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Optimality conditions for a unified vector optimization problem with not necessarily preordering relations

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Abstract

This paper studies a general vector optimization problem which encompasses those related to efficiency, weak efficiency, strict efficiency, proper efficiency and approximate efficiency among others involving non necessarily preordering relations. Based on existing results about complete characterization by scalarization of the solution set obtained by the same authors, several properties of (generalized) convexity and lower semicontinuity of the composition of the scalarizing functional and the objective vector function are studied. Finally, some optimality conditions are presented through subdifferentials in the convex and nonconvex case.

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References

  1. Benoist J., Borwein J.M., Popovici N.: A characterization of quasiconvex vector-valued functions. Proc. Am. Math. Soc. 131, 1109–1113 (2003)

    Article  Google Scholar 

  2. Bianchi M., Hadjisavvas N., Schaible S.: Vector equilibrium problems with generalized monotone bifunctions. J. Optim. Theory Appl. 92, 527–542 (1997)

    Article  Google Scholar 

  3. Bonnisseau J.-M, Cornet B.: Existence of equilibria when firms follows bounded losses pricing rules. J. Math. Econ. 17, 119–147 (1988)

    Article  Google Scholar 

  4. Breckner W.W., Kassay G.: A systematization of convexity concepts for sets and functions. J. Convex Anal. 4, 109–127 (1997)

    Google Scholar 

  5. Cambini R. et al.: Generalized concavity for bicriteria functions. In: Crouzeix, J.P. (ed.) Generalized Convexity, Generalized Monotonicity: Recent Results. Non-convex Optimization and its Applications, vol. 27, pp. 439–451. Kluwer, Dordrecht (1998)

    Chapter  Google Scholar 

  6. Chen G.-Y., Huang X., Yang X.: Vector Optimization. Set-Valued and Variational Analysis. Lecture Notes in Economics and Mathematical Systems, vol. 541. Springer, Berlin (2005)

    Google Scholar 

  7. Durea M., Dutta J., Tammer C.: Lagrange multipliers for \({\varepsilon}\) -Pareto solutions in vector optimization with nonsolid cones in Banach spaces. J. Optim. Theory Appl. 145, 196–211 (2010)

    Article  Google Scholar 

  8. Durea M., Tammer C.: Fuzzy necessary optimality conditions for vector optimization problems. Optimization 58, 449–467 (2009)

    Article  Google Scholar 

  9. Eichfelder G.: Adaptive Scalarization Methods in Multiobjective Optimization. Springer, Berlin (2008)

    Book  Google Scholar 

  10. Ferro F.: Minimax type theorems for n-valued functions. Annali di Matematica Pura ed Applicata 32, 113–130 (1982)

    Article  Google Scholar 

  11. Flores-Bazán F.: Ideal, weakly efficient solutions for vector optimization problems. Math. Program. Ser. A 93, 453–475 (2002)

    Article  Google Scholar 

  12. Flores-Bazán F.: Radial epiderivatives and asymptotic functions in nonconvex vector optimization. SIAM J. Optim. 14, 284–305 (2003)

    Article  Google Scholar 

  13. Flores-Bazán F.: Semistrictly quasiconvex mappings and nonconvex vector optimization. Math. Methods Oper. Res. 59, 129–145 (2004)

    Article  Google Scholar 

  14. Flores-Bazán F., Hadjisavvas N., Vera C.: An optimal alternative theorem and applications to mathematical programming. J. Glob. Optim. 37, 229–243 (2007)

    Article  Google Scholar 

  15. Flores-Bazán, F., Hernández, E.: A unified vector optimization problem: complete scalarizations and applications. Optimization 60(12), 1399–1419 (2011)

    Google Scholar 

  16. Flores-Bazán F., Hernández E., Novo V.: Characterizing efficiency without linear structure: a unified approach. J. Glob. Optim. 41, 42–60 (2008)

    Article  Google Scholar 

  17. Flores-Bazán, F., Laengle, S., Loyola, G.: Characterizing the efficient points without closedness or free-disposability. Cent. Eur. J. Oper. Res. doi:10.1007/s10100-011-0233-4

