Skip to main content
SpringerLink
Log in

Image Space Analysis for Set Optimization Problems with Applications

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

In this paper, we consider a set optimization problem with a partial order relation, which is defined by Minkowski difference. By using the image space analysis, we establish the relationships among the set optimization problem, a vector optimization problem and a set-valued optimization with vector criterion related to the image of the set optimization problem. In addition, two nonlinear regular weak separation functions are proposed for the set optimization problem. Based on the two nonlinear regular weak separation functions, saddle point sufficient optimality conditions, gap functions and error bounds for the set optimization problem, are obtained. Finally, we explore some applications of the obtained results to investigate robust multi-objective optimization problems and verify the validity of the results in shortest path problems with data uncertainty and multi-criteria traffic network equilibrium problems with interval-valued cost functions.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Ansari, Q.H., Köbis, E., Sharma, P.K.: Characterizations of multiobjective robustness via oriented distance function and image space analysis. J. Optim. Theory Appl. 181, 817–839 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ansari, Q.H., Sharma, P.K., Qin, X.: Characterizations of robust optimality conditions via image space analysis. Optimization 69, 2063–2083 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ansari, Q.H., Sharma, P.K., Yao, J.C.: Minimal elements theorems and Ekelands variational principle with new set order relations. J. Nonlinear Convex Anal. 19, 1127–1139 (2018)

    MathSciNet  MATH  Google Scholar 

  4. Bao, T.Q., Mordukhovich, B.S.: Set-valued optimization in welfare economics. Adv. Math. Econ. 13, 113–153 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bianchi, M., Pini, R.: Sensitivity for parametric vector equilibria. Optimization 55, 221–230 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cao, J.D., Li, R.X., Huang, W., Guo, J.H., Wei, Y.: Traffic network equilibrium problems with demands uncertainty and capacity constraints of arcs by scalarization approaches. Sci. China Technol. Sci. 61, 1642–1653 (2018)

    Article  Google Scholar 

  7. Castellani, G., Giannessi, F.: Decomposition of mathematical programs by means of theorems of alternative for linear and nonlinear systems. In: Proceedings of Ninth International Mathematical Programming Symposium, Budapest. Survey of Mathematical Programming, pp. 423–439. North-Holland, Amsterdam (1979)

  8. Chen, G.Y., Huang, X.X., Yang, X.Q.: Vector Optimization. Set-Valued and Variational Analysis. Springer, Heidelberg (2005)

    MATH  Google Scholar 

  9. Chen, J.W., Köbis, E., Yao, J.C.: Optimality conditions and duality for robust nonsmooth multiobjective optimization problems with constraints. J. Optim. Theory Appl. 181, 411–436 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chinaie, M., Zafarani, J.: A new approach to constrained optimization via image space analysis. Positivity 20, 99–114 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. Doagooei, A.R.: Minimum type functions, plus-cogauges and applications. J. Optim. Theory Appl. 164, 551–564 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ehrgott, M., Ide, J., Schöbel, A.: Minmax robustness for multi-objective optimization problems. Eur. J. Oper. Res. 239, 17–31 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Eichfelder, G., Krüger, C., Schöbel, A.: Decision uncertainty in multiobjective optimization. J. Global Optim. 69, 485–510 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. Giannessi, F.: Theorems of the alternative and optimality conditions. J. Optim. Theory Appl. 42, 331–365 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  15. Giannessi, F.: Constrained Optimization and Image Space Analysis, Volume 1: Separation of Sets and Optimality Conditions. Springer, Berlin (2005)

  16. Giannessi, F.: Some perspectives on vector optimization via image space analysis. J. Optim. Theory Appl. 177, 906–912 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  17. Giannessi, F., Mastroeni, G.: Separation of sets and Wolfe duality. J. Global Optim. 42, 401–412 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Göpfert, A., Riahi, H., Tammer, C., Zalinescu, C.: Variational Methods in Partially Ordered Spaces. CMS Book in Mathematics. Springer, New York (2003)

