Canonical Duality Theory for Solving Non-monotone Variational Inequality Problems

  • Guoshan Liu
  • David Yang Gao
  • Shouyang Wang
Part of the Advances in Mechanics and Mathematics book series (AMMA, volume 37)


This paper presents a canonical dual approach for solving a class of non-monotone variational inequality problems. It shows that by using the canonical dual transformation, these challenging problems can be reformulated as a canonical dual problem, which is equivalent to the primal problems in the sense that they have the same set of KKT points. Existence theorem for global optimal solutions is obtained. Based on the canonical duality theory, this dual problem can be solved via well-developed convex programming methods. Applications are illustrated with several examples.



The research of the first author (Guoshan Liu) was supported by the National Natural Science Foundation of China under its grand # 70771106 and the New Century Excellent Scholarship of Ministry of Education, China. The research of the second author (David Gao) was supported by US Air Force Office of Scientific Research under the grants FA2386-16-1-4082 and FA9550-17-1-0151.


  1. 1.
    Allen, G.: Variational inequalities, complementarity problems, and duality theorems. J. Math. Analy. Appl. 58, 1–10 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Duvaut, G., Lions, J.L.: Inequalities in Mechanics and Physics. Springer, Berlin (1976)CrossRefzbMATHGoogle Scholar
  3. 3.
    Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976)zbMATHGoogle Scholar
  4. 4.
    Fichera, G.: Sul problema elastostatico di Signorini con ambigue condizioni al contorno (On the elastostatic problem of Signorini with ambiguous boundary conditions), Rendiconti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, 8 34(2), 138–142 (1963)Google Scholar
  5. 5.
    Friedman, A.: Variational and Free Boundary Problems. Springer, New York (1993)CrossRefGoogle Scholar
  6. 6.
    Gao, D.Y.: Duality theory in nonlinear buckling analysis for von Karman equations. Stud. Appl. Math. 94, 423–444 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gao, D.Y.: Nonlinear elastic beam theory with application in contact problems and variational approaches. Mech. Res. Commun. 23(1), 11–17 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gao, D.Y.: Dual extremum principles in finite deformation theory with applications in post-buckling analysis of nonlinear beam model. Appl. Mech. Rev. ASME 50(11), 64–71 (1997)CrossRefGoogle Scholar
  9. 9.
    Gao, D.Y.: Bi-complementarity and duality: a framework in nonlinear equilibria with applications to the contact problem of elastoplastic beam theory. J. Math. Anal. Appl. 221, 672–697 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gao, D.Y.: Duality Principles in Nonconvex Systems: Theory. Methods and Applications. Kluwer Academic Publishers, Dordrecht (2000)CrossRefzbMATHGoogle Scholar
  11. 11.
    Gao, D.Y.: Nonconvex semi-linear problems and canonical dual solutions. In: Gao, D.Y., Ogden, R.W. (eds.) Advances in Mechanics and Mathematics, vol. II, pp. 261–312. Kluwer Academic, Dordrecht (2003)CrossRefGoogle Scholar
  12. 12.
    Gao, D.Y., Strang, G.: Geometric nonlinearity: potential energy, complementary energy, and the gap function. Quart. Appl. Math. XLVII(3), 487–504 (1989)Google Scholar
  13. 13.
    Gao, D.Y., Yu, H.F.: Multi-scale modelling and canonical dual finite element method in phase transitions of solids. Int. J. Solids Struct. 45, 3660–3673 (2008)CrossRefzbMATHGoogle Scholar
  14. 14.
    Glowinski, R., Lions, J.L., Trémolières, R.: Numerical Analysis of Variational Inequalities. North-Holland, Amsterdam (1981)zbMATHGoogle Scholar
  15. 15.
    Isac, G.: Complementarity Problems. Lecture Notes in Mathematics. Springer, New York (1992)CrossRefzbMATHGoogle Scholar
  16. 16.
    Kuttler, K.L., Purcell, J., Shillor, M.: Analysis and simulations of a contact problem for a nonlinear dynamic beam with a crack. Q. J. Mech. Appl. Mech. (2011). doi: 10.1093/qjmam/hbr018 zbMATHGoogle Scholar
  17. 17.
    MBengue, M.F. Shillor, M.: Regularity result for the problem of vibrations of a nonlinear beam. Electron. J. Differ. Equ., vol. 2008, No. 27, p. 112. or (2008)
  18. 18.
    Mosco, U.: Dual variational inequalities. J. Math. Analy. Appl. 40, 202–206 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Panagiotopoulos, P.D.: Inequality Problems in Mechanics and Applications. Birkhäuser, Boston (1985)CrossRefzbMATHGoogle Scholar
  20. 20.
    Santos, H.A.F.A., Gao, D.Y.: Canonical dual finite element method for solving post-buckling problems of a large deformation elastic beam. Int. J. Nonlinear Mech. 47, 240–247 (2012). doi: 10.1016/j.ijnonlinmec.2011.05.012
  21. 21.
    Stampacchia, G.: Formes bilineaires coercitives sur les ensembles convexes. Comptes rendus hebdomadaires des séances de l’Académie des sci. 258, 4413–4416 (1964)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Yau, S.T., Gao, D.Y.: Obstacle problem for von Karman equations. Adv. Appl. Math. 13, 123–141 (1992)CrossRefzbMATHGoogle Scholar

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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.School of BusinessRenmin University of ChinaBeijingChina
  2. 2.Faculty of Science and TechnologyFederation University AustraliaBallaratAustralia
  3. 3.Laboratory of ManagementDecision and Information Systems Chinese Academy of SciencesBeijingChina

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