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Affine Variational Inequalities on Normed Spaces

  • Nguyen Dong YenEmail author
  • Xiaoqi Yang
Article

Abstract

This paper studies infinite-dimensional affine variational inequalities on normed spaces. It is shown that infinite-dimensional quadratic programming problems and infinite-dimensional linear fractional vector optimization problems can be studied by using affine variational inequalities. We present two basic facts about infinite-dimensional affine variational inequalities: the Lagrange multiplier rule and the solution set decomposition.

Keywords

Infinite-dimensional affine variational inequality Infinite-dimensional quadratic programming Infinite-dimensional linear fractional vector optimization Generalized polyhedral convex set Solution set 

Mathematics Subject Classification

49J40 49J50 49K40 90C20 90C29 

Notes

Acknowledgements

The first author was supported by the joint research project from RFBR and VAST.HTQT.NGA-02/16-17. The second author was supported by the Research Grants Council of Hong Kong (PolyU 152167/15E). The authors would like to thank Professor Franco Giannessi for his helpful comments and suggestions.

References

  1. 1.
    Cottle, R.W., Pang, J.S., Stone, R.E.: The Linear Complementarity Problem. SIAM, Philadelphia (2009)CrossRefzbMATHGoogle Scholar
  2. 2.
    Gowda, M.S., Pang, J.-S.: On the boundedness and stability of solutions to the affine variational inequality problem. SIAM J. Control Optim. 32, 421–441 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Lee, G.M., Tam, N.N., Yen, N.D.: Quadratic Programming and Affine Variational Inequalities: A Qualitative Study. Springer, New York (2005)zbMATHGoogle Scholar
  4. 4.
    Ha, C.D.: Stability of the linear complementarity problem at a solution point. Math. Program. 31, 327–338 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ha, C.D., Narula, S.C.: Tolerance approach to sensitivity analysis in linear complementarity problems. J. Optim. Theory Appl. 73, 197–203 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gowda, M.S.: On the continuity of the solution map in linear complementarity problems. SIAM J. Optim. 2, 619–634 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gowda, M.S., Pang, J.-S.: On solution stability of the linear complementarity problem. Math. Oper. Res. 17, 77–83 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Phung, H.T.: On continuity properties of the solution map in linear complementarity problems. Vietnam J. Math. 30, 251–258 (2002)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Phung, H.T.: A geometrical approach to the linear complementarity problem. Vietnam J. Math. 32, 141–153 (2004)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Lee, G.M., Tam, N.N., Yen, N.D.: Continuity of the solution map in parametric affine variational inequalities. Set Valued Anal. 15, 105–123 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Robinson, S.M.: Solution continuity in monotone affine variational inequalities. SIAM J. Optim. 18, 1046–1060 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lu, S., Robinson, S.M.: Variational inequalities over perturbed polyhedral convex sets. Math. Oper. Res. 33, 689–711 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dontchev, A.L., Rockafellar, R.T.: Characterizations of strong regularity for variational inequalities over polyhedral convex sets. SIAM J. Optim. 6, 1087–1105 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Henrion, R., Mordukhovich, B.S., Nam, N.M.: Second-order analysis of polyhedral systems in finite and infinite dimensions and applications to robust stability of variational inequalities. SIAM J. Optim. 20, 2199–2227 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    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)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Yao, J.-C., Yen, N.D.: Coderivative calculation related to a parametric affine variational inequality, Part 2: applications. Pac. J. Optim. 5, 493–506 (2009)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Qui, N.T.: Linearly perturbed polyhedral normal cone mappings and applications. Nonlinear Anal. 74, 1676–1689 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Qui, N.T.: New results on linearly perturbed polyhedral normal cone mappings. J. Math. Anal. Appl. 381, 352–364 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Qui, N.T.: Nonlinear perturbations of polyhedral normal cone mappings and affine variational inequalities. J. Optim. Theory Appl. 153, 98–122 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Trang, N.T.Q.: A note on Lipschitzian stability of variational inequalities over perturbed polyhedral convex sets. Optim. Lett. 10, 1221–1231 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Huyen, D.T.K., Yen, N.D.: Coderivatives and the solution map of a linear constraint system. SIAM J. Optim. 26, 986–1007 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Huyen, D.T.K., Yao, J.-C.: Solution stability of a linearly perturbed constraint system and Applications. Set Valued Var. Anal. OnlineFirst (2017).  https://doi.org/10.1007/s11228-017-0442-7
