Skip to main content
SpringerLink
Log in

Optimality Conditions for Several Types of Efficient Solutions of Set-Valued Optimization Problems

  • Chapter
  • First Online:
Nonlinear Analysis and Variational Problems

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 35))

Abstract

A simple unified framework is presented for the study of strong efficient solutions, weak efficient solutions, positive proper efficient solutions, Henig global proper efficient solutions, Henig proper efficient solutions, super efficient solutions, Benson proper efficient solutions, Hartley proper efficient solutions, Hurwicz proper efficient solutions and Borwein proper efficient solutions of set-valued optimization problem with/or without constraints. Some versions of the Lagrange claim, the Fermat rule and the Lagrange multiplier rule are formulated in terms of the first- and second-order radial derivatives, the Ioffe approximate coderivative and the Clarke coderivative.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. J.P. Aubin, I. Ekeland, Applied Nonlinear Analysis, Wiley, New York, 1984.

    MATH  Google Scholar 

  2. T.Q. Bao, B.S. Mordukhovich, Necessary conditions for super minimizers in constrained multiobjective optimization, manuscript (2008).

    Google Scholar 

  3. H.P. Benson, An improved definition of proper efficiency for vector minimization with respect to cones, J. Math. Anal. Appl., 71 (1978) 232–241.

    Article  MathSciNet  Google Scholar 

  4. J.M. Borwein, The geometry of Pareto efficiency over cones, Mathematische Operationforschung and Statistik, Serie Optimization, 11(1980), 235–248.

    MATH  Google Scholar 

  5. J.M. Borwein, D. Zhuang, Super efficiency in vector optimization, Trans. Amer. Math. Society, 338(1993), 105–122.

    Article  MATH  MathSciNet  Google Scholar 

  6. G. Bouligand, Sur l’existence des demi-tangentes à une courbe de Jordan, Fundamenta Mathematicae 15 (1030), 215–218.

    Google Scholar 

  7. G.Y. Chen, J. Jahn, Optimality conditions for set-valued optimization problems, Math. Methods Oper. Res. 48 (1998) 1187–1200.

    MathSciNet  Google Scholar 

  8. F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983.

    MATH  Google Scholar 

  9. H.W. Corley, Optimality conditions for maximization of set-valued functions, J. Optim. Theory Appl. 58 (1988) 1–10.

    Article  MathSciNet  Google Scholar 

  10. G.P. Crepsi, I. Ginchev, M. Rocca, First-order optimality conditions in set-valued optimization, Math. Meth. Oper. Res. 63, (2006), 87–106.

    Article  Google Scholar 

  11. M. Durea, First and second-order Lagrange claims for set-valued maps, J. Optim. Theory Appl., 133 (2007), 111–116.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  13. F. Flores-Bazán, Optimality conditions in non-convex set-valued optimization, Math. Meth. Oper. Res. 53 (2001), 403–417.

    Article  MATH  Google Scholar 

  14. X.H. Gong, Optimality conditions for Henig and globally proper efficient solutions with ordering cone has empty interior, J. Math. Anal. Appl. 307 (2005) 12–37.

    Article  MATH  MathSciNet  Google Scholar 

  15. X.H. Gong, H.B. Dong, S.Y. Wang, Optimality conditions for proper efficient solutions of vector set-valued optimization, J. Math. Anal. Appl. 284 (2003) 332–350.

    Article  MATH  MathSciNet  Google Scholar 

  16. A. Gotz, J. Jahn, The Lagrange multiplier rule in set-valued optimization, SIAM J. Optim. 10 (1999), 331–344.

    Article  MathSciNet  Google Scholar 

  17. E. Goursat, Courses d’Analyse Mathématiques, Courcier, Paris (1813).

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  19. T. X. D. Ha, Lagrange multipliers for set-valued optimization problems associated with coderivatives, J. Math. Anal. Appl. 311 (2005) 647–663.

    Article  MATH  MathSciNet  Google Scholar 

  20. R. Hartley, On cone efficiency, cone convexity, and cone compactness, SIAM Journal on Applied Mathematics 34 (1978), 211–222.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  23. H. Huang, The Lagrange multiplier rule for super efficiency in vector optimization, J. Math. Anal. Appl. 342 (2008), no. 1, 503–513.

    Article  MATH  MathSciNet  Google Scholar 

  24. L. Hurwicz, Programming in linear spaces, Edited by K.J. Arrow, L. Hurwicz, H. Uzawa, Standford University Press, Stanford, CA, 1958.

    Google Scholar 

  25. A.D. Ioffe, Approximate subdifferentials and applications 3: Metric theory, Mathematica 36 (1989), 1–38.

    MATH  MathSciNet  Google Scholar 

  26. G. Isac, A.A. Khan, Dubovitskii-Milyutin approach in set-valued optimization, SIAM J. Control Optim. 47 (2008), no. 1, 144–162.

