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Separations and Optimality of Constrained Multiobjective Optimization via Improvement Sets

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Abstract

In this paper, we investigate the separations and optimality conditions for the optimal solution defined by the improvement set of a constrained multiobjective optimization problem. We introduce a vector-valued regular weak separation function and a scalar weak separation function via a nonlinear scalarization function defined in terms of an improvement set. The nonlinear separation between the image of the multiobjective optimization problem and an improvement set in the image space is established by the scalar weak separation function. Saddle point type optimality conditions for the optimal solution of the multiobjective optimization problem are established, respectively, by the nonlinear and linear separation methods. We also obtain the relationships between the optimal solution and approximate efficient solution of the multiobjective optimization problem. Finally, sufficient and necessary conditions for the (regular) linear separation between the approximate image of the multiobjective optimization problem and a convex cone are also presented.

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References

  1. Chen, J.W., Cho, Y.J., Kim, J., Li, J.: Multiobjective optimization problems with modified objective functions and cone constraints and applications. J. Glob. Optim. 49, 137–147 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Chen, J.W., Cho, Y.J., Wan, Z.: Optimality conditions for cone constrained nonsmooth multiobjective optimization. J. Nonlinear Convex Anal. 17, 1627–1642 (2016)

    MathSciNet  MATH  Google Scholar 

  3. Chen, J.W., Wan, Z., Cho, Y.J.: Nonsmooth multiobjective optimization problems and weak vector quasi-variational inequalities. Comput. Appl. Math. 32, 291–301 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Long, X.J., Huang, N.J.: Optimality conditions for efficiency on nonsmooth multiobjective programming problems. Taiwan. J. Math. 18, 687–699 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chicco, M., Mignanego, F., Pusillo, L., Tijs, S.: Vector optimization problems via improvement sets. J. Optim. Theory Appl. 150, 516–529 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Gutiérrez, C., Jiménez, B., Novo, V.: Improvement sets and vector optimization. Eur. J. Oper. Res. 223, 304–311 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Zhao, K.Q., Yang, X.M.: E-Benson proper efficiency in vector optimization. Optimization 64, 739–752 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Carathéodory, C.: Calculus of Variations and Partial Differential Equations of the First Order. Chelsea, New York (1982). Translation of the volume Variationsrechnung und Partielle Differential Gleichungen Erster Ordnung. B.G. Teubner, Berlin (1935)

  9. Bellman, R.: Dynamic Programming. Princeton University Press, Princeton (1957)

    MATH  Google Scholar 

  10. Hestenes, M.R.: Optimization Theory: The Finite Dimensional Case. Wiley, New York (1975)

    MATH  Google Scholar 

  11. Castellani, G., Giannessi, F.: Decomposition of mathematical programs by means of theorems of alternative for linear and nonlinear systems. In: Proceedings of the Ninth International Mathematical Programming Symposium, Hungarian Academy of Sciences, Budapest, pp. 423–439 (1979)

  12. Giannessi, F.: Theorems of alternative, quadratic programs and complementarity problems. In: Cottle, R.W., Giannessi, F., Lions, J.L. (eds.) Variational Inequalities and Complementarity Problems, pp. 151–186. Wiley, New York (1980)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Giannessi, F.: Constrained Optimization and Image Space Analysis, Seperation of Sets and Optimality Conditions, vol. 1. Springer, Berlin (2005)

    MATH  Google Scholar 

  15. Chen, J.W., Li, S.J., Wan, Z., Yao, J.C.: Vector variational-like inequalities with constraints: separation and alternative. J. Optim. Theory Appl. 166, 460–479 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Chen, J.W., Köbis, E., Köbis, M., Yao, J.C.: Image space analysis for constrained inverse vector variational inequalities via multiobjective optimization. J. Optim. Theory Appl. (2017). https://doi.org/10.1007/s10957-017-1197-x

    MATH  Google Scholar 

  17. Li, S.J., Xu, Y.D., Zhu, S.K.: Nonlinear separation approach to constrained extremum problems. J. Optim. Theory Appl. 154, 842–856 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Giannessi, F., Mastroeni, G., Pellegrini, L.: On the theory of vector optimization and variational inequalities. Image space analysis and separation. In: Giannessi, F. (ed.) Vector Variational Inequalities and Vector Equilibria, pp. 153–215. Kluwer, Dordrech (2000)

    Chapter  Google Scholar 

  19. 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 

  20. You, M.X., Li, S.J.: Nonlinear separation concerning E-optimal solution of constrained multi-objective optimization problems. Optim. Lett. 12, 123–136 (2018)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  23. Khan, A.A., Tammer, C., Zălinescu, C.: Set-Valued Optimization: An Introduction with Applications. Springer, Berlin (2015)

    Book  MATH  Google Scholar 

  24. Lalitha, C.S., Chatterjee, P.: Stability and scalarization in vector optimization using improvement sets. J. Optim. Theory Appl. 166, 825–843 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. Gutiérrez, C., Jiménez, B., Miglierina, E., Molho, E.: Scalarization in set optimization with solid and nonsolid ordering cones. J. Glob. Optim. 61, 525–552 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  28. You, M.X., Li, S.J.: Separation functions and optimality conditions in vector optimization. J. Optim. Theory Appl. 175, 527–544 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the anonymous referees for their valuable suggestions, which have helped to improve the paper. This research was partially supported by the Natural Science Foundation of China (11401487, 11571055), the Basic and Advanced Research Project of Chongqing (cstc2016jcyjA0239), the China Postdoctoral Science Foundation (2015M582512), the Fundamental Research Funds for the Central Universities (XDJK2018).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China

    Jiawei Chen & La Huang

  2. College of Computer Science, Chongqing University, Chongqing, 400044, China

    Jiawei Chen

  3. College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China

    Shengjie Li

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  1. Jiawei Chen
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  2. La Huang
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  3. Shengjie Li
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Correspondence to Shengjie Li.

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Chen, J., Huang, L. & Li, S. Separations and Optimality of Constrained Multiobjective Optimization via Improvement Sets. J Optim Theory Appl 178, 794–823 (2018). https://doi.org/10.1007/s10957-018-1325-2

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  • DOI: https://doi.org/10.1007/s10957-018-1325-2

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