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On the Theory of Vector Optimization and Variational Inequalities. Image Space Analysis and Separation

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Vector Variational Inequalities and Vector Equilibria

Part of the book series: Nonconvex Optimization and Its Applications ((NOIA,volume 38))

Abstract

By exploiting recent results, it is shown that the theories of Vector Optimization and of Vector Variational Inequalities can be based on the image space analysis and theorems of the alternative or separation theorems. It is shown that, starting from such a general scheme, several theoretical aspects can be developed - like optimality conditions, duality, penalization - as well as methods of solution - like scalarization.

Sections 1 and 2 are due to F. Giannessi; Sections 3,5,6,9,11, are due to G. Mastroeni; Sections 4,7,8,10 are due to L. Pellegrini.

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References

  1. Abadie J., “On the Kuhn-Tucker Theorem”. In “Nonlinear programming”. North-Holland, Amsterdam, 1967, pp. 19–36.

    Google Scholar 

  2. Benson H.P., “Hybrid approach for solving multiple-objective linear programs in outcome space”. Jou. of Optimiz. Theory Appls., Vol. 98, No. 1, 1998, pp. 17–35.

    Article  MathSciNet  MATH  Google Scholar 

  3. Benson H.P. and Lee D., “Outcome-based algorithm for optimizing over the efficient set of a bicriteria linear programming problem”. Jou. of Optimiz. Theory Appls., Vol. 88, No. 1, 1996, pp. 77–105.

    Article  MathSciNet  MATH  Google Scholar 

  4. Bigi G., “Lagrangian Functions and Saddle Points in Vector Optimization”. Submitted to Optimization.

    Google Scholar 

  5. Bigi G. and Pappalardo M., “Regularity conditions in Vector Optimization”. Jou. of Optimiz. Theory Appls., Plenum, New York, Vol. 102, No. 1, 1999, pp. 83–96.

    Article  MathSciNet  MATH  Google Scholar 

  6. Castellani M., Mastroeni G. and Pappalardo M., “On regularity for generalized systems and applications”. In “Nonlinear Optimization and Applications”, G. Di Pillo et al. (Eds.), Plenum Press, New York, 1996, pp. 13–26.

    Google Scholar 

  7. Conti R. et al. (Eds.), “Optimization and related fields”. Lecture Notes in Mathematics No. 1190, Springer-Verlag, Berlin, 1986, pp. 57–93.

    Google Scholar 

  8. Craven B.D., “Nonsmooth multiobjective programming”. Nu-mer. Funct. Anal. and Optimiz., Vol. 10, 1989, pp. 49–64.

    Article  MathSciNet  MATH  Google Scholar 

  9. Dauer J.P., “Analysis of the objective space in multiple objective linear programming”. Jou. Mathem. Analysis and Appls., Vol. 126, 1987, pp. 579–593.

    Article  MathSciNet  MATH  Google Scholar 

  10. Dauer J.P., “Solving multiple objective linear programs in the objective space”. Europ. Jou. of Operational Research, Vol. 46, 1990, pp. 350–357.

    Article  MathSciNet  MATH  Google Scholar 

  11. Dien P.H., Mastroeni G., Pappalardo M. and Quang P.H., “Regularity conditions for constrained extremum problems via image space: the linear case”. In Lecture Notes in Sc. and Mathem. Systems, No. 405, Komlosi, Rapcsâck, Schaible Eds., Springer-Verlag, 1994, pp. 145–152.

    Google Scholar 

  12. Dinh The Luc, “Theory of Vector Optimization”. Lecture Notes in Ec. and Mathem. Systems No. 319, Springer-Verlag, Berlin, 1989.

    Google Scholar 

  13. Di Pillo G. et al. (Eds.), “Nonlinear optimization and Applications”. Plenum, New York, 1996, pp. 13–26 and 171–179.

    Google Scholar 

  14. Favati P. and Pappalardo M., “On the reciprocal vector optimization problems”. Jou. Optimiz. Theory Appls., Plenum, New York, Vol. 47, No. 2, 1985, pp. 181–193.

