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Tangential extremal principles for finite and infinite systems of sets II: applications to semi-infinite and multiobjective optimization

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This paper contains selected applications of the new tangential extremal principles and related results developed in Mordukhovich and Phan (Math Program 2011) to calculus rules for infinite intersections of sets and optimality conditions for problems of semi-infinite programming and multiobjective optimization with countable constraints.

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References

  1. Bao T.Q., Mordukhovich B.S.: Relative Pareto minimizers to multiobjective problems: existence and optimality conditions. Math. Program. 122, 301–347 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  3. Bauschke H.H., Borwein J.M., Li W.: Strong conical hull intersection property, bounded linear regularity, Jameson’s property (G), and error bounds in convex optimization. Math. Program. 86, 135–160 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Borwein J.M., Zhu Q.J.: Techniques of Variational Analysis. Springer, New York (2005)

    MATH  Google Scholar 

  5. Cánovas M.J., López M.A., Mordukhovich B.S., Parra J.: Variational analysis in semi-infinite and infinite programming, I: stability of linear inequality systems of feasible solutions. SIAM J. Optim. 20, 1504–1526 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cánovas M.J., López M.A., Mordukhovich B.S., Parra J.: Variational analysis in semi-infinite and infinite programming, II: necessary optimality conditions. SIAM J. Optim. 20, 2788–2806 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chui C.K., Deutsch F., Ward J.D.: Constrained best approximation in Hilbert space. Constr. Approx. 6, 35–64 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  8. Deutsch F., Li W., Ward J.D.: A dual approach to constrained interpolation from a convex subset of Hilbert space. J. Approx. Theory 90, 385–414 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dinh N., Mordukhovich B.S., Nghia T.T.A.: Qualification and optimality conditions for DC programs with infinite constraints. ACTA Math. Vietnam. 34, 125–155 (2009)

    MathSciNet  MATH  Google Scholar 

  10. Dinh N., Mordukhovich B.S., Nghia T.T.A.: Suddifferentials of value functions and optimality conditions for DC and bilivel infinite and semi-infinite programs. Math. Program. 123, 101–138 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ernst E., Théra M.: Boundary half-strips and the strong CHIP. SIAM. J. Optim. 18, 834–852 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fang D.H., Li C., Ng K.F.: Constraint qualifications for optimality conditions and total Lagrange dualities in convex programming. Nonlinear Anal. 73, 1143–1159 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  14. Goberna M.A., Jeyakumar V., Lopez M.A.: Necessary and sufficient constraint qualifications for solvability of systems of infinite convex inequalities. Nonlinear Anal. 68, 1184–1194 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Goberna M.A., López M.A.: Linear Semi-Infinite Optimization. Wiley, Chichester (1998)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  17. Li C., Ng K.F.: Constraint qualification, the strong CHIP and best approximation with convex constraints in Banach spaces. SIAM J. Optim. 14, 584–607 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Li C., Ng K.F.: Strong CHIP for infinite system of closed convex sets in normed linear spaces. SIAM J. Optim. 16, 311–340 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. Li C., Ng K.F., Pong T.K.: The SECQ, linear regularity, and the strong CHIP for an infinite system of closed convex sets in normed linear spaces. SIAM. J. Optim. 18, 643–665 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Li W., Nahak C., Singer I.: Constraint qualifications for semi-infinite systems of convex inequalities. SIAM J. Optim. 11, 31–52 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mordukhovich B.S.: Analysis and Generalized Differentiation, I: Basic Theory. Springer, Berlin (2006)

    Google Scholar 

  22. Mordukhovich B.S.: Variational Analysis and Generalized Differentiation, II: Applications. Springer, Berlin (2006)

    Google Scholar 

  23. Mordukhovich, B.S., Phan, H.M.: Tangential extremal principle for finite and infinite set systems, I: basic theory. Math. Program. (2011). doi:10.1007/s10107-012-0549-4

  24. Puente R., Verade Serio V.N.: Locally Farkas-Minkowski linear semi-infinite systems. TOP 7, 103–121 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  25. Rockafellar R.T., Wets R.-J.: Variational Analysis. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  26. Schirotzek W.: Nonsmooth Analysis. Springer, Berlin (2007)

    Book  MATH  Google Scholar 

  27. Song W., Zang R.: Bounded linear regularity of convex sets in Banach spaces and its applications. Math. Program. 106, 59–79 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA

    Boris S. Mordukhovich & Hung M. Phan

Authors
  1. Boris S. Mordukhovich
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  2. Hung M. Phan
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Corresponding author

Correspondence to Boris S. Mordukhovich.

Additional information

This research was partially supported by the US National Science Foundation under grants DMS-0603846 and DMS-1007132 and by the Australian Research Council under grant DP-12092508.

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Mordukhovich, B.S., Phan, H.M. Tangential extremal principles for finite and infinite systems of sets II: applications to semi-infinite and multiobjective optimization. Math. Program. 136, 31–63 (2012). https://doi.org/10.1007/s10107-012-0550-y

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