, Heidelberg, 2006), pp. 315–326

    Google Scholar 

  99. E.K. Karaskal, W. Michalowski, Incorporating wealth information into a multiple criteria decision making model. Eur. J. Oper. Res. 150, 204–219 (2003)

    Google Scholar 

  100. R. Kasimbeyli, A nonlinear cone separation theorem and scalarization in nonconvex vector optimization. SIAM J. Optim. 20, 1591–1619 (2010)

    MathSciNet  Google Scholar 

  101. P.Q. Khanh, L.M. Luu, On the existence of solutions to vector quasi-variational inequalities and quasi-complementarity problems with applications to traffic equilibria. J. Optim. Theory Appl. 123, 533–548 (2004)

    MATH  MathSciNet  Google Scholar 

  102. P. Korhonen, J. Wallenius, S. Zionts, Solving the discrete multiple criteria problem using convex cones. Manag. Sci. 30, 1336–1345 (1984)

    MATH  MathSciNet  Google Scholar 

  103. M.M. Kostreva, W. Ogryczak, A. Wierzbicki, Equitable aggregations in multiple criteria analysis. Eur. J. Oper. Res. 158, 362–377 (2004)

    MATH  MathSciNet  Google Scholar 

  104. M.A. Krasnosel’skij, Positive Solutions of Operator Equations (Noordhoff, Groningen, 1964)

    Google Scholar 

  105. D. Kuroiwa, Convexity for set-valued maps. Appl. Math. Lett. 9, 97–101 (1996)

    MATH  MathSciNet  Google Scholar 

  106. D. Kuroiwa, Natural criteria of set-valued optimization. Manuscript, Shimane University, 1998

    Google Scholar 

  107. G.M. Lee, D.S. Kim, H. Kuk, Existence of solutions for vector optimization problems. J. Math. Anal. Appl. 220, 90–98 (1998)

    MATH  MathSciNet  Google Scholar 

  108. G.M. Lee, D.S. Kim, B.S. Lee, On noncooperative vector equilibrium. Indian J. Pure Appl. Math. 27, 735–739 (1996)

    MATH  MathSciNet  Google Scholar 

  109. S.J. Li, M.H. Li, Levitin-Polyak well-posedness of vector equilibrium problems. Math. Methods Oper. Res. 69, 125–140 (2009)

    MATH  MathSciNet  Google Scholar 

  110. C.G. Liu, K.F. Ng, W.H. Yang, Merit functions in vector optimization. Math. Program. Ser. A 119, 215–237 (2009)

    MATH  MathSciNet  Google Scholar 

  111. J. Liu, W. Sond, On proper efficiencies in locally convex spaces—a survey. Acta Math. Vietnam. 26(3), 301–312 (2001)

    MATH  MathSciNet  Google Scholar 

  112. D.T. Luc, Scalarization of vector optimization problems. J. Optim. Theory Appl. 55, 85–102 (1987)

    MATH  MathSciNet  Google Scholar 

  113. D.T. Luc, Theory of Vector Optimization (Springer, Berlin, 1989)

    Google Scholar 

  114. D.T. Luc, J.P. Penot, Convergence of asymptotic directions. Trans. Am. Math. Soc. 353, 4095–4121 (2001)

    MATH  MathSciNet  Google Scholar 

  115. G. Mastroeni, On the image space analysis for vector quasi-equilibrium problems with a variable ordering relation. J. Glob. Optim. 53(2), 203–214 (2012)

    MATH  MathSciNet  Google Scholar 

  116. K.M. Miettinen, Nonlinear Multiobjective Optimization (Kluwer, Boston, 1999)

    MATH  Google Scholar 

  117. B.S. Mordukhovich, Maximum principle in problems of time optimal control with nonsmooth constraints. J. Appl. Math. Mech. 40, 960–969 (1976)

    MATH  MathSciNet  Google Scholar 

  118. B.S. Mordukhovich, Metric approximations and necessary optimality conditions for general classes of nonsmooth extremal problems. Sov. Math. Dokl. 22, 526–530 (1980)

    MATH  Google Scholar 

  119. B.S. Mordukhovich, Variational Analysis and Generalized Differentiation, I: Basic Theory, Grundlehren Series (Fundamental Principles of Mathematical Sciences), vol. 330 (Springer, Berlin, 2006)

    Google Scholar 

  120. B.S. Mordukhovich, Variational Analysis and Generalized Differentiation, II: Applications, Grundlehren Series (Fundamental Principles of Mathematical Sciences), vol. 331 (Springer, Berlin, 2006)

    Google Scholar 

  121. B.S. Mordukhovich, Multiobjective optimization problems with equilibrium constraints. Math. Program. Ser. B 117, 331–354 (2009)

