On the Solution Existence of Nonconvex Quadratic Programming Problems in Hilbert Spaces


We propose conditions for the existence of solutions for nonconvex quadratic programming problems whose constraint set is defined by finitely many convex linear-quadratic inequalities in Hilbert spaces. In order to obtain our results, we use either properties of the Legendre form or properties of compact operators with closed range. The results are established without requesting the convexity of the objective function or the compactness of the constraint set. As a special case, we obtain some on the existence of solutions results for the quadratic programming problems under linear constraints in Hilbert spaces.

This is a preview of subscription content, log in to check access.


  1. 1.

    Belousov, E.G.: Introduction to Convex Analysis and Integer Programming. Moscow University Publisher, Moscow (1977)

    Google Scholar 

  2. 2.

    Belousov, E.G., Klatte, D.: A Frank-Wolfe type theorem for convex polynomial programs. Comput. Optim. Appl. 22, 37–48 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Bertsekas, D.P., Tseng, P.: Set intersection theorems and existence of optimal solutions. Math. Program. 110, 287–314 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Blum, E., Oettli, W.: Direct proof of the existence theorem in quadratic programming. Oper. Res. 20, 165–167 (1972)

    Article  MATH  Google Scholar 

  5. 5.

    Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer (2000)

  6. 6.

    Borwein, J.M.: Necessary and sufficient conditions for quadratic minimality. Numer. Funct. Anal. Appl. 5, 127–140 (1982)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Dong, V.V., Tam, N.N.: On the solution existence for convex quadratic programming problems in Hilbert spaces. Taiwanese J. Math. 20(6), 1417–1436 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Eaves, B.C.: On quadratic programming. Manag. Sci. 17, 698–711 (1971)

    Article  MATH  Google Scholar 

  9. 9.

    Frank, M., Wolfe, P.: An algorithm for quadratic programming. Naval Res. Logist. Quarter. 3, 95–110 (1956)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Hestenes, M.R.: Applications of the theory of quadratic forms in Hilbert space to the calculus of variations. Pac. J. Math. 1, 525–581 (1951)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Hauser, R.: The S-Procedure via dual cone calculus. arXiv:1305.2444 (2013)

  12. 12.

    Ioffe, A.D., Tihomirov, V.M.: Theory of Extremal Problems. North-Holand Publishing Company, Amsterdam-New York-Oxford (1979)

    Google Scholar 

  13. 13.

    Kim, D.S., Tam, N.N., Yen, N.D.: Solution existence and stability of quadratically constrained convex quadratic programs. Optim. Lett. 6, 363–373 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Kurdila, A.J., Zabarankin, M.: Convex Functional Analysis. Birkhauser Verlag (2005)

  15. 15.

    Luenberger, D.G.: Optimization by Vector Space Methods. Wiley, New York-London-Sydney-Toronto (1969)

    Google Scholar 

  16. 16.

    Luo, Z.Q., Zhang, S.: On extensions of the Frank-Wolfe theorems. Comput. Optim. Appl. 13, 87–110 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Schochetman, I.E., Smith, R.L., Tsui, S.K.: Solution existence for infinite quadratic programming. Technical Report 97–10 (1997)

  18. 18.

    Semple, J.: Infinite positive-definite quadratic programming in a Hilbert space. J. Optim. Theory Appl. 88, 743–749 (1996)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Rudin, W.: Functional Analysis. McGraw-Hill Inc (1991)

  20. 20.

    Yakubovich, V.A.: Nonconvex optimization problem: The infinite-horizon linear-quadratic control problem with quadratic constraints. Sys. Control Lett. 19, 13–22 (1992)

    MathSciNet  Article  MATH  Google Scholar 

Download references


This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2014.39. The authors would like to thank Prof. N. D. Yen for valuable remarks and suggestions. The authors would like to express our sincere thanks to the anonymous referees and editors for insightful comments and useful suggestions.

Author information



Corresponding author

Correspondence to Vu Van Dong.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Dong, V.V., Tam, N.N. On the Solution Existence of Nonconvex Quadratic Programming Problems in Hilbert Spaces. Acta Math Vietnam 43, 155–174 (2018). https://doi.org/10.1007/s40306-017-0237-9

Download citation


  • Quadratic program in Hilbert spaces
  • Convex quadratic constraints
  • Solution existence
  • Legendre form
  • Recession cone
  • Compact operator with closed range

Mathematics Subject Classification (2010)

  • 90C20
  • 90C26
  • 90C30