  18. Flores-Bazán F., Vera C.: Characterization of the nonemptiness and compactness of solution sets in convex and nonconvex vector optimization. J. Optim. Theory Appl. 130, 185–207 (2006)

    Article  Google Scholar 

  19. Flores-Bazán F., Vera C.: Unifying efficiency and weak efficiency in generalized quasiconvex vector minimization on the real-line. Int. J. Optim. Theory Methods Appl. 1, 247–265 (2009)

    Google Scholar 

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

    Article  Google Scholar 

  21. Göpfert A., Riahi H., Tammer C., Zălinescu C.: Variational Methods in Partially Ordered Spaces, CMS Books in Mathematics. Springer, New York (2003)

    Google Scholar 

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

    Article  Google Scholar 

  23. Gutiérrez C., Jiménez B., Novo V.: Optimality conditions via scalarization for a new \({\varepsilon}\) -efficiency concept in vector optimization problems. Eur. J. Oper. Res. 201, 11–22 (2010)

    Article  Google Scholar 

  24. Ha T.X.D.: Optimality conditions for several types of efficient solutions of set-valued optimization problems. In: Pardalos, P., Rassias, Th.M., Khan, A.A. (eds) Nonlinear Analysis and Variational Problems, pp. 305–324. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  25. Hernández E., Rodríguez-Marín L.: Nonconvex scalarization in set optimization with set-valued maps. J. Math. Anal. Appl. 325, 1–18 (2007)

    Article  Google Scholar 

  26. Hiriart-Urruty J.-B, Moussaoui M., Seeger A., Volle M.: Subdifferential calculus without qualification conditions, using approximate subdifferentials: a survey. Nonlinear Anal. 24, 1727–1754 (1995)

    Article  Google Scholar 

  27. Jahn J.: Vector Optimization. Theory, Applications and Extensions. Springer, Berlin (2004)

    Book  Google Scholar 

  28. Jahn J., Sachs E.: Generalized quasiconvex mappings and vector optimization. SIAM Control Optim. 24(2), 306–322 (1986)

    Article  Google Scholar 

  29. Jeyakumar V., Oettli W., Natividad M.: A solvability theorem for a class of quasiconvex mappings with applications to optimization. J. Math. Anal. Appl. 179, 537–546 (1993)

    Article  Google Scholar 

  30. Luc D.T.: Theory of Vector Optimization, Lecture Notes in Economics and Mathematical Systems, vol. 319. Springer, New york, Berlin (1989)

    Google Scholar 

  31. Luc D.T.: On three concepts of quasiconvexity in vector optimization. Acta Math. Vietnam. 15, 3–9 (1990)

    Google Scholar 

  32. Makarov V.L., Levin M.J., Rubinov A.M.: Mathematical Economic Theory: Pure and Mixed Types of Economic Mechanisms. North-Holland, Amsterdam (1995)

    Google Scholar 

  33. Mordukhovich B.S.: Variational Analysis and Generalized Differentiation, Vol I: Basic Theory, Vol. II: Applications. Springer, Berlin (2006)

    Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  37. Tanaka T.: General quasiconvexities, cones saddle points and minimax theorem for vector-valued functions. J. Optim. Theory Appl. 81, 355–377 (1994)

    Article  Google Scholar 

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Authors and Affiliations

  1. CI2MA, Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile

    Fabián Flores-Bazán

  2. Departamento de Matemática Aplicada, Universidad Nacional de Educación a Distancia, c/Juan del Rosal, 12, 28040, Madrid, Spain

    Elvira Hernández

Authors
  1. Fabián Flores-Bazán
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  2. Elvira Hernández
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Correspondence to Fabián Flores-Bazán.

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Flores-Bazán, F., Hernández, E. Optimality conditions for a unified vector optimization problem with not necessarily preordering relations. J Glob Optim 56, 299–315 (2013). https://doi.org/10.1007/s10898-011-9822-y

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  • DOI: https://doi.org/10.1007/s10898-011-9822-y

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