    MATH  Google Scholar 

  19. Gupta, M., Srivastava, M.: Approximate solutions and Levitin-Polyak well-posedness for set optimization using weak efficiency. J. Optim. Theory Appl. 186, 191–208 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hamel, A.H., Heyde, F., Löhne, A., Rudloff, B., Schrage, C.: Set Optimization and Applications-The State of the Art. Springer-Verlag, New York (2015)

    Book  MATH  Google Scholar 

  21. Hasan, A.Q., Elisabeth, K., Kumar, S.P.: Characterizations of set relations with respect to variable domination structures via oriented distance function. Optimization 67, 1389–1407 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hestenes, M.: Optimization Theory: The Finite Dimensional Case. Wiley, London (1975)

    MATH  Google Scholar 

  23. Hiriart-Urruty, J.B.: Tangent cone, generalized gradients and mathematical programming in Bananch spaces. Math. Oper. Res. 4, 79–97 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ide, J., Köbis, E.: Concepts of efficiency for uncertain multi-objective optimization problems based on set order relations. Math. Methods Oper. Res. 80, 99–127 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  25. Ide, J., Schöbel, A.: Robustness for uncertain multi-objective optimization: a survey and analysis of different concepts. OR Spectr. 38, 235–271 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  27. Jahn, J., Ha, T.X.D.: New order relations in set optimization. J. Optim. Theory Appl. 148, 209–236 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  28. Karaman, E., Soyertem, M., Güvenç, İA., Tozkan, D., Küçük, M., Küçük, Y.: Partial order relations on family of sets and scalarizations for set optimization. Positivity 22, 783–802 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  29. Khan, A., Tammer, C., Zǎlinescu, C.: Set-Valued Optimization: An Introduction with Applications. Springer, Heidelberg (2015)

    Book  MATH  Google Scholar 

  30. Khushboo, Lalitha, C.S.: Scalarizations for a set optimization problem using generalized oriented distance function. Positivity 23, 1195–1213 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  31. Kuroiwa, D.: The natural criteria in set-valued optimization. RIMS Kokyuroku Kyto Univ 1031, 85–90 (1998)

    MathSciNet  MATH  Google Scholar 

  32. Kuroiwa, D.: On set-valued optimization. Nonlinear Anal. 47, 1395–1400 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  33. Li, J., Mastroeni, G.: Refinements on gap functions and optimality conditions for vector quasi-equilibrium problems via image space analysis. J. Optim. Theory Appl. 177, 696–716 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  34. Li, S.J., Xu, Y.D., You, M.X., Zhu, S.K.: Constrained extremum problems and image space analysis-part I: optimality conditions. J. Optim. Theory Appl. 177, 609–636 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  35. Li, S.J., Xu, Y.D., You, M.X., Zhu, S.K.: Constrained extremum problems and image space analysis-part II: duality and penalization. J. Optim. Theory Appl. 177, 637–659 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  36. Li, S.J., Xu, Y.D., You, M.X., Zhu, S.K.: Constrained extremum problems and image space analysis-part III: generalized systems. J. Optim. Theory Appl. 177, 660–678 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  37. Lin, Z.: The study of traffic equilibrium problems with capacity constraints of arcs. Nonlinear Anal.-Real World Appl. 11, 2280–2284 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  38. Luc, D.T., Phuong, T.T.T.: Equilibrium in multi-criteria transportation networks. J. Optim. Theory Appl. 169, 116–147 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  39. Luc, D.T., Raţiu, A.: Vector optimization: basic concepts and solution methods. In: Al-Mezel, S.A.R., Al-Solamy, F.R.M., Ansari, Q.H. (eds.) Fixed Point Theory, Variational Analysis and Optimization, pp. 249–306. CRC Press, Taylor and Francis Group, Boca Raton (2014)

