Journal of Global Optimization

, Volume 73, Issue 1, pp 223–237 | Cite as

Numbers of the connected components of the solution sets of monotone affine vector variational inequalities

  • Vu Trung HieuEmail author


This paper establishes several upper and lower estimates for the maximal number of the connected components of the solution sets of monotone affine vector variational inequalities. Our results give a partial solution to Question 2 in Yen and Yao (Optimization 60:53–68, 2011) and point out that the number depends not only on the number of the criteria but also on the number of variables of the vector variational inequality under investigation.


Monotone affine vector variational inequality Solution set Number of connected components Scalarization formula Skew-symmetric matrix 

Mathematics Subject Classification

49J40 47H05 90C29 90C33 



The author is indebted to Professor Nguyen Dong Yen for many stimulating conversations.


  1. 1.
    Facchinei, F., Pang, J.-S.: Finite-dimensional variational inequalities and complementarity problems, vol. I and II. Springer, New York (2003)zbMATHGoogle Scholar
  2. 2.
    Giannessi, F.: Theorems of alternative, quadratic programs and complementarity problems. In: Cottle, R.W., Giannessi, F., Lions, J.-L. (eds.) Variational Inequality and Complementarity Problems, pp. 151–186. Wiley, New York (1980)Google Scholar
  3. 3.
    Hoa, T.N., Phuong, T.D., Yen, N.D.: Number of connected components of the solution sets in linear fractional vector optimization, Preprint 2002/41, Institute of Mathematics, HanoiGoogle Scholar
  4. 4.
    Hieu, V.T.: The Tarski -Seidenberg theorem with quantifiers and polynomial vector variational inequalities (2018).
  5. 5.
    Huong, N.T.T., Hoa, T.N., Phuong, T.D., Yen, N.D.: A property of bicriteria affine vector variational inequalities. Appl. Anal. 91, 1867–1879 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Huong, N.T.T., Yao, J.-C., Yen, N.D.: Connectedness structure of the solution sets of vector variational inequalities. Optimization 66, 889–901 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Huong, N.T.T., Yao, J.-C., Yen, N.D.: Polynomial vector variational inequalities under polynomial constraints and applications. SIAM J. Optim. 26, 1060–1071 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Lax, P.D.: Linear Algebra and Its Applications. Wiley, Hoboken (2007)zbMATHGoogle Scholar
  9. 9.
    Lee, G.M., Kim, D.S., Lee, B.S., Yen, N.D.: Vector variational inequalities as a tool for studying vector optimization problems. Nonlinear Anal. 34, 745–765 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Lee, G.M., Yen, N.D.: A result on vector variational inequalities with polyhedral constraint sets. J. Optim. Theory Appl. 109, 193–197 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lee, G.M., Tam, N.N., Yen, N.D.: Quadratic Programming and Affine Variational Inequalities: A Qualitative Study, Series: Nonconvex Optimization and its Applications, vol. 78. Springer, New York (2005)Google Scholar
  12. 12.
    Yao, J.-C., Yen, N.D.: Monotone affine vector variational inequalities. Optimization 60, 53–68 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Yen, N.D.: Linear fractional and convex quadratic vector optimization problems. Vector Optimization. In: Ansari, Q., Yao, J.-C. (eds.) Recent Developments in Vector Optimization, vol. 1. Springer, Berlin (2012)CrossRefGoogle Scholar
  14. 14.
    Yen, N.D.: An introduction to vector variational inequalities and some new results. Acta Math. Vietnam 41, 505–529 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Yen, N.D., Phuong, T.D.: Connectedness and stability of the solution sets 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

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

Authors and Affiliations

  1. 1.Division of MathematicsPhuong Dong UniversityHanoiVietnam

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