Non-Convex Quadratic Programming Problems in Short Wave Antenna Array Optimization

  • Anton V. EremeevEmail author
  • Nikolay N. Tyunin
  • Alexander S. Yurkov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11548)


In this paper, we describe a non-convex constrained quadratic programming problem arising in short wave transmitting antenna array synthesis and provide preliminary computational results. We consider problem instances for three different antenna designs including up to 25 radiators. In the computational experiments, BARON package is compared to the gradient optimization method, applied to the unconstrained problem formulation using the penalty function method. Global optimality of the obtained solutions is established using BARON package the smallest instances of 4 radiators. On small instances, both methods have demonstrated similar results, while on larger instances significant difference has been observed. The set of local optima is studied experimentally. It is established that even though the problem instances have numerous local optima, the objective function in many local optima has the same value.


Quadratic programming Local optima Antenna array Gradient optimization Computational experiment 



The work on Sect. 2 was funded in accordance with the state task of the Omsk Scientific Center SB RAS (project number FWEF-2019-0006).


  1. 1.
    Akdagli, A., Guney, K.: Shaped-beam pattern synthesis of equally and unequally spaced linear antenna arrays using a modified tabu search algorithm. Microwave Opt. Technol. Lett. 36(1), 16–20 (2003)CrossRefGoogle Scholar
  2. 2.
    Aoki, M.: Introduction to optimization techniques. fundamentals and applications of nonlinear programming. Technical report, California Univ Los Angeles Dept of System Science (1971)Google Scholar
  3. 3.
    Boriskin, A.V., Balaban, M.V., Galan, O.Y., Sauleau, R.: Efficient approach for fast synthesis of phased arrays with the aid of a hybrid genetic algorithm and a smart feed representation. In: 2010 IEEE International Symposium on Phased Array Systems and Technology, pp. 827–832. IEEE (2010)Google Scholar
  4. 4.
    Burke, G.J., Poggio, A.J., Logan, J.C., Rockway, J.W.: Numerical electromagnetic code (\(\text{NEC}\)). In: 1979 IEEE International Symposium on Electromagnetic Compatibility, pp. 1–3. IEEE (1979)Google Scholar
  5. 5.
    Eberhart, R., Kennedy, J.: Particle swarm optimization. In: Proceedings of the IEEE international Conference on Neural Networks, vol. 4, pp. 1942–1948. IEEE (1995)Google Scholar
  6. 6.
    Echeveste, J.I., de Aza, M.A.G., Zapata, J.: Shaped beam synthesis of real antenna arrays via finite-element method, floquet modal analysis, and convex programming. IEEE Trans. Antennas Propag. 64(4), 1279–1286 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Fuchs, B.: Application of convex relaxation to array synthesis problems. IEEE Trans. Antennas Propag. 62(2), 634–640 (2014)CrossRefGoogle Scholar
  8. 8.
    Hansen, R.C.: Phased Array Antennas, vol. 213. Wiley, Hoboken (2009)CrossRefGoogle Scholar
  9. 9.
    Himmelblau, D.M.: Applied Nonlinear Programming. McGraw-Hill Companies, New York (1972)zbMATHGoogle Scholar
  10. 10.
    Horn, R.A., Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1990)zbMATHGoogle Scholar
  11. 11.
    Horst, R., Pardalos, P.M.: Handbook of Global Optimization, vol. 2. Springer, Dordrecht (2013)zbMATHGoogle Scholar
  12. 12.
    Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches. Springer, New York (2013)zbMATHGoogle Scholar
  13. 13.
    Indenbom, M., Izhutkin, V., Sharapov, A., Zonov, A.: Synthesis of conical phased antenna arrays optimization of amplitude distribution parameters. DEStech Transactions on Computer Science and Engineering (optim) (2018)Google Scholar
  14. 14.
    Kudzin, V.P., Lozovsky, V.N., Shlyk, N.I.: The compact linear antenna array system of the short-wave band consisting of “butterfly” radiators. In: 2013 IX Internatioal Conference on Antenna Theory and Techniques, pp. 252–253. IEEE (2013)Google Scholar
  15. 15.
    Obukhovets, V.A.: Antenna array iterative synthesis algorithm. In: 2017 Radiation and Scattering of Electromagnetic Waves (RSEMW), pp. 58–60. IEEE (2017)Google Scholar
  16. 16.
    Storn, R., Price, K.: Differential evolution-a simple and efficient heuristic for global optimization over continuous spaces. J. Global Optim. 11(4), 341–359 (1997)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Strekalovsky, A.S.: Global optimality conditions in nonconvex optimization. J. Optim. Theory Appl. 173(3), 770–792 (2017)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Tawarmalani, M., Sahinidis, N.V.: Global optimization of mixed-integer nonlinear programs: a theoretical and computational study. Math. Program. 99(3), 563–591 (2004)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Villegas, F.J.: Parallel genetic-algorithm optimization of shaped beam coverage areas using planar \(\text{2-D }\) phased arrays. IEEE Trans. Antennas Propag. 55(6), 1745–1753 (2007)CrossRefGoogle Scholar
  20. 20.
    Wilensky, R.: High-power, broad-bandwidth \(\text{ HF }\) dipole curtain array with extensive vertical and azimuthal beam control. IEEE Trans. Broadcast. 34(2), 201–209 (1988)CrossRefGoogle Scholar
  21. 21.
    Yin, Y., Deng, J.: Design of short wave communication system with phased array antenna. Electronic Eng. 33(9), 31–33 (2007). in ChineseGoogle Scholar
  22. 22.
    Yurkov, A.S.: O vliyanii poter v zemle na rabotu chetyrehelementnoi FAR KV diapazona. Tehnika radiosvyazi 1, 78–81 (2014). in RussianGoogle Scholar
  23. 23.
    Yurkov, A.S.: Optimizatsiya vozbuzhdeniya peredayushih fazirovannyh antennyh reshotok dekametrovogo diapazona dlin voln. ONIIP, Omsk (2014). in RussianGoogle Scholar
  24. 24.
    Yurkov, A.S.: Directivity maximization of the short wave band phased antenna array. Tehnika radiosvyazi 2, 46–53 (2016). in RussianGoogle Scholar

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Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsOmskRussia
  2. 2.Dostoevsky Omsk State UniversityOmskRussia
  3. 3.Institute of Radiophysics and Physical Electronics Omsk Scientific Center SB RASOmskRussia

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