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A nonconvex separation property and some applications

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Abstract

In this paper we proved a nonconvex separation property for general sets which coincides with the Hahn-Banach separation theorem when sets are convexes. Properties derived from the main result are used to compute the subgradient set to the distance function in special cases and they are also applied to extending the Second Welfare Theorem in economics and proving the existence of singular multipliers in Optimization.

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References

  1. Arrow, K., Hahn, F.: General competitive analysis. Holden Day, San Francisco, 1971

  2. Attouch, H.: Variational convergence for functions and operators. Applicable Mathematics Series, Pitman, London, 1984

  3. Bonnans, J.F., Shapiro, A.: Optimization problem with perturbations, a guided tour. SIAM Review 40, 202–227 (1998)

    Article  MathSciNet  Google Scholar 

  4. Bonnisseau, J.M., Cornet, B.: Valuation of equilibrium and Pareto optimum in nonconvex economies. J. Math. Econ. 17, 293–315 (1998)

    Article  MathSciNet  Google Scholar 

  5. Borwein, J., Jofre, A.: A nonconvex separation property in Banach spaces. J Oper. Res. Appl. Math. 48, 169–180 (1997)

    Article  Google Scholar 

  6. Clarke, F.: Generalized gradients and applications. Trans. Am. Math. Soc. 205, 247–262 (1975)

    Article  MATH  Google Scholar 

  7. Clarke, F.: Optimization and nonsmooth analysis. Wiley, New York, 1983

  8. Cominetti, R.: Metric regularity, tangent cones, and second-order optimality conditions. J. Appl. Math. Optim. 21, 265–287 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  9. Cornet, B., Rockafellar, R.T.: Separation theorems and supporting price theorems for nonconvex sets. Preprint. Verbal communication, 1989

  10. Debreu, G.: Theory of value, an axiomatic analysis of economic equilibrium. Wiley, New York, 1959

  11. Dontchev, A.L., Zolezzi, T.: Well-Posed Optimization Problems. Lectures Notes in Mathematics, Springer-Verlag, 1993

  12. Ioffe, A.D.: Approximate subdifferentials and applications. I: The finite dimensional theory. Trans. Am. Math. Soc. 281, 389–416 (1984)

    MATH  MathSciNet  Google Scholar 

  13. Ioffe, A.D.: Approximate subdifferentials and applications II. Mathematika, 33, 111–128 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  14. Ioffe, A.D.: Approximate subdifferentials and applications III: The metric theory. Mathematika, 71, 1–38 (1989)

    MathSciNet  Google Scholar 

  15. Jofré, A., Rivera, J.: The second welfare theorem with public goods in nonconvex nontransitive economies with externalities. Technical Report, CMM, Universidad de Chile, 2001

  16. Jourani, A., Thibault, L.: Metric regularity for strongly compactly lipschitzian mappings. Nonlinear Analysis: Theory, Methods and Applications, 24, 229–240 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  17. Jourani, A., Thibault, L.: Verifiable conditions for openness and regularity of multivalued mappings in Banach spaces. Trans. Am. Math. Soc. 347, 1255–1268 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kruger, A., Mordukhovich, B.: Extremal points and the Euler equation in nonsmooth optimization. Dokl. Akad. Nauk BSSR. 24, 684–687 (1980)

    MATH  MathSciNet  Google Scholar 

  19. Mordukhovich, B.: Maximum principle in problems of time optimal control with nonsmooth constrains. J. Appl. Math. Mech. 40, 960–969 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  20. Mordukhovich, B.: Metric approximations and necessary optimality conditions for general classes of nonsmooth extremal problems. Dokl. Akad. Nauk BSSR. 254, 526–530 (1980). (English transl. in Soviet Math. Dokl. 22 (1980))

    MATH  Google Scholar 

  21. Mordukhovich, B.: Generalized differential calculus for nonsmooth and set-valued mappings. J. Math. Anal. Appl. 183, 250–288 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  22. Mordukhovich, B.: The extremal principle and its applications to optimization and economics. In: Rubinov, A., Glover, M. (eds). Nonlinear Optimization and Related Topics, Kluwer Academic Publishers, 47, 343–369 (2001)

    MATH  MathSciNet  Google Scholar 

  23. Mordukhovich, B., Shao, Y.: Extremal characterizations of Asplund spaces. Proceedings of the American Mathematical Society, 124, 197–205 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  24. Mordukhovich, B., Shao, Y.: Nonconvex differential calculus for infinite directional multifunction. Set-Valued Analysis, 4, 205–256 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  25. Mordukhovich, B., Shao, Y.: Nonsmooth sequential analysis in Asplund spaces. Transactions of the American Mathematical Society, 348, 1235–1280 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  26. Robinson, S.: Stability theory for systems of inequalities, part II: differentiable nonlinear systems. SIAM J. Numer. Anal. 13, 497–513 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  27. Rockafellar, R.T.: Convex analysis. Princeton University Press, Princeton, NJ, 1970

  28. Rockafellar, R.T.: The theory of subgradients and its applications to problems of optimization convex and nonconvex functions. Helderman Verlag, Berlin, 1981

  29. Rockafellar, R.T.: Extensions of subgradient calculus with applications to optimization. Nonlinear Analysis: Theory, Methods and Applications, 9, 665-698 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  30. Rockafellar, R.T., Wets, R.J.: Variational analysis. Springer-Verlag, Berlin, 1998

  31. Thibault, L.: Personal communication. 1998

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Authors and Affiliations

  1. Centre for Mathematical Modelling and Departamento de Ingeniería Matemática, Universidad de Chile, Casilla 170, correo 3, Santiago, Chile

    Alejandro Jofré

  2. Departamento de Economía, Universidad de Chile, Diagonal Paraguay 257, Torre 26, Of. 1502, Santiago, Chile

    Jorge Rivera Cayupi

Authors
  1. Alejandro Jofré
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  2. Jorge Rivera Cayupi
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Correspondence to Alejandro Jofré.

Additional information

This work was partially supported by FONDECYT, ICM Complex Engineering System, CEE-ECOS.

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Jofré, A., Cayupi, J. A nonconvex separation property and some applications. Math. Program. 108, 37–51 (2006). https://doi.org/10.1007/s10107-006-0703-y

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  • DOI: https://doi.org/10.1007/s10107-006-0703-y

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