Exact Dual Bounds for Some Nonconvex Minimax Quadratic Optimization Problems


The nonconvex separable minimax quadratic optimization problem is analyzed. Two approaches to the problem solution are described, namely, by using SOCP relaxation and by using Lagrangian relaxation of a quadratic extremum analog problem. A condition is obtained that guarantees finding the value and the global extremum point of the problem of the considered class by calculating the dual bound of the equivalent quadratic extremum problem.

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  1. 1.

    N. Z. Shor, Nondifferentiable Pptimization and Polynomial Problems, Kluwer Acad. Publ., Boston–London–Dordrecht (1998).

  2. 2.

    P. M. Pardalos and J. B. Rosen, “Constrained global optimization: Algorithms and applications,” in: Lecture Notes in Computer Science, Vol. 268, Springer-Verlag, Berlin–Heidelberg (1987).

  3. 3.

    J. A. Momoh, Electric Power System Applications of Optimization, CRC Press, New York (2017).

    Google Scholar 

  4. 4.

    J. S. Arora, Optimization of Structural and Mechanical Systems, World Scientific, Singapore (2007).

    Google Scholar 

  5. 5.

    N. Z. Shor, I. V. Sergienko, V. P. Shylo, P. I. Stetsyuk, I. M. Parasyuk, T. T. Lebedeva, Yu. P. Laptin, M. G. Zhurbenko, T. O. Bardadym, F. A. Sharifov, O. P. Lykhovyd, O. A. Berezovskyi, and V. M. Myroshnychenko, Problems of Optimal Design of Reliable Networks [in Ukrainian], Naukova Dumka, Kyiv (2005).

    Google Scholar 

  6. 6.

    P. I. Stetsyuk, G. Bortis, J.-F. Emmenegger, L. B. Koshlai, O. A. Berezovskyi, T. A. Bardadym, E. L. Pervukhina, V. V. Golikova, K. N. Osipov, E. P. Karpets, and A. V. Pilipovskii, Institutional and Technological Changes in the Countries with Market and Transition Economy [in Russian], Vyd. Dim “Kyivo-Mogylyans’ka Akademiya” (2015).

  7. 7.

    G. M. Guisewite, “Network problems,” in: R. Horst and P. M. Pardalos (eds.), Handbook of Global Optimization, Nonconvex Optimization and Its Applications, Vol. 2, Springer, Boston (1995), pp. 609–648.

  8. 8.

    Yu. I. Zhuravlev, Yu. P. Laptin, A. P. Vinogradov, N. G. Zhurbenko, O. P. Lykhovyd, and O. A. Berezovskyi, “Linear classifiers and selection of informative features,” Pattern Recognition and Image Analysis, Vol. 27, Iss. 3, 426–432 (2017).

  9. 9.

    V. I. Muntiyan, O. V. Prokopenko, M. M. Petrushenko, L. I. Aveskulova, and B. I. Adasovskii, Economic Security of a State: Strategy, Power Energy, Information Technologies [in Ukrainian], OOO Yurka Lyubchenka, Kyiv (2014).

  10. 10.

    A. R. Pankov, E. N. Platonov, and K. V. Semenikhin, “Minimax quadratic optimization and its application to investment planning,” Automation and Remote Control, Vol. 62, Iss. 12, 1978–1995 (2001).

  11. 11.

    N. Z. Shor and O. A. Berezovskii, “Using the method of dual quadratic solutions to solve systems of polynomial equations in the complex domain,” Cybern. Syst. Analysis, Vol. 30, No. 5, 686–692 (1994). https://doi.org/10.1007/BF02367749.

    Article  MATH  Google Scholar 

  12. 12.

    Y. Nesterov, H. Wolkowicz, and Y. Ye, “Semidefinite programming relaxations of nonconvex quadratic optimization,” in: Handbook of Semidefinite Programming, Springer US, New York (2000), pp. 361–419.

  13. 13.

    S. Kim and M. Kojima, “Second order cone programming relaxation of nonconvex quadratic optimization problems,” Optimization Methods and Software, Vol. 15, Iss. 3, 201–224 (2001).

    MathSciNet  Article  Google Scholar 

  14. 14.

    C. Lemaréchal, “Lagrangian relaxation,” in: M. Junger and D. Naddef (eds.), Computational Combinatorial Optimization, Lecture Notes in Computer Science, Vol. 2241, Springer, Berlin–Heidelberg (2001), pp. 112–156. https://doi.org/10.1007/3-540-45586-8_4.

  15. 15.

    K. Anstreicher and H. Wolkowicz, “On Lagrangian relaxation of quadratic matrix constraints,” SIAM J. on Matrix Analysis and Applications, Vol. 2, Iss. 1, 41–55 (2000).

