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Nonlinear separation functions, optimality conditions and error bounds for Ky Fan quasi-inequalities

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Abstract

In this paper, we employ the image space analysis to investigate a Ky Fan quasi-inequality with cone constraints. By means of the oriented distance function, a new nonlinear weak (regular) separation function is introduced. Some necessary and sufficient optimality conditions, especially, a saddle-point sufficient optimality condition for the Ky Fan quasi-inequality with cone constraints, are obtained. By virtue of the nonlinear regular weak separation function, a gap function for the Ky Fan quasi-inequality with cone constraints is obtained. Moreover, we get an error bound for the solution set of the Ky Fan quasi-inequality with respect to the gap function under strongly monotone assumptions.

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References

  1. 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)

  2. Fan, K.: Extensions of two fixed point theorems of F.E. Browder. Math. Z. 112, 234–240 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  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.: Constrained Optimization and Image Space Analysis, vol. 1: separation of sets and optimality conditions. Springer, Berlin (2005)

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Giannessi, F.: On the theory of Lagrangian duality. Optim. Lett. 1, 9–20 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chinaie, M., Zafarani, J.: Image space analysis and scalarization of multivalued optimization. J. Optim. Theory Appl. 142, 451–467 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dien, P.H., Mastroeni, G., Pappalardo, M., Quang, P.H.: Regularity conditions for constrained extremum problems via image space. J. Optim. Theory Appl. 80, 19–37 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  9. 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 

  10. 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 

  11. 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 

  12. Mastroeni, G., Pellegrini, L.: Conic separation for vector optimization problems. Optimization 60, 129–142 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Pappalardo, M.: Image space approach to penalty methods. J. Optim. Theory Appl. 64, 141–152 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  15. Rubinov, A.M., Uderzo, A.: On global optimality conditions via separation functions. J. Optim. Theory Appl. 109, 345–370 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  16. Tardella, F.: On the image of a constrained extremum problem and some applications to existence of a minimum. J. Optim. Theory Appl. 69, 93–104 (1989)

    Article  MathSciNet  Google Scholar 

  17. Li, J., Feng, S.Q., Zhang, Z.: A unified approach for constrained extremum problems: image space analysis. J. Optim. Theory Optim. 159, 69–92 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  19. Li, J., Huang, N.J.: Image space analysis for variational inequalities with cone constraints applications to traffic equilibria. Sci. China Math. 55, 851–868 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. 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)

    MathSciNet  MATH  Google Scholar 

  21. Mastroeni, G.: On the image space analysis for vector quasi-equilibrium problems with a variable ordering relation. J. Glob. Optim. 53, 203–214 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. 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 

  23. 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 

  24. 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 

  25. Xu, Y.D., Li, S.J.: Optimality conditions for generalized Ky Fan quasi-inequalities with applications. J. Optim. Theory Appl. 157, 663–684 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. Xu, Y.D., Li, S.J.: Gap functions and error bounds for weak vector variational inequalities. Optimization 63, 1339–1352 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  27. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  29. Amahroq, T., Taa, A.: On Lagrange Kuhn-Tucker multipliers for multiobjective optimization problems. Optimization 41, 159–172 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  30. Miglierina, E., Molho, E.: Scalarization and stability in vector optimization. J. Optim. Theory Appl. 114, 657–670 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

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

    Y. D. Xu

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  1. Y. D. Xu
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Correspondence to Y. D. Xu.

Additional information

The author thanks the two anonymous reviewers for their valuable comments and suggestions, which helped to improve the paper. This research was supported by the National Natural Science Foundation of China (Grant numbers: 11426055, 11401058, 61472056), Basic and Advanced Research Project of CQ CSTC (Grant number: cstc2014jcyjA00044) and the Doctor Start-up Foundation of University of Posts and Telecommunications (Grant number: A2014-15).

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Xu, Y.D. Nonlinear separation functions, optimality conditions and error bounds for Ky Fan quasi-inequalities. Optim Lett 10, 527–542 (2016). https://doi.org/10.1007/s11590-015-0879-2

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  • DOI: https://doi.org/10.1007/s11590-015-0879-2

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