  23. 23.
    Roy, M., Pang, J.-S.: Error bounds for the linear complementarity problem with a \(P\)-matrix. Linear Algebra Appl. 132, 123–136 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Luo, Z.-Q., Tseng, P.: Error bound and convergence analysis of matrix splitting algorithms for the affine variational inequality problem. SIAM J. Optim. 2, 43–54 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)CrossRefzbMATHGoogle Scholar
  26. 26.
    Jeyakumar, V., Yang, X.Q.: Convex composite multi-objective nonsmooth programming. Math. Program. Ser. A 59, 325–343 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Zheng, X.Y., Yang, X.Q.: Conic positive definiteness and sharp minima of fractional orders in vector optimization problems. J. Math. Anal. Appl. 391, 619–629 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Ioffe, A.D., Tihomirov, V.M.: Theory of Extremal Problems. North-Holland Publishing Company, Amsterdam (1979)zbMATHGoogle Scholar
  29. 29.
    Luan, N.N., Yao, J.-C., Yen, N.D.: On some generalized polyhedral convex constructions. Numer. Funct. Anal. Optim. 39, 537–570 (2018)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Luan, N.N., Yen, N.D.: A representation of generalized convex polyhedra and applications. Preprint [arXiv:submit/1896080] (2015)
  31. 31.
    Luan, N.N.: Efficient solutions in generalized linear vector optimization. Appl. Anal. FirstOnline [ https://doi.org/10.1080/00036811.2018.1441992] (2018)
  32. 32.
    Bartl, D.: A short algebraic proof of the Farkas lemma. SIAM J. Optim. 19, 234–239 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Choo, E.U., Atkins, D.R.: Bicriteria linear fractional programming. J. Optim. Theory Appl. 36, 203–220 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Choo, E.U., Atkins, D.R.: Connectedness in multiple linear fractional programming. Manag. Sci. 29, 250–255 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Steuer, R.E.: Multiple Criteria Optimization: Theory, Computation and Application. Wiley, New York (1986)zbMATHGoogle Scholar
  36. 36.
    Malivert, C.: Multicriteria fractional programming. In: Sofonea, M., Corvellec, J.N. (eds.) Proceedings of the 2nd Catalan Days on Applied Mathematics, pp. 189–198. Presses Universitaires de Perpinan (1995)Google Scholar
  37. 37.
    Malivert, C., Popovici, N.: Bicriteria linear fractional optimization. In: “Optimization”, Lecture Notes in Economic and Mathematical Systems, vol. 481, pp. 305–319. Springer, Berlin (2000)Google Scholar
  38. 38.
    Hoa, T.N., Phuong, T.D., Yen, N.D.: Linear fractional vector optimization problems with many components in the solution sets. J. Ind. Manag. Optim. 1, 477–486 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Hoa, T.N., Huy, N.Q., Phuong, T.D., Yen, N.D.: Unbounded components in the solution sets of strictly quasiconcave vector maximization problems. J. Glob. Optim. 37, 1–10 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Yen, N.D., Yao, J.-C.: Monotone affine vector variational inequalities. Optimization 60, 53–68 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Yen, N.D.: Linear fractional and convex quadratic vector optimization problems. In: Ansari, Q.H., Yao, J.-C. (eds.) Recent Developments in Vector Optimization, pp. 297–328. Springer, Berlin (2012)CrossRefGoogle Scholar
  42. 42.
    Jeyakumar, V., Yang, X.Q.: On characterizing the solution sets of pseudolinear programs. J. Optim. Theory Appl. 87, 747–755 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)CrossRefzbMATHGoogle Scholar
  44. 44.
    Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980)zbMATHGoogle Scholar
  45. 45.
    Yen, N.D., Phuong, T.D.: Connectedness and stability of the solution set in linear fractional vector optimization problems. In: Giannessi, F. (ed.) Vector Variational Inequalities and Vector Equilibria, pp. 479–489. Kluwer Academic Publishers, Dordrecht (2000)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of MathematicsVietnam Academy of Science and TechnologyHanoiVietnam
  2. 2.Department of Applied Mathematics, The Hong Kong Polytechnic UniversityHong KongChina

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