    Article  MATH  MathSciNet  Google Scholar 

  27. J. Jahn, Mathematical vector optimization in partially ordered linear spaces, Peter Lang, Frankfurt, 1986.

    MATH  Google Scholar 

  28. J. Jahn, Vector Optimization, Springer-Verlag, Berlin, 2004.

    MATH  Google Scholar 

  29. J. Jahn, A. A. Khan, Generalized contingent epiderivatives in set-valued optimization: optimality conditions, Numer. Funct. Anal. Optim. 23 (2002) 807–831.

    Article  MATH  MathSciNet  Google Scholar 

  30. J. Jahn, R. Rauh, Contingent epiderivatives and set-valued optimization, Math. Methods Oper. Res. 46 (1997) 193–211.

    Article  MATH  MathSciNet  Google Scholar 

  31. M.A. Krasnoselski, Positive Solutions of Operator Equations, Nordhoff, Groningen, 1964.

    Google Scholar 

  32. J.L. Lagrange, Théorie des Fonctions Analytiques, Impr.de la République Paris, 1797.

    Google Scholar 

  33. S. J. Li, X. Q. Yang, G.Y. Chen, Nonconvex vector optimization of set-valued mappings, J. Math. Anal. Appl. 283 (2003) 337–350.

    Article  MATH  MathSciNet  Google Scholar 

  34. D.T. Luc, Theory of Vector Optimization, Springer-Verlag, Berlin, 1989.

    Google Scholar 

  35. E.K. Makarov, N.N. Rachkovski, Unified representation of proper efficiency by means of dilating cones, J. Optim. Theory Appl., 101 (1999),141–165.

    Article  MATH  MathSciNet  Google Scholar 

  36. J.-H. Qiu, Dual characterization and scalarization for Benson proper efficiency, SIAM J. Optim., 19 (2008), 144–162.

    Article  MATH  MathSciNet  Google Scholar 

  37. P.H. Sach, Nearly subconvexlike set-valued maps and vector optimization problems, J. Optim. Theory Appl., 119 (2003), 335–356.

    Article  MATH  MathSciNet  Google Scholar 

  38. A. Taa, Set-valued derivatives of multifunctions and optimality conditions, Numer. Func. Anal. Optimiz. 19 (1998), 121–140.

    Article  MATH  MathSciNet  Google Scholar 

  39. A. Taa, Subdifferentials of multifunctions and Lagrange multipliers for multiobjective optimization, J. Math. Anal. Appl. 283 (2003) 398–415.

    Article  MATH  MathSciNet  Google Scholar 

  40. X.M. Yang, D. Li, S.Y. Wang, Nearly subconvexlikeness in vector optimization with set- valued functions, J. Optim. Theory Appl., 110 (2001), 413–427.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  42. X.Y. Zheng, Scalarization of Henig proper efficient points in a normed space, J. Optim. Theory Appl., 105 (2000), 233–247.

    Article  MATH  MathSciNet  Google Scholar 

  43. X.Y. Zheng, Proper efficiency in local convex topological vector spaces, J. Optim. Theory Appl., 94 (1997), 469–486.

    Article  MATH  MathSciNet  Google Scholar 

  44. X.Y. Zheng, K.F. Ng, Fermat rule for multifunctions in Banach spaces, Math. Program, 104 (2005) 69–90.

    Article  MATH  MathSciNet  Google Scholar 

  45. X.Y. Zheng, K.F. Ng, The Lagrange multiplier rule for multifunctions in Banach spaces, SIAM J. Optim., 17 (2006), 1154–1175.

    Article  MATH  MathSciNet  Google Scholar 

  46. D. Zhuang, Density results for proper efficiency, SIAM J. Control Optim., 32 (1994), 51–58.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgment

This research was initiated during the author’s stay at the Institute of Applied Mathematics of the University of Erlangen-Nürnberg under the Georg Forster grant of the Alexander von Humboldt Foundation. The author thanks Professor J. Jahn for hospitality, advice, and help in her work.

Author information

Authors and Affiliations

  1. Hanoi Institute of Mathematics, Hanoi, Vietnam

    T.X.D. Ha (Researcher)

Authors
  1. T.X.D. Ha
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Dept. Industrial & Systems, Engineering, University of Florida, Weil Hall 303, Gainesville, 32611-6595, Florida, USA

    Panos M. Pardalos

  2. Zagoras Street 4, Athens, 151 25, Greece

    Themistocles M. Rassias

  3. School of Mathematical Sciences, Rochester Institute of Technology, Lomb Memorial Drive 85, Rochester, 14623-5602, USA

    Akhtar A. Khan

Additional information

Dedicated to the memory of Professor George Isac

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Ha, T. (2010). Optimality Conditions for Several Types of Efficient Solutions of Set-Valued Optimization Problems. In: Pardalos, P., Rassias, T., Khan, A. (eds) Nonlinear Analysis and Variational Problems. Springer Optimization and Its Applications, vol 35. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-0158-3_21

Download citation

Publish with us

Policies and ethics

Search

Navigation