    MathSciNet  Article  Google Scholar 

  16. 16.

    A. Qualizza, P. Belotti, and F. Margot, “Linear programming relaxations of quadratically constrained quadratic programs,” in: J. Lee and S. Leyffer (eds.), Mixed Integer Nonlinear Programming, The IMA Volumes in Mathematics and its Applications, Vol. 154, Springer, New York (2012), pp. 407–426. https://doi.org/10.1007/978-1-4614-1927-3_14.

  17. 17.

    V. Jeyakumar and G. Li, “Exact second-order cone programming relaxations for some nonconvex minimax quadratic optimization problems,” SIAM J. on Optimization, Vol. 28, Iss. 1, 760–787 (2018).

    MathSciNet  Article  Google Scholar 

  18. 18.

    H. Wolkowicz, R. Saigal, and L. Vandenberghe (eds.), Handbook of Semidefinite Programming: Theory, Algorithms, and Applications, Springer Sci. & Business Media, New York (2012).

    Google Scholar 

  19. 19.

    J. B. Lasserre, “An explicit exact SDP relaxation for nonlinear 0-1 programs,” in: Proc. Intern. Conf. on Integer Programming and Combinatorial Optimization, Springer, Berlin (2001), pp. 293–303.

    Google Scholar 

  20. 20.

    S. Burer and K. M. Anstreicher, “Second-order-cone constraints for extended trust-region subproblems,” SIAM J. on Optimization, Vol. 23, Iss. 1, 432–451 (2013).

    MathSciNet  Article  Google Scholar 

  21. 21.

    A. Beck and Y. C. Eldar, “Strong duality in nonconvex quadratic optimization with two quadratic constraints,” SIAM J. on Optimization, Vol. 17, Iss. 3, 844–860 (2006).

    MathSciNet  Article  Google Scholar 

  22. 22.

    O. A. Berezovskyi, “On the accuracy of dual bounds for quadratic extremum problems,” Cybern. Syst. Analysis, Vol. 48, No. 1, 26–30 (2012). https://doi.org/10.1007/s10559-012-9389-8.

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    V. Jeyakumar and G. Li, “Trust-region problems with linear inequality constraints: Exact SDP relaxation, global optimality and robust optimization”, Mathematical Programming, Vol. 147, Iss. 1, 171–206 (2014).

    MathSciNet  Article  Google Scholar 

  24. 24.

    V. Jeyakumar and G.Y. Li, “Strong duality in robust convex programming: Complete characterizations,” SIAM J. on Optimization, Vol. 20, Iss. 6, 3384–3407 (2010).

    MathSciNet  Article  Google Scholar 

  25. 25.

    O. A. Berezovskyi, “On solving of a special optimization problem connected with determination of invariant sets of dynamical systems,” J. Autom. Inform. Sci., Vol. 47, Iss. 5, 69–77 (2015).

    Article  Google Scholar 

  26. 26.

    S. Haines, J. Loeppky, P. Tseng, and X. Wang, “Convex relaxations of the weighted maxmin dispersion problem,” SIAM J. on Optimization, Vol. 23, Iss. 4, 2264–2294 (2013).

    MathSciNet  Article  Google Scholar 

  27. 27.

    A. Ben-Tal, L. E. Ghaoui, and A. Nemirovski, Robust Optimization, Princeton Ser. Appl. Math. Princeton Univ. Press, Princeton (2009).

  28. 28.

    F. Alizadeh, “Optimization over the positive-definite cone: Interior point methods and combinatorial applications,” in: P. M. Pardalos (ed.), Advances in Optimization and Parallel Computing, Elsevier Science, Amsterdam (1992), pp. 1–25.

  29. 29.

    T. Fujit and M. Kojima, “Semidefinite programming relaxation for nonconvex quadratic problems,” J. of Global Optimization, Vol. 10, Iss. 4, 367–380 (1997).

    MathSciNet  Article  Google Scholar 

  30. 30.

    O. A. Berezovskyi, “Exactness criteria of SDP-relaxations of quadratic extremum problems,” Cybern. Syst. Analysis, Vol. 52, No. 6, 915–920 (2016). https://doi.org/10.1007/s10559-016-9893-3.

    Article  MATH  Google Scholar 

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Correspondence to O. A. Berezovskyi.

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Translated from Kibernetyka ta Systemnyi Analiz, No. 1, January–February, 2021, pp. 115–122.

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Berezovskyi, O.A. Exact Dual Bounds for Some Nonconvex Minimax Quadratic Optimization Problems. Cybern Syst Anal 57, 101–107 (2021). https://doi.org/10.1007/s10559-021-00333-1

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  • minimax quadratic optimization problem
  • SOCP relaxation
  • Lagrangian relaxation
  • exact dual bound
  • positive definite matrix