    Article  MathSciNet  MATH  Google Scholar 

  15. Ferreira P.A.V. and Machado M.E.S., “Solving multiple-objective problems in the objectice space”. Jou. Optimiz. Theory Appls., Plenum, New York, Vol. 89, No. 3, 1996, pp. 659–680.

    Article  MathSciNet  Google Scholar 

  16. Galperin E.A., “Nonscalarized multiobjective global optimization”. Jou. Optimiz. Theory Appls., Plenum, New York, Vol. 75, No. 1, 1992, pp. 69–85.

    Article  MathSciNet  MATH  Google Scholar 

  17. Galperin E.A., “Pareto analysis vis-à-vis balance space approach in multiobjective global optimization”. Jou. Optimiz. Theory Appls., Vol. 93, No. 3, 1997, pp. 533–545.

    Article  MathSciNet  MATH  Google Scholar 

  18. Giannessi F., “Theorems of the alternative, quadratic programs and complementarity problems”. In “Variational Inequalities and complementarity problems”, R.W. Cottle et al. Eds., J. Wiley, 1980, pp. 151–186.

    Google Scholar 

  19. Giannessi F., “Theorems of the alternative and optimality conditions”. Jou. Optimiz. Theory Appls., Plenum, New York, Vol. 42, No. 11, 1984. pp. 331–365.

    Article  MathSciNet  MATH  Google Scholar 

  20. Giannessi F., “Semidifferentiable functions and necessary optimality conditions”. Jou. Optimiz. Theory Appls., Vol. 60, 1989, pp. 191–241.

    Article  MathSciNet  MATH  Google Scholar 

  21. Giannessi F., “On Minty variational principle”. In “New trends in mathematical programming”, F. Giannessi, S. Komlosi and T. Rapcsâck Eds., Kluwer Acad. Publ., Dordrecht, 1998, pp. 93–99.

    Google Scholar 

  22. Giannessi F. and Maugeri A. (Eds.), “Variational Inequalities and Networks equilibrium problems”. Plenum, New York, 1995, pp. 1–7, 21–31, 101–121 and 195–211.

    Google Scholar 

  23. Giannessi F. and Pellegrini L., “Image space Analysis for Vector Optimization and Variational Inequalities. Scalarization”. In “Advances in Combinatorial and Global Optimization”, A. Migdalas, P. Pardalos and R. Burkard, Eds., Worlds Scientific Publ., To appear.

    Google Scholar 

  24. Isermann H., “On some relations between a dual pair of multiple objective linear programs”. Zeitschrift für Operations Research, Vol. 22, 1978, pp. 33–41.

    Article  MathSciNet  MATH  Google Scholar 

  25. Kinderleherer D. and Stampacchia G., “An introduction to Variational inequalities”. Academic Press, New York, 1980.

    Google Scholar 

  26. Komlosi S., “On the Stampacchia and Minty Vector Variational Inequalities”. In “Generalized Convexity and Optimization for economic and financial decisions”, G. Giorgi and F. Rossi Eds., Editrice Pitagora, Bologna, Italy, 1999, pp. 231–260.

    Google Scholar 

  27. Leitmann G., “The Calculus of Variations and Optimal Control”. Plenum Press, New York, 1981.

    Google Scholar 

  28. Maeda T., “Constraints Qualifications in Multiobjective Optimization Problems: Differentiable Case”. Jou. of Optimiz. Theory and Appls., Vol. 80, No. 3, 1994, pp. 483–500.

    Article  MATH  Google Scholar 

  29. Mangasarian O.L., “Nonlinear Programming”. Series “Classics in Applied Mathematics”, No. 10, SIAM, Philadelphia, 1994.

