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Nonlinear separation concerning E-optimal solution of constrained multi-objective optimization problems

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Abstract

This paper aims at investigating optimality conditions in terms of E-optimal solution for constrained multi-objective optimization problems in a general scheme, where E is an improvement set with respect to a nontrivial closed convex point cone with apex at the origin. In the case where E is not convex, nonlinear vector regular weak separation functions and scalar weak separation functions are introduced respectively to realize the separation between the two sets in the image space, and Lagrangian-type optimality conditions are established. These results extend and improve the convex ones in the literature.

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References

  1. 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 

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

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Giannessi, F., Mastroeni, G., Pellegrini, L.: On the Theory of Vector Optimization and Variational Inequalities: Image Space Analysis and Separation, Vector Variational Inequalities and Vector Equilibria, Mathematical Theories. F. Giannessi, Kluwer, Dordrecht (2000)

    Book  MATH  Google Scholar 

  5. Giannessi, F.: Constrained Optimization and Image Space Analysis. Volume 1: Separation of Sets and Optimality Conditions, 1st edn. Mathematical Concepts and Methods in Science and Engineering. Springer, Berlin (2005)

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

    Article  MathSciNet  Google Scholar 

  7. Gupta, D., Mehra, A.: Two types of approximate saddle points. Numer. Funct. Anal. Optim. 29, 532–550 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Guu, S.-M., Li, J.: Vector quasi-equilibrium problems: separation, saddle points and error bounds for the solution set. J. Glob. Optim. 58, 751–767 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. 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 

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

    Book  MATH  Google Scholar 

  11. Khan, A.A., Tammer, C., Zalinescu, C.: Set-Valued Optimization. Springer, Berlin (2015)

    Book  MATH  Google Scholar 

  12. 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 

  13. Mastroeni, G.: Some applications of the image space analysis to the duality theory for constrained extremum problems. J. Glob. Optim. 46, 603–614 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Mastroeni, G.: Optimality conditions and image space analysis for vector optimization problems. In: Ansari, Q.H., Yao, J.-C. (eds.) Recent Developments in Vector Optimization, Vector Optimization, vol. 1, pp. 169–220. Springer, Dordrecht (2012)

    Chapter  Google Scholar 

  15. Mastroeni, G., Panicucci, B., Passacantando, M., Yao, J.C.: A separation approach to vector quasi-equilibrium problems: saddle point and gap function. Taiwan. J. Math. 13, 657–673 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Xu, Y.D., Li, S.J.: Nonlinear separation functions and constrained extremum problems. Optim. Lett. 8, 1149–1160 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the anonymous referees for valuable comments and suggestions, which helped to improve the paper. This research was supported by the National Natural Science Foundation of China (Grant Numbers: 11171362, 11571055).

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

    M. X. You & S. J. Li

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  1. M. X. You
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  2. S. J. Li
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Correspondence to S. J. Li.

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You, M.X., Li, S.J. Nonlinear separation concerning E-optimal solution of constrained multi-objective optimization problems. Optim Lett 12, 123–136 (2018). https://doi.org/10.1007/s11590-017-1109-x

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  • DOI: https://doi.org/10.1007/s11590-017-1109-x

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