    MATH  MathSciNet  Google Scholar 

  122. B.S. Mordukhovich, Methods of variational analysis in multiobjective optimization. Optimization 58, 413–430 (2009)

    MATH  MathSciNet  Google Scholar 

  123. K. Nikodem, D. Popa, On single-valuedness of set-valued maps satisfying linear inclusions. Banach J. Math. Anal. 3, 44–51 (2009)

    MathSciNet  Google Scholar 

  124. Z.G. Nishnianidze, Fixed points of monotonic multiple-valued operators. Bull. Georgian Acad. Sci. 114, 489–491 (1984)

    MATH  MathSciNet  Google Scholar 

  125. W. Ogryczak, T. Sliwinski, On solving linear programs with the ordered weighted averaging objective. Eur. J. Oper. Res. 148, 80–91 (2003)

    MATH  MathSciNet  Google Scholar 

  126. W. Ogryczak, A. Wierzbicki, On multi-criteria approaches to bandwith allocation. Control Cybern. 33, 427–448 (2004)

    MATH  MathSciNet  Google Scholar 

  127. V. Pareto, Manuale di economia politica (Societa Editrice Libraria, Milano, 1906). (English translation: V. Pareto, Manual of Political Economy, translated by A.S. Schwier, M. Augustus, Kelley, New York, 1971)

    Google Scholar 

  128. A. Pascoletti, P. Serafini, Scalarizing vector optimization problems. J. Optim. Theory Appl. 42, 499–524 (1984)

    MATH  MathSciNet  Google Scholar 

  129. M. Petschke, On a theorem of Arrow, Barankin, and Blackwell. SIAM J. Control Optim. 28, 395–401 (1990)

    MATH  MathSciNet  Google Scholar 

  130. M. Pruckner, Kegelwertige Abbildungen in der Optimierung. Diploma thesis, Universität Erlangen-Nürnberg, 2011

    Google Scholar 

  131. J.H. Qiu, Y. Hao, Scalarization of Henig properly efficient points in locally convex spaces. J. Optim. Theory Appl. 147, 71–92 (2010)

    MATH  MathSciNet  Google Scholar 

  132. R. Ramesh, M.H. Karwan, S. Zionts, Preference structure representation using convex cones in multicriteria integer programming. Manag. Sci. 35, 1092–1105 (1989)

    MATH  MathSciNet  Google Scholar 

  133. A.M. Rubinov, R.N. Gasimov (Kasimbeyli), Scalarization and nonlinear scalar duality for vector optimization with preferences that are not necessarily a pre-order relation. J. Glob. Optim. 29, 455–477 (2004)

    Google Scholar 

  134. Y. Sawaragi, H. Nakayama, T. Tanino, Theory of Multiobjective Optimization (Academic, London, 1985)

    MATH  Google Scholar 

  135. P.K. Shukla, C. Hirsch, H. Schmeck, In search of equitable solutions using multi-objective evolutionary algorithms, in Parallel Problem Solving from Nature - PPSN XI, ed. by R. Schaefer et al. Lecture Notes in Computer Science, vol. 6238 (Springer, Heidelberg, 2011), pp. 687–696

    Google Scholar 

  136. P.K. Shukla, M.A. Braun, Indicator based search in variable orderings: theory and algorithms, in EMO 2013, ed. by R.C. Purshouse et al. Lecture Notes in Computer Science, vol. 7811 (Springer, Heidelberg, 2013), pp. 66–80

    Google Scholar 

  137. W. Stadler, Fundamentals of multicriteria optimization, in Multicriteria Optimization in Engineering and in the Sciences, ed. by W. Stadler (Plenum, New York, 1988)

    Google Scholar 

  138. A. Sterna-Karwat, Continuous dependence of solutions on a parameter in a scalarization method. J. Optim. Theory Appl. 55(3), 417–434 (1987)

    MATH  MathSciNet  Google Scholar 

  139. A. Sterna-Karwat, Lipschitz and differentiable dependence of solutions on a parameter in a scalarization method. J. Aust. Math. Soc. A 42, 353–364 (1987)

    MATH  MathSciNet  Google Scholar 

  140. B. Soleimani, C. Tammer, Approximate solutions of vector optimization problem with variable ordering structure, in AIP Conference Proceedings of Numerical Analysis and Applied Mathematics, ICNAAM 2012: International Conference of Numerical Analysis and Applied Mathematics, vol. 1479, 2012, pp. 2363–2366

    Google Scholar 

  141. B. Soleimani, C. Tammer, Concepts for approximate solutions of vector optimization problems with variable order structure. Preprint-Series of the Institute of Mathematics, Martin-Luther-Universität Halle-Wittenberg, 2013