  40. Luo, H.Z., Mastroeni, G., Wu, H.X.: Separation approach for augmented Lagrangians in constrained nonconvex optimization. J. Optim. Theory Appl. 144, 275–290 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  41. Mastroeni, G.: Nonlinear separation in the image space with applications to penalty methods. Appl. Anal. 91, 1901–1914 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  42. Moldovan, A., Pellegrini, L.: On regularity for constrained extremum problems, part 1: suffficient optimality conditions. J. Optim. Theory Appl. 142, 147–163 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  43. Moldovan, A., Pellegrini, L.: On regularity for constrained extremum problems, part 2: necessary optimality conditions. J. Optim. Theory Appl. 142, 165–183 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  44. Pallaschke, D., Urbanski, R.: Pairs of Compact Convex Sets. Kluwer Academic Publishers, Dordrecht (2002)

    Book  MATH  Google Scholar 

  45. Pellegrini, L.: Some perspectives on set-valued optimization via image space analysis. J. Optim. Theory Appl. 177, 811–815 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  46. Phuong, T.T.T.: Smoothing method in multi-criteria transportation network equilibrium problem. Optimization 68, 1577–1598 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  47. Studniarski, M., Michalak, A., Stasiak, A.: Necessary and sufficient conditions for robust minimal solutions in uncertain vector optimization. J. Optim. Theory Appl. 186, 375–397 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  48. Wei, H.Z., Chen, C.R., Li, S.J.: A unified characterization of multiobjective robustness via separation. J. Optim. Theory Appl. 179, 86–102 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  49. Wei, H.Z., Chen, C.R., Li, S.J.: Characterizations of multiobjective robustness on vectorization counterparts. Optimization 69, 493–518 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  50. Wei, H.Z., Chen, C.R., Li, S.J.: Robustness characterizations for uncertain optimization problems via image space analysis. J. Optim. Theory Appl. 186, 459–479 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  51. Wei, H.Z., Chen, C.R., Wu, B.W.: Vector network equilibrium problems with uncertain demands and capacity constraints of arcs. Optim. Lett. 15, 1113–1131 (2021)

    Article  MathSciNet  Google Scholar 

  52. Xu, Y.D., Li, S.J., Teo, K.L.: Vector network equilibrium problems with capacity constraints of arcs. Trans. Res. Part E 48, 567–577 (2012)

    Article  Google Scholar 

  53. Xu, Y.D., Zhang, P.P.: Gap functions for constrained vector variational inequalities with applications. Optimization 66, 2171–2191 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  55. Zhu, S.K., Li, S.J.: Unified duality theory for constrained extremum problems, part I: image space analysis. J. Optim. Theory Appl. 161, 738–762 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  56. Zhu, S.K., Li, S.J.: Unified duality theory for constrained extremum problems, part II: special duality schemes. J. Optim. Theory Appl. 161, 763–782 (2014)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to express their sincere gratitude to Professor Jafar Zafarani and the anonymous reviewers for their constructive suggestions toward the improvement of this paper. This research was supported by the National Natural Science Foundation of China (Grant Numbers: 11801051, 11601437) and the Natural Science Foundation of Chongqing (Grant Number: cstc2019jcyj-msxmX0075).

Author information

Authors and Affiliations

  1. College of Science, Chongqing University of Posts and Telecommunications, Chongqing, 400065, China

    Yang-Dong Xu & Cheng-Ling Zhou

  2. Department of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, 611130, China

    Sheng-Kun Zhu

Authors
  1. Yang-Dong Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Cheng-Ling Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sheng-Kun Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yang-Dong Xu.

Additional information

Communicated by Jafar Zafarani.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xu, YD., Zhou, CL. & Zhu, SK. Image Space Analysis for Set Optimization Problems with Applications. J Optim Theory Appl 191, 311–343 (2021). https://doi.org/10.1007/s10957-021-01939-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-021-01939-3

Keywords

Search

Navigation