Advertisement

On quadratically constrained quadratic optimization problems and canonical duality theory

  • Constantin ZălinescuEmail author
Original Paper

Abstract

In this work we provide a rigorous study of quadratically constrained quadratic optimization problems using the method suggested by the canonical duality theory (CDT). Then we compare our results with those obtained using CDT. We also point out unnecessarily strong assumptions (e.g. Ruan and Gao, in: Gao, Latorre, Ruan (eds) Canonical duality theory. Advances in mechanics and mathematics. Springer, Cham, vol 37, pp 315–338, 2017), non-standard definitions (e.g. Ruan and Gao, in: Gao, Latorre, Ruan (eds) Canonical duality theory. Advances in mechanics and mathematics. Springer, Cham, vol 37, pp 187–201, 2017), unclear statements (e.g. Gao and Ruan in J Glob Optim 47:463–484, 2010), false results (e.g. Gao et al. in J Glob Optim 45:473–497, 2009; Gao and Ruan 2010).

Keywords

Quadratic optimization problem Dual function Canonical duality theory Counterexample 

Notes

Acknowledgements

We thank Prof. Marius Durea for reading a previous version of the paper and for his useful remarks, as well as Prof. Oleg A. Prokopyev, Editor in chief of Optimization Letters, and the two anonymous referees for their useful remarks and suggestions. The results of this paper were announced at the 4th International Conference on Nonlinear Analysis and Optimization (NAOP 2018), June 18–20, 2018, Zanjan, Iran. The work was funded by CNCS-UEFISCDI (Romania) under Grant Number PN-III-P4-ID-PCE-2016-0188.

