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Argument division based branch-and-bound algorithm for unit-modulus constrained complex quadratic programming

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Abstract

This paper proposes a branch-and-bound algorithm for solving the unit-modulus constrained complex quadratic programming problems (CQPP). We study the convex hull of a unit-modulus complex variable with argument constraints, derive new valid linear inequalities from the convex hull, construct an improved semidefinite relaxation of CQPP, and then design an efficient algorithm for solving CQPP globally. The proposed algorithm branches on the sets of argument constraints and derives new valid inequalities from the partitioned sets of arguments. Numerical results are included to support the effectiveness of the proposed algorithm for finding a global solution to CQPP.

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References

  1. Bandeira, A.S., Boumal, N., Singer, A.: Tightness of the maximum likelihood semidefinite relaxation for angular synchronization. Math. Progr. 163, 145–167 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  2. Boumal, N.: Nonconvex phase synchronization. SIAM J. Optim. 26, 2355–2377 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Burer, S., Vandenbussche, D.: A finite branch-and-bound algorithm for nonconvex quadratic programming via semidefinite relaxations. Math. Progr. 113, 259–282 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cucuringu, M.: Sync-rank: Robust ranking, constrained ranking and rank aggregation via eigenvector and SDP synchronization. IEEE Trans. Netw. Serv. Eng. 3(1), 58–79 (2016)

    Article  MathSciNet  Google Scholar 

  5. Goemans, M., Williamson, D.: Improved approximation algorithms for maximumcut and satisfiability problems using semidefinite programming. J. ACM 42, 1115–1145 (1995)

    Article  MATH  Google Scholar 

  6. Goemans, M., Williamson, D.: Approximation algorithms for Max-3-Cut and other problems via complex semidefinite programming. J. Comput. Syst. Sci. 68, 442–470 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Jaldén, J., Martin, C., Ottersten, B.: Semidefinite programming for detection in linear systems—optimality conditions and space-time decoding. In: IEEE International Conference on Acoustics, Speech, and Signal Processing, pp. 9–12 (2003)

  8. Linderoth, J.: A simplicial branch-and-bound algorithm for solving quadratically constrained quadratic programs. Math. Progr. 103, 251–282 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Lu, C., Liu, Y.-F.: An efficient global algorithm for single-group multicast beamforming. IEEE Trans. Signal Proces. 65, 3761–3774 (2017)

    Article  MathSciNet  Google Scholar 

  10. Luo, Z.-Q., Ma, W.K., So, A.M.C., Ye, Y., Zhang, S.: Semidefinite relaxation of quadratic optimization problems: from its practical deployments and scope of applicability to key theoretical results. IEEE Signal. Proc. Mag. 27, 20–34 (2010)

    Article  Google Scholar 

  11. Ma, W.K., Ching, P.C., Ding, Z.: Semidefinite relaxation based multiuser detection for M-ary PSK multiuser systems. IEEE Trans. Signal Proces. 52, 2862–2871 (2004)

    Article  Google Scholar 

  12. Maio, A., Huang, Y., Piezzo, M., Zhang, S., Farina, A.: Design of optimized radar codes with a peak to average power ratio constraint. IEEE Trans. Signal Proces. 59, 2683–2696 (2011)

    Article  MathSciNet  Google Scholar 

  13. Maio, A., Nicola, S., Huang, Y., Luo, Z.Q.: Design of phase codes for radar performance optimization with a similarity constraint. IEEE Trans. Signal Proces. 57, 610–621 (2009)

    Article  MathSciNet  Google Scholar 

  14. Palomar, D., Eldar, Y.: Convex Optimization in Signal Processing and Communications. Cambridge University, Cambridge (2012)

    MATH  Google Scholar 

  15. So, A.M.C.: Probabilistic analysis of the semidefinite relaxation detector in digital communications. In: ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, Austin, Texas, USA, January, pp. 698–711 (2010)

  16. So, A.M.C., Zhang, J., Ye, Y.: On approximating complex quadratic optimization problems via semidefinite programs. Math. Progr. 110, 93–110 (2008)

    Article  MATH  Google Scholar 

  17. Sturm, J.F.: Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim. Methods Softw. 11/12, 625–653 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  18. Soltanalian, M., Stoica, P.: Designing unimodular codes via quadratic optimization. IEEE Trans. Signal Proces. 62, 1221–1234 (2014)

    Article  MathSciNet  Google Scholar 

  19. Tan, P., Rasmussen, L.: The application of semidefinite programming for detection in CDMA. IEEE J. Sel. Area Comm. 19, 1442–1449 (2001)

    Article  Google Scholar 

  20. Tawarmalan, M., Sahinidis, N.: A polyhedral branch-and-cut approach to global optimization. Math. Program. (Ser. B) 103, 225–249 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  21. Waldspurger, I., Aspremont, A., Mallat, S.: Phase recovery, maxcut and complex semidefinite programming. Math. Program. 149, 47–81 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Zhang, S., Huang, Y.: Complex quadratic optimization and semidefinite programming. SIAM J. Optim. 16, 871–890 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Economics and Management, North China Electric Power University, Beijing, 102206, China

    Cheng Lu

  2. School of Economics and Management, University of Chinese Academy of Sciences, Key Laboratory of Big Data Mining and Knowledge Management, Chinese Academy of Sciences, Beijing, 100190, China

    Zhibin Deng

  3. Department of Electronic Engineering, Tsinghua University, Beijing, 100084, China

    Wei-Qiang Zhang

  4. Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC, 27606, USA

    Shu-Cherng Fang

Authors
  1. Cheng Lu
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  2. Zhibin Deng
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  3. Wei-Qiang Zhang
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  4. Shu-Cherng Fang
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Corresponding author

Correspondence to Zhibin Deng.

Additional information

Lu’s research has been supported by the Fundamental Research Funds for the Central Universities 2017MS058. Deng’s research has been supported by National Natural Science Foundation of China under Grant No. 11501543, Reseach Foundation of UCAS Grant No. Y65201VY00, Y551037Y00 and Y65302V1G4. Zhang’s research has been supported by the National Natural Science Foundation of China under Grant No. 61370034, and in part supported by China Scholarship Council. Fang’s research has been supported by US Army Research Office Grant # W911NF-15-1-0223.

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Lu, C., Deng, Z., Zhang, WQ. et al. Argument division based branch-and-bound algorithm for unit-modulus constrained complex quadratic programming. J Glob Optim 70, 171–187 (2018). https://doi.org/10.1007/s10898-017-0551-8

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  • DOI: https://doi.org/10.1007/s10898-017-0551-8

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