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Coderivatives of implicit multifunctions and stability of variational systems

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Abstract

We establish formulas for computing/estimating the regular and Mordukhovich coderivatives of implicit multifunctions defined by generalized equations in Asplund spaces. These formulas are applied to obtain conditions for solution stability of parametric variational systems over perturbed smooth-boundary constraint sets.

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References

  1. Dontchev, A.L., Rockafellar, R.T.: Characterizations of strong regularity for variational inequalities over polyhedral convex sets, SIAM. J. Optim. 6, 1087–1105 (1996)

    MathSciNet  MATH  Google Scholar 

  2. Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings. Springer, Dordrecht (2009)

    Book  MATH  Google Scholar 

  3. Fabian, M., Mordukhovich, B.S.: Sequential normal compactness versus topological normal compactness in variational analysis. Nonlinear Anal. 54, 1057–1067 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Henrion, R., Mordukhovich, B.S., Nam, N.M.: Second-order analysis of polyhedral systems in finite and infinite dimensions with applications to robust stability of variational inequalities. SIAM J. Optim. 20, 2199–2227 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Lang, S.: Real and Functional Analysis. Springer, New York (1993)

    Book  MATH  Google Scholar 

  6. Le Thi, H.A., Pham Dinh, T.: Solving a class of linearly constrained indefinite quadratic problems by d.c. algorithms. J. Global Optim. 11, 253–285 (1997)

  7. Le Thi, H.A., Pham Dinh, T., Yen, N.D.: Behavior of DCA sequences for solving the trust-region subproblem. J. Global Optim. 53, 317–329 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Le Thi, H.A., Pham Dinh, T., Yen, N.D.: Properties of two DC algorithms in quadratic programming. J. Global Optim. 49, 481–495 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ledyaev, Y.S., Zhu, Q.J.: Implicit multifunctions theorems. Set-Valued Anal. 7, 209–238 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Lee, G.M., Yen, N.D.: Fréchet and normal coderivatives of implicit multifunctions. Appl. Anal. 90, 1011–1027 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Lee, G.M., Yen, N.D.: Coderivatives of a Karush–Kuhn–Tucker point set map and applications. Nonlinear Anal. 95, 191–201 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Levy, A.B., Mordukhovich, B.S.: Coderivatives in parametric optimization. Math. Program. Ser. A. 99, 311–327 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Mordukhovich, B.S.: Coderivative analysis of variational systems. J. Global Optim. 28, 347–362 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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

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

  16. Nam, N.M.: Coderivatives of normal mappings and the Lipschitzian stability of parametric variational inequalities. Nonlinear Anal. 73, 2271–2282 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Pham Dinh, T., Le Thi, H.A.: A DC optimization algorithm for solving the trust-region subproblem. SIAM J. Optim. 8, 476–505 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  18. Qui, N.T.: Generalized differentiation of a class of normal cone operators. J. Optim. Theory Appl. 161, 398–429 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Qui, N.T.: Linearly perturbed polyhedral normal cone mappings and applications. Nonlinear Anal. 74, 1676–1689 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Qui, N.T.: New results on linearly perturbed polyhedral normal cone mappings. J. Math. Anal. Appl. 381, 352–364 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Qui, N.T.: Nonlinear perturbations of polyhedral normal cone mappings and affine variational inequalities. J. Optim. Theory Appl. 153, 98–122 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Qui, N.T.: Stability for trust-region methods via generalized differentiation. J. Global Optim. 59, 139–164 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Qui, N.T.: Variational inequalities over Euclidean balls. Math. Methods Oper. Res. 78, 243–258 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Qui, N.T., Yen, N.D.: A class of linear generalized equations. SIAM J. Optim. 24, 210–231 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  25. Robinson, S.M.: Generalized equations and their solutions. I. Basic theory. Math. Program. Stud. 10, 128–141 (1979)

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

    Book  MATH  Google Scholar 

  27. Rudin, W.: Principles of Mathematical Analysis. McGraw-Hill, New Delhi (1976)

    MATH  Google Scholar 

  28. Yao, J.-C., Yen, N.D.: Coderivative calculation related to a parametric affine variational inequality, part 1: basic calculations. Acta Math. Vietnam. 34, 157–172 (2009)

    MathSciNet  MATH  Google Scholar 

  29. Yao, J.-C., Yen, N.D.: Coderivative calculation related to a parametric affine variational inequality, part 2: applications. Pacific J. Optim. 5, 493–506 (2009)

    MathSciNet  MATH  Google Scholar 

  30. Yao, J.-C., Yen, N.D.: Parametric variational system with a smooth-boundary constraint set. In: Mordukhovich, B.S., Burachik, R.S., Yao, J.-C. (eds.) Variational Analysis and Generalized Differentiation in Optimization and Control, vol. 47, pp. 205–221. Springer, New York (2010). Series “Optimization and Its Applications”

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Acknowledgments

A part of this work was done when the author was visiting the Vietnam Institute for Advanced Study in Mathematics (VIASM). The author would like to thank the VIASM for hospitality and kind support. The author is indebted to the handling Editors and the anonymous referees for their valuable remarks and detailed suggestions that have greatly improved the original version of the paper.

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Correspondence to Nguyen Thanh Qui.

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This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2014.56.

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Qui, N.T. Coderivatives of implicit multifunctions and stability of variational systems. J Glob Optim 65, 615–635 (2016). https://doi.org/10.1007/s10898-015-0387-z

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  • DOI: https://doi.org/10.1007/s10898-015-0387-z

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