    Google Scholar 

  142. C. Tammer, C. Zălinescu, Lipschitz properties of the scalarization function and applications. Optimization 59, 305–319 (2010)

    MATH  MathSciNet  Google Scholar 

  143. C. Tammer, C. Zălinescu, Vector variational principles for set-valued functions, in Recent Developments in Vector Optimization, Chap. 11, ed. by Q.H. Ansari, J.-C. Yao (Springer, Heidelberg, 2011)

    Google Scholar 

  144. M. Tanaka, GA-based decision support system for multi-criteria optimization. Proc. Int. Conf. Syst. Man Cybern. 2, 1556–1561 (1995)

    Google Scholar 

  145. L. Thibault, On subdifferentials of optimal value functions. SIAM J. Control Optim. 29, 1019–1036 (1991)

    MATH  MathSciNet  Google Scholar 

  146. M. Wacker, Multikriterielle Optimierung bei der Registrierung medizinischer Daten. Diploma thesis, Universität Erlangen-Nürnberg, 2008

    Google Scholar 

  147. M. Wacker, F. Deinzer, Automatic robust medical image registration using a new democratic vector optimization approach with multiple measures, in Medical Image Computing and Computer-Assisted Intervention - MICCAI 2009, ed. by G.-Z. Yang et al., 2009, pp. 590–597

    Google Scholar 

  148. D.W. Walkup, R.J.-B. Wets, Continuity of some convex-cone-valued maps. Proc. Am. Math. Soc. 18, 229–235 (1967)

    MATH  MathSciNet  Google Scholar 

  149. P. Weidner, Dominanzmengen und Optimalitätsbegriffe in der Vektoroptimierung. Wiss. Z. Techn. Hochschule Ilmenau 31, 133–146 (1985)

    MATH  MathSciNet  Google Scholar 

  150. P. Weidner, Problems in models and methods of vector optimization. Wiss. Schriftenr. TU Karl-Marx-Stadt 5, 47–57 (1989)

    Google Scholar 

  151. P. Weidner, Ein Trennungskonzept und seine Anwendung auf Vektoroptimierungsverfahren. Habilitation thesis, Martin-Luther-Universität Halle-Wittenberg, 1990

    Google Scholar 

  152. P. Weidner, An approach to different scalarizations in vector optimization. Wiss. Z. Techn. Hochschule Ilmenau 36, 103–110 (1990)

    MATH  MathSciNet  Google Scholar 

  153. P. Weidner, Problems in scalarizing multicriteria approaches, in Multiple Criteria Decision Making in the New Millennium, ed. by M. Köksalen, S. Zionts (Springer, Heidelberg, 2001), pp. 199–209

    Google Scholar 

  154. M.M. Wiecek, Advances in cone-based preference modeling for decision making with multiple criteria. Decis. Mak. Manuf. Serv. 1, 153–173 (2007)

    MATH  MathSciNet  Google Scholar 

  155. G. Xiao, H. Xiao, S. Liu, Scalarization and pointwise well-posedness in vector optimization problems. J. Glob. Optim. 49(4), 561–574 (2011)

    MATH  MathSciNet  Google Scholar 

  156. Y.M. Younes, Studies on discrete vector optimization. Dissertation, University of Demiatta, 1993

    Google Scholar 

  157. R.C. Young, The algebra of many-valued quantities. Math. Ann. 104, 260–290 (1931)

    MathSciNet  Google Scholar 

  158. P.L. Yu, Cone convexity, cone extreme points, and nondominated solutions in decision problems with multiobjectives. J. Optim. Theory Appl. 14, 319–377 (1974)

    MATH  Google Scholar 

  159. P.L. Yu, Multiple-Criteria Decision Making: Concepts, Techniques and Extensions (Plenum Press, New York, 1985)

    MATH  Google Scholar 

  160. A. Zaffaroni, Degrees of efficiency and degrees of minimality. SIAM J. Control Optim. 42, 1071–1086 (2003)

    MATH  MathSciNet  Google Scholar 

  161. F. Zheng, Vector variational inequalities with semi-monotone operators. J. Glob. Optim. 32, 633–642 (2005)

    MATH  Google Scholar 

  162. M. Ziegler, Numerische Verfahren zur Bestimmung optimaler Elemente bei variablen Ordnungsstrukturen. Diploma thesis, Universität Erlangen-Nürnberg, 2012

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Eichfelder, G. (2014). Linear Scalarizations. In: Variable Ordering Structures in Vector Optimization. Vector Optimization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54283-1_4

Download citation

Publish with us

Policies and ethics