References

  1. 1.
    Fang, S.C., Gao, D.Y., Sheu, R.L., Wu, S.Y.: Canonical dual approach to solving 0–1 quadratic programming problems. J. Ind. Manag. Optim. 4, 125–142 (2008)MathSciNetzbMATHGoogle Scholar
  2. 2.
    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. Springer, Berlin (2003)CrossRefGoogle Scholar
  3. 3.
    Gao, D.Y.: Canonical duality theory and solutions to constrained nonconvex quadratic programming. J. Glob. Optim. 29, 377–399 (2004)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Gao, D.Y.: Sufficient conditions and perfect duality in nonconvex minimization with inequality constraints. J. Ind. Manag. Optim. 1, 59–69 (2005)MathSciNetGoogle Scholar
  5. 5.
    Gao, D.Y.: Solutions and optimality to box constrained nonconvex minimization problems. J. Ind. Manag. Optim. 3, 293–304 (2007)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Gao, D.Y.: Advances in canonical duality theory with applications to global optimization. In: Proceedings of the Fifth International Conference on Foundations of Computer-Aided Process Operations, pp. 73–82. Omni Press, Cambridge (2008)Google Scholar
  7. 7.
    Gao, D.Y.: Canonical duality theory: unified understanding and generalized solution for global optimization problems. Comput. Chem. Eng. 13, 1964–1972 (2009)CrossRefGoogle Scholar
  8. 8.
    Gao, D.Y.: Canonical duality theory for topology optimization. In: Gao, D.Y., Latorre, V., Ruan, N. (eds.) Canonical Duality Theory. Advances in Mechanics and Mathematics, vol. 37, pp. 263–276. Springer, Cham (2017)CrossRefGoogle Scholar
  9. 9.
    Gao, D.Y., Latorre, V., Ruan, N. (eds.): Canonical Duality Theory. Unified Methodology for Multidisciplinary Study. Advances in Mechanics and Mathematics, vol. 37. Springer, Cham (2017)zbMATHGoogle Scholar
  10. 10.
    Gao, D.Y., Ruan, N.: Solutions to quadratic minimization problems with box and integer constraints. J. Glob. Optim. 47, 463–484 (2010)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Gao, D.Y., Ruan, N., Latorre, V.: RETRACTED: Canonical duality-triality theory: bridge between nonconvex analysis/mechanics and global optimization in complex system. Math. Mech. Solids 21(3), NP5–NP36 (2016)Google Scholar
  12. 12.
    Gao, D.Y., Ruan, N., Latorre, V.: Canonical duality-triality theory: bridge between nonconvex analysis/mechanics and global optimization in complex system. In: Gao, D.Y., Latorre, V., Ruan, N. (eds.) Canonical Duality Theory. Advances in mechanics and mathematics, vol. 37, pp. 1–47. Springer, Cham (2017) CrossRefGoogle Scholar
  13. 13.
    Gao, D.Y., Ruan, N., Sherali, H.: Solutions and optimality criteria for nonconvex constrained global optimization problems with connections between canonical and Lagrangian duality. J. Glob. Optim. 45, 473–497 (2009)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gao, D.Y., Ruan, N., Sherali, H.: Canonical dual solutions for fixed cost quadratic program. In: Chinchuluun, A., Pardalos, P.M., Enkhbat, R., Tseveendorj, I. (eds.) Optimization and Optimal Control. Springer Optimization and Its Applications, vol. 39, pp. 139–156. Springer, New York (2010)CrossRefGoogle Scholar
  15. 15.
    Gao, D.Y., Sherali, H.: Canonical duality theory: connections between nonconvex mechanics and global optimization. In: Gao, D.Y., Sherali, H. (eds.) Advances in Applied Mathematics and Global Optimization. Advances in Mechanics and Mathematics, vol. 17, pp. 257–326. Springer, New York (2009)CrossRefGoogle Scholar
  16. 16.
    Gao, D.Y., Watson, L.T., Easterling, D.R., Thacker, W.I., Billups, S.C.: Solving the canonical dual of box- and integer-constrained nonconvex quadratic programs via a deterministic direct search algorithm. Optim. Methods Softw. 28, 313–326 (2013)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Hiriart-Urruty, J.B.: Conditions for global optimality 2. J. Glob. Optim. 13, 349–367 (1998)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Jeyakumar, V., Rubinov, A.M., Wu, Z.Y.: Non-convex quadratic minimization problems with quadratic constraints: global optimality conditions. Math. Program. 110, 521–541 (2007)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Penot, J.-P.: On the existence of Lagrange multipliers in nonlinear programming in Banach spaces. In: Optimization and Optimal Control (Proceedings of the Conference on Mathematical Research Institute at Oberwolfach, 1980), Lecture Notes in Control and Information Science, pp. 89–104, vol. 30. Springer, Berlin-New York (1981)Google Scholar
  20. 20.
    Ruan, N., Gao, D.Y.: RETRACTED: Canonical duality theory for solving nonconvex/discrete constrained global optimization problems. Math. Mech. Solids 21(3), NP194–NP205 (2016)CrossRefGoogle Scholar
  21. 21.
    Ruan, N., Gao, D.Y.: Canonical duality theory for solving nonconvex/discrete constrained global optimization problems. In: Gao, D.Y., Latorre, V., Ruan, N. (eds.) Canonical Duality Theory. Advances in Mechanics and Mathematics, vol. 37, pp. 187–201. Springer, Cham (2017)CrossRefGoogle Scholar
  22. 22.
    Ruan, N., Gao, D.Y.: Global optimal solution to quadratic discrete programming problem with inequality constraints. In: Gao, D.Y., Latorre, V., Ruan, N. (eds.) Canonical Duality Theory. Advances in Mechanics and Mathematics, pp. 315–338. Springer, Cham (2017)CrossRefGoogle Scholar
  23. 23.
    Voisei, M.D., Zalinescu, C.: On three duality results. arXiv:1008.4329
  24. 24.
    Voisei, M.D., Zălinescu, C.: Counterexamples to some triality and tri-duality results. J. Glob. Optim. 49, 173–183 (2011)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Wang, Z.B., Fang, S.C., Gao, D.Y., Xing, W.X.: Global extremal conditions for multi-integer quadratic programming. J. Ind. Manag. Optim. 4, 213–225 (2008)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Zălinescu, C.: On Gwinner’s paper “Results of Farkas type”. Numer. Funct. Anal. Optim. 10, 199–210 (1989)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Zălinescu, C.: On two triality results. Optim. Eng. 12, 477–487 (2011)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Zălinescu, C.: On V. Latorre and D.Y. Gao’s paper “Canonical duality for solving eneral nonconvex constrained problems”. Optim. Lett. 10, 1781–1787 (2016)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Zhang, X., Zhu, J., Gao, D.Y.: Solution to nonconvex quadratic programming with both inequality and box constraints. Optim. Eng. 10, 183–191 (2009)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2020

Authors and Affiliations

  1. 1.Octav Mayer Institute of MathematicsIaşiRomania
  2. 2.Faculty of MathematicsUniversity “Al. I. Cuza” IaşiIaşiRomania

Personalised recommendations