    Google Scholar 

  30. Martein L., “Stationary points and necessary conditions in Vector extremum problems”. Tech. Report No. 133, Dept. of Mathem., Optimiz. Group, Univ. of Pisa, 1986. Published with the same title in Jou. of Informations and Optimization Sciences, Vol. 10, No. 1, 1989, pp. 105–128.

    MathSciNet  MATH  Google Scholar 

  31. Martein L., “Lagrange multipliers and generalized differentiable functions in vector extremum problems”. Tech. Report No. 135, Dept. of Mathem., Optimiz. Group, Univ. of Pisa, 1986. Published with the same title in Jou. of Optimiz. Theory Appls., Vol. 63, No. 2, 1989, pp. 281–297.

    MathSciNet  MATH  Google Scholar 

  32. Mastroeni G., “Separation methods for Vector Variational Inequalities. Saddle-point and gap function”. To appear in “Nonlinear Optimization and Applications 2”, G. Di Pillo at al. (Eds.), Kluwer Acad. Publ., Dordrecht, 1999.

    Google Scholar 

  33. Mastroeni G. and Rapcsak T., “On Convex Generalized Systems”. Jou. of Optimiz. Theory and Appls., 1999. To appear.

    Google Scholar 

  34. Pappalardo M., “Some Calculus Rules for semidifferentiable functions and related topics”. In “Nonsmooth Optimization. Methods and Applications”, F. Giannessi Ed., Gordon & Breach, 1992, pp. 281–294.

    Google Scholar 

  35. Pappalardo M., “Stationarity in Vector Optimization”. Rendiconti del Circolo Matematico di Palermo, Serie II, No. 48, 1997, pp. 195–200.

    MathSciNet  Google Scholar 

  36. Pascoletti A. and Serafini P., “Scalarizing Vector Optimization problems”. Jou. Optimiz. Theory Appls., Plenum, New York, Vol. 42, No. 4, 1984, pp. 499–523.

    Article  MathSciNet  MATH  Google Scholar 

  37. Rapcsack T., “Smooth Nonlinear Optimization in Ilan”. Series “Nonconvex Optimization and its Applications”, No. 19, Kluwer Acad. Publ., Dordrecht, 1997.

    Google Scholar 

  38. Song W., “Duality for Vector Optimization of Set Valued Functions”. Jou. of Mathem. Analysis and Appls., Vol. 201, 1996, pp. 212–225.

    Article  MATH  Google Scholar 

  39. Sawaragi Y., Nakayama H. and Tanino T., “Theory of multiobjective Optimization”. Academic Press, New York, 1985.

    Google Scholar 

  40. Tardella F., “On the image of a constrained extremum problem and some applications to the existence of a minimum”. Jou. of Optimiz. Theory Appls., Plenum, New York, Vol. 60, No. 1, 1989, pp. 93–104.

    Article  MathSciNet  MATH  Google Scholar 

  41. Wang S. and Li Z., “Scalarization and Lagrange duality in multiobjective optimization”. Optimization, Vol. 26, Gordon and Breach Publ., 1992, pp. 315–324.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Pisa, Pisa, Italy

    Franco Giannessi & Giandomenico Mastroeni

  2. Institute of Mathematics, University of Verona, Verona, Italy

    Letizia Pellegrini

Authors
  1. Franco Giannessi
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  2. Giandomenico Mastroeni
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  3. Letizia Pellegrini
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Editors and Affiliations

  1. Department of Mathematics, University of Pisa, Pisa, Italy

    Franco Giannessi

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© 2000 Kluwer Academic Publishers

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Giannessi, F., Mastroeni, G., Pellegrini, L. (2000). On the Theory of Vector Optimization and Variational Inequalities. Image Space Analysis and Separation. In: Giannessi, F. (eds) Vector Variational Inequalities and Vector Equilibria. Nonconvex Optimization and Its Applications, vol 38. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0299-5_11

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  • DOI: https://doi.org/10.1007/978-1-4613-0299-5_11

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-7985-0

  • Online ISBN: 978-1-4613-0299-5

  • eBook Packages: Springer Book Archive

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